2.1. Diffusion due to interactions

The magnetic force and torque on a particle with magnetic moment $\boldsymbol{M}$ in a magnetic field $\boldsymbol{H}$ in a suspension of particles with number density $n$ are

(2.1)\begin{gather} \boldsymbol{F}^{m} = \boldsymbol{\nabla} (\boldsymbol{M} \boldsymbol{\cdot} \mu_0 (\boldsymbol{H} + n \boldsymbol{M})), \end{gather}

(2.2)\begin{gather}\boldsymbol{T}^{m} = \mu_0 \boldsymbol{M} \times \boldsymbol{H}, \end{gather}

where $\mu _0$ is the magnetic permeability. In the expression for the force, (2.1), the magnetic flux density is expressed as $\boldsymbol{B} = \mu _0 (\boldsymbol{H} + n \boldsymbol{M})$, where $n \boldsymbol{M}$ is the magnetic moment per unit volume. This term is an effective-medium approximation for the variations in the magnetic field due to variations in the number density of particles. The contribution to the magnetic moment per volume in the expression for the torque, (2.2), is zero, since it contains the cross product of the particle moment at a location with itself.

The hydrodynamic force and torque are

(2.3)\begin{gather} \boldsymbol{F}^{h} = 3 {\rm \pi}\eta d (\boldsymbol{v} - \boldsymbol{u}), \end{gather}

(2.4)\begin{gather}\boldsymbol{T}^{h} = {\rm \pi}\eta d^{3} (\tfrac{1}{2} \boldsymbol{\omega} - \boldsymbol{\varOmega}), \end{gather}

where $d$ is the particle diameter, $\boldsymbol{v}$ and $\boldsymbol{\omega }$ are the fluid velocity and vorticity at the particle centre, and $\boldsymbol{u}$ and $\boldsymbol{\varOmega }$ are the particle linear and angular velocities. The imposed magnetic field is denoted $\bar {\boldsymbol{H}}$, and the vorticity due to the imposed shear flow at the particle centre is $\bar {\boldsymbol{\omega }}$. The particle magnetic moment is expressed as $\boldsymbol{M} = M \boldsymbol{o}$, where $M$ is the magnitude of the magnetic moment and $\boldsymbol{o}$ is the orientation vector. Linear models are used to incorporate the effect of the variations in the volume fraction on the viscosity and the magnetic permeability,

(2.5)\begin{equation} \eta = \eta_0 (1 + \eta^{\prime} \phi^{\prime}), \end{equation}

where $\phi ^{\prime }=\phi -\bar {\phi }$, with $\phi$ the local volume fraction of the particles, $\bar {\phi }$ the average volume fraction in the uniform suspension, and $\eta _0$ the viscosity for the uniform suspension. For a viscous suspension of spherical particles in the limit of low volume fraction, the result $\eta ^{\prime } = \frac {5}{2}$ is obtained for the Einstein correction to the viscosity.

The orientation vector of the particle $\boldsymbol{o}$ is determined from the torque balance equation, $\boldsymbol{T}^{m} + \boldsymbol{T}^{h} = 0$:

(2.6)\begin{equation} {\rm \pi}\eta d^{3} (\tfrac{1}{2} \boldsymbol{\omega} - \boldsymbol{\varOmega}) + \mu_0 M (\boldsymbol{o} \times \boldsymbol{H}) = 0. \end{equation}

The magnetic force on the particle (2.1) is zero in a uniform suspension. The force balance requires that the hydrodynamic force is also zero, that is, the particle moves with the same velocity as the fluid, $\bar {\boldsymbol{u}} = \bar {\boldsymbol{v}}$.

For a stable stationary state where the particle orientation is steady, the component of the angular velocity $\boldsymbol{\varOmega }$ perpendicular to the orientation vector is zero. However, the particle does spin around the orientation vector, and the components of the particle angular velocity and fluid rotation rate (one-half of the vorticity) along the orientation vector are equal. This is because the torque exerted by the magnetic field is perpendicular to the direction of the magnetic moment and the magnetic field, and consequently there is no magnetic torque along the orientation vector. Therefore, the particle angular velocity is necessarily along the orientation vector, $\boldsymbol{\varOmega } = \varOmega \boldsymbol{o}$. It is necessary to consider the torque balance equations parallel and perpendicular to the orientation vector in order to determine the particle orientation. The torque balance equation along the orientation vector is obtained by taking the dot product of (2.6) with $\boldsymbol{o}$, and this is solved to obtain

(2.7)\begin{equation} \varOmega = \tfrac{1}{2} \boldsymbol{\omega} \boldsymbol{\cdot} \boldsymbol{o}. \end{equation}

The above solution (2.7) for the angular velocity is substituted into the torque balance equation (2.6) to obtain

(2.8)\begin{equation} \tfrac{1}{2} {\rm \pi}\eta d^{3} (\boldsymbol{I} - \boldsymbol{o} \boldsymbol{o}) \boldsymbol{\cdot} \boldsymbol{\omega} + \mu_0 M (\boldsymbol{o} \times \boldsymbol{H}) = 0, \end{equation}

where $\boldsymbol{I}$ is the identity tensor, and $(\boldsymbol{I} - \boldsymbol{o} \boldsymbol{o}) \boldsymbol {\cdot }$ is the transverse projection operator that projects a vector on to the plane perpendicular to $\boldsymbol{o}$. One relation between the vorticity, magnetic field and orientation vector is obtained by taking the dot product of (2.8) with $\boldsymbol{H}$:

(2.9)\begin{equation} \boldsymbol{H} \boldsymbol{\cdot} \boldsymbol{\omega} - (\boldsymbol{o} \boldsymbol{\cdot} \boldsymbol{H})(\boldsymbol{o} \boldsymbol{\cdot} \boldsymbol{\omega}) = 0. \end{equation}

The torque balance for the base state in the absence of interactions is obtained by substituting $\bar {\boldsymbol{o}}$, $\bar {\boldsymbol{\omega }}$ and $\bar {\boldsymbol{H}}$ for $\boldsymbol{o}$, $\boldsymbol{\omega }$ and $\boldsymbol{H}$ in (2.8).

There is a disturbance to the magnetic field $\boldsymbol{H}'$ and the fluid vorticity $\boldsymbol{\omega }'$ due to a number density fluctuation ${n'}(\boldsymbol{x})$ imposed on a uniform suspension with number density $\bar {n}$. The orientation vector of the particle is expressed as $\boldsymbol{o} = \bar {\boldsymbol{o}} + \boldsymbol{o}'(\boldsymbol{x})$, where $\boldsymbol{o}'(\boldsymbol{x})$ is the disturbance to the orientation vector due to hydrodynamic and magnetic interactions resulting from concentration inhomogeneities; the latter is calculated from the torque balance equation. In the linear approximation where terms quadratic in the primed quantities are neglected, the mean and disturbance to the orientation vector satisfy

(2.10a,b)\begin{equation} |\bar{\boldsymbol{o}}| = 1, \quad \bar{\boldsymbol{o}} \boldsymbol{\cdot} \boldsymbol{o}' = 0. \end{equation}

The reason for the above relations is that both $\bar {\boldsymbol{o}}$ and $(\bar {\boldsymbol{o}} + \boldsymbol{o}')$ are unit vectors, that is, $\bar {\boldsymbol{o}} \boldsymbol{\cdot } \bar {\boldsymbol{o}} = 1$ and $(\bar {\boldsymbol{o}} + \boldsymbol{o}') \boldsymbol{\cdot } (\bar {\boldsymbol{o}} + \boldsymbol{o}') = 1$. In the linear approximation, this implies that $(\bar {\boldsymbol{o}} \boldsymbol{\cdot } \bar {\boldsymbol{o}} + 2 \bar {\boldsymbol{o}} \boldsymbol{\cdot } \boldsymbol{o}') = 1$. Therefore, $\bar {\boldsymbol{o}} \boldsymbol{\cdot } \boldsymbol{o}' = 0$.

The Fourier transform $\hat {\boldsymbol{o}}_{\boldsymbol{k}}$ of the disturbance to the orientation vector is defined using (1.4). The Fourier transform of the force on a particle due to interactions is determined from (2.1) as

(2.11)\begin{equation} \hat{\boldsymbol{F}}_{\boldsymbol{k}} ={-} {{\rm{i}}} \boldsymbol{k} \mu_0 M [\bar{\boldsymbol{o}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}} + \bar{\boldsymbol{H}} \boldsymbol{\cdot} \hat{\boldsymbol{o}}_{\boldsymbol{k}} + M \hat{n}_{\boldsymbol{k}}], \end{equation}

where $M$ is the magnitude of the magnetic moment. The last term in the square brackets on the right in (2.11) is the force due to the variation in the number density in the expression (2.1).

The conservation equation for the particle number density is

(2.12)\begin{equation} \frac{\partial n}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} (\boldsymbol{u} n) = D_B\,\nabla^{2} n, \end{equation}

where $\boldsymbol{u}$ is the particle velocity and $D_B$ is the Brownian diffusion coefficient. The particle velocity is expressed as the sum of the fluid velocity $\boldsymbol{v}$ and the particle velocity relative to the fluid, $(\boldsymbol{u} - \boldsymbol{v}) + \boldsymbol{v}$. Since there is no net force on the particles in a uniform suspension, $\bar {\boldsymbol{u}} = \bar {\boldsymbol{v}}$. In the linear approximation, the particle conservation equation is expressed as

(2.13)\begin{equation} \frac{\partial {n'}}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} (\bar{\boldsymbol{v}} {n'} + \boldsymbol{v}^{\prime} \bar{n}) + \boldsymbol{\nabla} \boldsymbol{\cdot} ((\boldsymbol{u}^{\prime} -\boldsymbol{v}^{\prime}) \bar{n}) = D_B\,\nabla^{2} n. \end{equation}

The Fourier transform of the above equation is

(2.14)\begin{equation} \frac{\partial \hat{n}_{\boldsymbol{k}}}{\partial t} - {{\rm{i}}} \boldsymbol{k} \boldsymbol{\cdot} (\bar{\boldsymbol{v}} \hat{n}_{\boldsymbol{k}}) - {{\rm{i}}} \boldsymbol{k} \boldsymbol{\cdot} (\bar{n} \hat{\boldsymbol{v}}_{\boldsymbol{k}}) - {{\rm{i}}} \boldsymbol{k} \bar{n} (\hat{\boldsymbol{u}}_{\boldsymbol{k}} - \hat{\boldsymbol{v}}_{\boldsymbol{k}}) + D_B k^{2} \hat{n}_{\boldsymbol{k}} = 0. \end{equation}

The first two terms on the left of (2.14) are the rate of change and the convection of concentration fluctuations due to the base flow. The third term on the left is the rate of change of concentration disturbances due to fluid velocity perturbation. The perturbation to the fluid velocity at a test particle due to the rotation of neighbouring particles, which is calculated later in (2.22), is perpendicular to $\boldsymbol{k}$. Consequently, the third term on the left in (2.14) is zero. The fourth term on the left is the relative motion between the particles and the fluid due to the forces on the particle.

In a dilute suspension, the difference between the particle and fluid velocity disturbances, $\hat {\boldsymbol{u}}_{\boldsymbol{k}}$ and $\hat {\boldsymbol{v}}_{\boldsymbol{k}}$, is given by Stokes’ law,

(2.15)\begin{equation} \hat{\boldsymbol{u}}_{\boldsymbol{k}} - \hat{\boldsymbol{v}}_{\boldsymbol{k}} = \frac{\hat{\boldsymbol{F}}_{\boldsymbol{k}}}{3 {\rm \pi}\eta_0 d}. \end{equation}

The drift velocity does not depend on the coefficient $\eta ^{\prime }$ in (2.5) in the linear approximation, because the force on a particle is zero in a uniform suspension.

The expression (2.15) for the drift velocity is substituted into the linearised conservation equation (2.14):

(2.16)\begin{equation} \frac{\partial \hat{n}_{\boldsymbol{k}}}{\partial t} - {{\rm{i}}} \boldsymbol{k} \boldsymbol{\cdot} (\hat{n}_{\boldsymbol{k}} \bar{\boldsymbol{v}}) + D_B k^{2} \hat{n}_{\boldsymbol{k}} - \frac{k^{2} \bar{n} \mu_0 M (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}} + \bar{\boldsymbol{H}} \boldsymbol{\cdot} \hat{\boldsymbol{o}}_{\boldsymbol{k}} + M \hat{n}_{\boldsymbol{k}})}{3 {\rm \pi}\eta_0 d} = 0. \end{equation}

The disturbance $\hat {\boldsymbol{H}}_{\boldsymbol{k}}$ due to magnetic interactions is expressed as a function of $\hat {n}_{\boldsymbol{k}}$, and the orientation disturbance $\hat {\boldsymbol{o}}_{\boldsymbol{k}}$, currently unknown, is determined from the first correction to the torque balance. These are substituted into (2.16) to obtain an equation of the form

(2.17)\begin{equation} \frac{\partial \hat{n}_{\boldsymbol{k}}}{\partial t} - {{\rm{i}}} \boldsymbol{k} \boldsymbol{\cdot} (\hat{n}_{\boldsymbol{k}} \bar{\boldsymbol{u}}) + \boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{D} \boldsymbol{\cdot} \boldsymbol{k} \hat{n}_{\boldsymbol{k}} + D_B k^{2} \hat{n}_{\boldsymbol{k}} = 0, \end{equation}

where $\boldsymbol{D}$ is the magnetophoretic diffusion tensor due to hydrodynamic and magnetic interactions. In the following calculation, first the disturbances to the magnetic field and the vorticity due to particle interactions are discussed, and then the disturbance to the orientation vector is calculated using torque balance. This is inserted into (2.16) to extract the diffusion tensor $\boldsymbol{D}$.

The change in the magnetic field at the location $\boldsymbol{x}$ due to interactions with other particles is a generalisation of (1.3) which incorporates the disturbance to the orientation vector,

(2.18)\begin{equation} \boldsymbol{H}'(\boldsymbol{x}) = \frac{1}{4 {\rm \pi}} \int {\rm d}\kern0.06em \boldsymbol{x}' \left( \frac{3 (\boldsymbol{x} - \boldsymbol{x}')(\boldsymbol{x} - \boldsymbol{x}')}{|\boldsymbol{x} - \boldsymbol{x}'|^{5}} - \frac{\boldsymbol{I}}{|\boldsymbol{x} - \boldsymbol{x}'|^{3}} \right) \boldsymbol{\cdot} [M (\bar{\boldsymbol{o}} {n'}(\boldsymbol{x}') + \bar{n} \boldsymbol{o}'(\boldsymbol{x}'))], \end{equation}

where ${\rm d}\boldsymbol{x}'$ is the volume element. It is easily verified that the contribution to $\boldsymbol{H}'$ due to the product of the background constant particle density and magnetic field, $M \bar {n} \bar {\boldsymbol{o}}$, is zero. Equation (1.4) is used to determine the disturbance to magnetic field in reciprocal space,

(2.19)\begin{equation} \hat{\boldsymbol{H}}_{\boldsymbol{k}} ={-} \frac{\boldsymbol{k} \boldsymbol{k} \boldsymbol{\cdot} [M (\bar{\boldsymbol{o}} \hat{n}_{\boldsymbol{k}} + \bar{n} \hat{\boldsymbol{o}}_{\boldsymbol{k}})]}{k^{2}} = \hat{\boldsymbol{H}}'_{\boldsymbol{k}} + \hat{\boldsymbol{H}}''_{\boldsymbol{k}} \boldsymbol{\cdot} \hat{\boldsymbol{o}}_{\boldsymbol{k}}, \end{equation}

where

(2.20a,b)\begin{equation} \hat{\boldsymbol{H}}'_{\boldsymbol{k}} ={-} \frac{M \hat{n}_{\boldsymbol{k}} \boldsymbol{k} \boldsymbol{k} \boldsymbol{\cdot} \bar{\boldsymbol{o}}}{k^{2}}, \quad\hat{\boldsymbol{H}}''_{\boldsymbol{k}} ={-} \frac{M \bar{n} \boldsymbol{k} \boldsymbol{k}}{k^{2}}.\end{equation}

Note that $\hat {\boldsymbol{H}}'_{\boldsymbol{k}}$ is a vector, and $\hat {\boldsymbol{H}}''_{\boldsymbol{k}}$ is a second-order tensor.

For a particle located at $\boldsymbol{x}'$ in a fluid with vorticity $\bar {\boldsymbol{\omega }}$ in the absence of the particle, the fluid velocity disturbance $\boldsymbol{v}'$ at the location $\boldsymbol{x}$ is

(2.21)\begin{align} \boldsymbol{v}'(\boldsymbol{x}) & = \int {\rm d}\kern0.06em \boldsymbol{x}'\, n(\boldsymbol{x}') \left(\varOmega(\boldsymbol{x}') - \frac{1}{2} \boldsymbol{\omega}(\boldsymbol{x}')\right) \times \frac{d^{3} (\boldsymbol{x} - \boldsymbol{x}')}{8 |\boldsymbol{x} - \boldsymbol{x}'|^{3}} \nonumber\\ & ={-} \int {\rm d}\kern0.06em \boldsymbol{x}' \,n(\boldsymbol{x}') [\boldsymbol{I} - \boldsymbol{o}(\boldsymbol{x}')\,\boldsymbol{o}(\boldsymbol{x}')] \boldsymbol{\cdot} \boldsymbol{\omega}(\boldsymbol{x}') \times \frac{d^{3} (\boldsymbol{x} - \boldsymbol{x}')}{16 |\boldsymbol{x} - \boldsymbol{x}'|^{3}} \nonumber\\ & ={-} \int {\rm d}\kern0.06em \boldsymbol{x}' [ {n'}(\boldsymbol{x}')\,(\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \bar{\boldsymbol{\omega}} + \bar{n} (\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \boldsymbol{\omega}'(\boldsymbol{x}') \nonumber\\ & \quad - \bar{n} (\bar{\boldsymbol{o}} \bar{\boldsymbol{\omega}} + \boldsymbol{I} \bar{\boldsymbol{\omega}} \boldsymbol{\cdot} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \boldsymbol{o}'(\boldsymbol{x}')] \times \frac{d^{3} (\boldsymbol{x} - \boldsymbol{x}')}{16 |\boldsymbol{x} - \boldsymbol{x}'|^{3}}. \end{align}

The expression (2.7) for the particle angular velocity has been used to simplify in the second step in (2.21). The linearisation approximation has been used in the third step in (2.21), resulting in an equation that is linear in the disturbances due to particle interactions. The Fourier transform of the velocity field is

(2.22)\begin{equation} \hat{\boldsymbol{v}}_{\boldsymbol{k}} ={-} [\hat{n}_{\boldsymbol{k}} (\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \bar{\boldsymbol{\omega}} + \bar{n} (\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \hat{\boldsymbol{\omega}}_{\boldsymbol{k}} - \bar{n} \hat{\boldsymbol{o}}_{\boldsymbol{k}} \boldsymbol{\cdot} (\bar{\boldsymbol{\omega}} \bar{\boldsymbol{o}} + {\boldsymbol{I}} \bar{\boldsymbol{\omega}} \boldsymbol{\cdot} \bar{\boldsymbol{o}})] \times \frac{{\rm \pi} {d}^{3} {{\rm{i}}} \boldsymbol{k}}{4 k^{2}}. \end{equation}

The Fourier transform of the disturbance to the fluid vorticity is

(2.23)\begin{align} \hat{\boldsymbol{\omega}}_{\boldsymbol{k}} & ={-} {{\rm{i}}} \boldsymbol{k} \times \hat{\boldsymbol{v}}_{\boldsymbol{k}} \nonumber\\ & = \frac{{\rm \pi} {d}^{3} (\boldsymbol{k} \boldsymbol{k} - \boldsymbol{I} k^{2}) \boldsymbol{\cdot} [\hat{n}_{\boldsymbol{k}} (\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \bar{\boldsymbol{\omega}} + \bar{n} (\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \hat{\boldsymbol{\omega}}_{\boldsymbol{k}} - \bar{n} (\bar{\boldsymbol{o}} \bar{\boldsymbol{\omega}} + {\boldsymbol{I}} \bar{\boldsymbol{\omega}} \boldsymbol{\cdot} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \hat{\boldsymbol{o}}_{\boldsymbol{k}}]}{4 k^{2}}. \end{align}

The above implicit equation is solved for $\hat {\boldsymbol{\omega }}_{\boldsymbol{k}}$:

(2.24)\begin{equation} \left( \boldsymbol{I} - \frac{{\rm \pi} {d}^{3} \bar{n} (\boldsymbol{k} \boldsymbol{k} - \boldsymbol{I} k^{2}) \boldsymbol{\cdot} (\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}})}{4 k^{2}} \right) \boldsymbol{\cdot} \hat{\boldsymbol{\omega}}_{\boldsymbol{k}} = \hat{\boldsymbol{\omega}}' + \hat{\boldsymbol{\omega}}'' \boldsymbol{\cdot} \hat{\boldsymbol{o}}_{\boldsymbol{k}}, \end{equation}

where

(2.25)\begin{gather} \hat{\boldsymbol{\omega}}' = \frac{{\rm \pi} \hat{n}_{\boldsymbol{k}} {d}^{3} (\boldsymbol{k} \boldsymbol{k} - \boldsymbol{I} k^{2}) \boldsymbol{\cdot} (\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}}{4 k^{2}}, \end{gather}

(2.26)\begin{gather}\hat{\boldsymbol{\omega}}'' ={-} \frac{{\rm \pi} \bar{n} d^{3} (\boldsymbol{k} \boldsymbol{k} - \boldsymbol{I} k^{2}) \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \bar{\boldsymbol{\omega}} + \boldsymbol{I} \bar{\boldsymbol{\omega}} \boldsymbol{\cdot} \bar{\boldsymbol{o}})}{4 k^{2}}. \end{gather}

Note that $\hat {\boldsymbol{\omega }}'_{\boldsymbol{k}}$ is a vector, and $\hat {\boldsymbol{\omega }}''_{\boldsymbol{k}}$ is a second-order tensor. The latter is proportional to ${d}^{3} \bar {n}$ or the volume fraction. In the limit of small volume fraction, this term is small compared to $1$.

The velocity due to the force exerted by particles is higher order in gradients or higher power in $\boldsymbol{k}$, so it will result in a higher gradient of the concentration field. The contribution to the velocity due to the magnetic field perturbation, in Fourier space, is $|\hat {\boldsymbol{F}}_{\boldsymbol{k}}|/ 3 {\rm \pi}\eta d \sim k \mu _0 M \hat {\boldsymbol{H}}_{\boldsymbol{k}}/3 {\rm \pi}\eta d \sim k \mu _0 M^{2} \hat {n}_{\boldsymbol{k}}/3 {\rm \pi}\eta d$, where $k$ is the magnitude of the wavenumber. Here, (2.11) is used for the force, and (2.19), (2.20a,b) are used for the magnetic field perturbation. The magnitude of the fluid velocity due to particle rotation, from (2.22), is $|\hat {\boldsymbol{v}}_{\boldsymbol{k}}| \sim {\rm \pi}|\bar {\boldsymbol{\omega }}| \hat {n}_{\boldsymbol{k}} {d}^{3}/k$. The ratio of the velocities due to the force and torque is $k^{2} \mu _0 M^{2}/3 {\rm \pi}\eta {d}^{4} |\bar {\boldsymbol{\omega }}| \sim (k d)^{2} {M^{\ast }} \varSigma$, where the dimensionless groups $\varSigma$ and ${M^{\ast }}$ are defined later, in (2.43) and (2.44). The continuum approximation is valid only for $k d \ll 1$; that is, the length scale for the gradients in the concentration field is much larger than the particle diameter. In this limit, the velocity disturbance due to the particle force is neglected compared to that due to the torque exerted on the fluid.

The disturbance to the magnetic field and the vorticity at the particle location causes a disturbance to the particle orientation $\boldsymbol{o}$, which is determined from the correction to the torque balance equation in Fourier space. The torque balance equation, (2.8), is divided by $\mu _0 M$ and linearised in the primed quantities to obtain

(2.27)\begin{align} &\frac{{\rm \pi} d^{3} \eta_0}{2 \mu_0 M}\,[(\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \hat{\boldsymbol{\omega}}_{\boldsymbol{k}} - \bar{\boldsymbol{o}} (\hat{\boldsymbol{o}}_{\boldsymbol{k}} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}) - \hat{\boldsymbol{o}}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}) + \eta^{\prime} \hat{\phi}_{\boldsymbol{k}} (\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}] & \nonumber\\ &\quad + (\bar{\boldsymbol{o}} \times \hat{\boldsymbol{H}}_{\boldsymbol{k}} + \hat{\boldsymbol{o}}_{\boldsymbol{k}} \times \bar{\boldsymbol{H}}) = 0, \end{align}

where $\hat {\phi }_{\boldsymbol{k}} = \hat {n}_{\boldsymbol{k}} ({\rm \pi} d^{3}/6)$ is the Fourier transform of the fluctuation in the volume fraction. The expressions (2.19) and (2.24) for $\hat {\boldsymbol{H}}_{\boldsymbol{k}}$ and $\hat {\boldsymbol{\omega }}_{\boldsymbol{k}}$ are substituted into (2.27):

(2.28)\begin{align} &\frac{{\rm \pi} d^{3} \eta_0}{2 \mu_0 M}\,\{(\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \hat{\boldsymbol{\omega}}'_{\boldsymbol{k}} + [(\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \hat{\boldsymbol{\omega}}''_{\boldsymbol{k}} - \bar{\boldsymbol{o}} \bar{\boldsymbol{\omega}} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}) \boldsymbol{I}] \boldsymbol{\cdot} \hat{\boldsymbol{o}}_{\boldsymbol{k}} \nonumber\\ &\quad + \eta^{\prime} \hat{\phi}_{\boldsymbol{k}} (\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \bar{\boldsymbol{\omega}} \} + [\bar{\boldsymbol{o}} \times \hat{\boldsymbol{H}}'_{\boldsymbol{k}} + \hat{\boldsymbol{o}}_{\boldsymbol{k}} \times \bar{\boldsymbol{H}} + \bar{\boldsymbol{o}} \times (\hat{\boldsymbol{H}}''_{\boldsymbol{k}} \boldsymbol{\cdot} \hat{\boldsymbol{o}}_{\boldsymbol{k}})] = 0. \end{align}

The first term on the left-hand side of (2.28) is the correction to the hydrodynamic torque due to interactions and due to the concentration dependence of the viscosity and magnetic permeability. In this, the contribution proportional to $\eta ^{\prime } \hat {\phi }_{\boldsymbol{k}}$ is the correction to the hydrodynamic torque due to the concentration dependence of the viscosity. In the first term on the left in (2.28), within the braces, the prefactor of $\hat {\boldsymbol{o}}_{\boldsymbol{k}}$ contains one contribution proportional to $\hat {\boldsymbol{\omega }}''$, and a second that is proportional to $|\bar {\boldsymbol{\omega }}|$. From (2.26), the former is proportional to $\bar {n} d^{3} |\bar {\boldsymbol{\omega }}|$, which is small compared to the latter, because the volume fraction is small. Therefore, the term $(\boldsymbol{I} - \bar {\boldsymbol{o}} \bar {\boldsymbol{o}}) \boldsymbol{\cdot } \hat {\boldsymbol{\omega }}''_{\boldsymbol{k}}$ is neglected in the first term on the left in (2.28). The second term on the left in (2.28) is the contribution due to the magnetic torque. There are two terms containing $\hat {\boldsymbol{o}}_{\boldsymbol{k}}$ in the magnetic torque, the first proportional to $|\bar {\boldsymbol{H}}|$ and the second proportional to $|\hat {\boldsymbol{H}}''_{\boldsymbol{k}}|$. From (2.20a,b), the latter scales as $\bar {n} M$, where $M$ is the magnetic moment of a particle. From dimensional analysis, $\bar {n} M$ is the disturbance to the magnetic field at a distance $\bar {n}^{-1/3}$ from a particle, comparable to the inter-particle distance in the suspension. The weak interaction limit is considered here, where the magnetic field disturbance due to particle interactions is small compared to the applied field, $\bar {n} M \ll |\bar {\boldsymbol{H}}|$. Therefore, the term containing $\bar {\boldsymbol{o}} \times (\hat {\boldsymbol{H}}''_{\boldsymbol{k}} \boldsymbol{\cdot } \hat {\boldsymbol{o}}_{\boldsymbol{k}})$ is neglected in comparison to $\hat {\boldsymbol{o}}_{\boldsymbol{k}} \times \bar {\boldsymbol{H}}$ in (2.28).

Equation (2.28) is solved to obtain the unknown orientation disturbance $\hat {\boldsymbol{o}}_{\boldsymbol{k}}$. Since $\hat {\boldsymbol{o}}_{\boldsymbol{k}}$ is perpendicular to $\bar {\boldsymbol{o}}$ from (2.10a,b), $\hat {\boldsymbol{o}}_{\boldsymbol{k}}$ is expressed as

(2.29)\begin{equation} \hat{\boldsymbol{o}}_{\boldsymbol{k}} = \hat{o}^{{{\dagger}}}_{\boldsymbol{k}} (\bar{\boldsymbol{H}} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) \bar{\boldsymbol{o}}) + \hat{o}^{\ddagger}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}), \end{equation}

where $\hat {o}^{{{\dagger}} }_{\boldsymbol{k}}$ and $\hat {o}^{\ddagger }_{\boldsymbol{k}}$ are scalar functions of the wavenumber. The specific form (2.29) is chosen because it satisfies the requirement $\hat {\boldsymbol{o}}_{\boldsymbol{k}} \boldsymbol{\cdot } \bar {\boldsymbol{o}} = 0$. The expression (2.29) is inserted into the torque balance equation (2.28):

(2.30)\begin{align} &\frac{{\rm \pi} d^{3} \eta_0}{2 \mu_0 M}\,\{(\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \hat{\boldsymbol{\omega}}'_{\boldsymbol{k}} - \bar{\boldsymbol{o}} [\hat{o}^{{{\dagger}}}_{\boldsymbol{k}} \underline{(\bar{\boldsymbol{H}} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}} + \hat{o}^{\ddagger}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}] \nonumber\\ &\quad - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}) [\hat{o}^{{{\dagger}}}_{\boldsymbol{k}} (\bar{\boldsymbol{H}} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) \bar{\boldsymbol{o}}) + \hat{o}^{\ddagger}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}})] + \eta^{\prime} \hat{\phi}_{\boldsymbol{k}} (\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}\} \nonumber\\ &\quad + [\bar{\boldsymbol{o}} \times \hat{\boldsymbol{H}}'_{\boldsymbol{k}} - \hat{o}^{{{\dagger}}}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})(\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) - \hat{o}^{\ddagger}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} |\bar{\boldsymbol{H}}|^{2} - \bar{\boldsymbol{H}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}))] = 0. \end{align}

From (2.9), the underlined term in the above equation is zero. The scalar functions $\hat {o}^{{{\dagger}} }_{\boldsymbol{k}}$ and $\hat {o}^{\ddagger }_{\boldsymbol{k}}$ are evaluated by taking the dot product of (2.30) with $\bar {\boldsymbol{o}} \times \bar {\boldsymbol{H}}$ and $\bar {\boldsymbol{o}}$, respectively:

(2.31)\begin{align} &\frac{{\rm \pi} d^{3} \eta_0}{2 \mu_0 M}\,[\hat{\boldsymbol{\omega}}'_{\boldsymbol{k}} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) - \hat{o}^{\ddagger}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}})(|\bar{\boldsymbol{H}}|^{2} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2}) + \eta^{\prime} \hat{\phi}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}] \nonumber\\ &\quad + [\bar{\boldsymbol{H}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}'_{\boldsymbol{k}} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})(\hat{\boldsymbol{H}}'_{\boldsymbol{k}} \boldsymbol{\cdot} \bar{\boldsymbol{o}}) - \hat{o}^{{{\dagger}}}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})(|\bar{\boldsymbol{H}}|^{2} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2})] = 0, \end{align}

(2.32)\begin{align} &- \frac{{\rm \pi} d^{3} \eta_0}{2 \mu_0 M}\,\hat{o}^{\ddagger}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) \boldsymbol{\cdot} \bar{\boldsymbol{\omega}} - \hat{o}^{\ddagger}_{\boldsymbol{k}} (|\bar{\boldsymbol{H}}|^{2} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2}) = 0. \end{align}

These equations are solved to obtain

(2.33)\begin{align} \hat{o}^{{{\dagger}}}_{\boldsymbol{k}}& = \frac{{\rm \pi} \eta_0 d^{3} \hat{\boldsymbol{\omega}}'_{\boldsymbol{k}} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}})}{2 \mu_0 M (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})(|\bar{\boldsymbol{H}}|^{2} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2})} + \frac{\bar{\boldsymbol{H}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}'_{\boldsymbol{k}} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})(\bar{\boldsymbol{o}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}'_{\boldsymbol{k}})}{(\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) (|\bar{\boldsymbol{H}}|^{2} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2})} \nonumber\\ & \quad + \frac{{\rm \pi} d^{3} \eta_0 \eta^{\prime} \hat{\phi}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}}{2 \mu_0 M (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})(|\bar{\boldsymbol{H}}|^{2} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2})}, \end{align}

(2.34)\begin{gather} \hat{o}^{\ddagger}_{\boldsymbol{k}} = 0. \end{gather}

The magnetophoretic diffusion term in (2.16) is simplified by neglecting the term $\hat {\boldsymbol{H}}''_{\boldsymbol{k}} \boldsymbol{\cdot } \hat {\boldsymbol{o}}_{\boldsymbol{k}}$ in (2.19), since it is much smaller than $\hat {\boldsymbol{H}}'_{\boldsymbol{k}}$:

(2.35)\begin{align} \boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{D} \boldsymbol{\cdot} \boldsymbol{k}& ={-} \frac{\bar{n} \mu_0 M k^{2} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}} + \hat{\boldsymbol{o}}_{\boldsymbol{k}} \boldsymbol{\cdot} \bar{\boldsymbol{H}} + M \hat{n}_{\boldsymbol{k}}) }{3 {\rm \pi}\eta_0 d \hat{n}_{\boldsymbol{k}}} \nonumber\\ & ={-} \frac{\bar{n} k^{2}}{3 {\rm \pi}\eta_0 d \hat{n}_{\boldsymbol{k}}} \left[\vphantom{\left. {}+ \frac{\mu_0 M (\bar{\boldsymbol{H}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}} - \overbrace{(\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})(\hat{\boldsymbol{H}}_{\boldsymbol{k}} \boldsymbol{\cdot} \bar{\boldsymbol{o}})}^{\unicode{x2461}})}{(\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})} + \mu_0 M^{2} \hat{n}_{\boldsymbol{k}} \right]} \mu_0 M \overbrace{\bar{\boldsymbol{o}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}}}^{\unicode{x2460}} + \frac{{\rm \pi} d^{3} \eta_0 \hat{\boldsymbol{\omega}}_{\boldsymbol{k}} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}})}{2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})} + \frac{{\rm \pi} d^{3} \eta_0 \eta^{\prime} \hat{\phi}_{\boldsymbol{k}} \bar{\boldsymbol{\omega}} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}})}{2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})} \right. \nonumber\\ &\quad \left. {}+ \frac{\mu_0 M (\bar{\boldsymbol{H}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}} - \overbrace{(\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})(\hat{\boldsymbol{H}}_{\boldsymbol{k}} \boldsymbol{\cdot} \bar{\boldsymbol{o}})}^{\unicode{x2461}})}{(\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})} + \mu_0 M^{2} \hat{n}_{\boldsymbol{k}} \right]. \end{align}

Here, the right-hand sides of (2.29) and (2.33)–(2.34) have been substituted for $\hat {\boldsymbol{o}}_{\boldsymbol{k}}$ in the second step. In (2.35), the terms $\unicode{x2460}$ and $\unicode{x2461}$ cancel. The term $\unicode{x2460}$ is the contribution due to the interaction between the undisturbed particle magnetic moment and the disturbance to the magnetic field due to particle interactions. This cancels exactly with one of the terms resulting from the disturbance to the particle orientation; the remaining terms are due entirely to the interaction between the undisturbed magnetic field and the disturbance to the particle orientation due to interactions:

(2.36)\begin{align} \boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{D} \boldsymbol{\cdot} \boldsymbol{k}& ={-} \frac{\bar{n} k^{2} d^{2} \hat{\boldsymbol{\omega}}_{\boldsymbol{k}} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}})}{6 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) \hat{n}_{\boldsymbol{k}}} - \frac{{\rm \pi} \bar{n} k^{2} d^{5} \eta^{\prime} \bar{\boldsymbol{\omega}} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}})}{36 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})} \nonumber\\ & \quad - \frac{\bar{n} k^{2} \mu_0 M \bar{\boldsymbol{H}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}}}{3 {\rm \pi}\eta d (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) \hat{n}_{\boldsymbol{k}}} - \frac{k^{2} \bar{n} \mu_0 M^{2}}{3 {\rm \pi}\eta_0 d}. \end{align}

Here, the substitution $\hat {\phi }_{\boldsymbol{k}} = \hat {n}_{\boldsymbol{k}} ({\rm \pi} d^{3}/6)$ has been used to relate the volume fraction and number density in the third term on the right. The first term on the right is due to the disturbance to the orientation vector caused by the hydrodynamic interactions, and the third term results from the disturbance to the orientation vector caused by magnetic interactions. The expressions (2.20a,b) and (2.25) are substituted for the disturbance to the magnetic field and vorticity due to particle interactions, to obtain

(2.37)\begin{align} \boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{D} \boldsymbol{\cdot} \boldsymbol{k} & ={-} \frac{{\rm \pi} \bar{n} d^{5} \{ [\boldsymbol{k} \boldsymbol{\cdot} (\bar{\boldsymbol{\omega}} - \bar{\boldsymbol{o}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}))] [\boldsymbol{k} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}})] - k^{2} [\bar{\boldsymbol{\omega}} - \bar{\boldsymbol{o}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}})] \boldsymbol{\cdot} [\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}] \}}{24 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})} \nonumber\\ &\quad + \frac{\bar{n} \mu_0 M^{2} (\boldsymbol{k} \boldsymbol{\cdot} \bar{\boldsymbol{o}})(\boldsymbol{k} \boldsymbol{\cdot} \bar{\boldsymbol{H}})}{3 {\rm \pi}\eta_0 d (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})} - \frac{{\rm \pi} \bar{n} k^{2} d^{5} \eta^{\prime} \bar{\boldsymbol{\omega}} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}})}{36 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})} \nonumber\\ & \quad - \frac{k^{2} \bar{n} \mu_0 M^{2}}{3 {\rm \pi}\eta_0 d}. \end{align}

The cross product $\bar {\boldsymbol{o}} \times \bar {\boldsymbol{H}}$ in (2.37) is expressed in terms of the vorticity using the torque balance equation (2.8) for the base state:

(2.38)\begin{align} \boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{D} \boldsymbol{\cdot} \boldsymbol{k}& = \frac{{\rm \pi} \bar{\phi} d^{5} \eta_0 \{ [\boldsymbol{k} \boldsymbol{\cdot} (\bar{\boldsymbol{\omega}} - \bar{\boldsymbol{o}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}))][\boldsymbol{k} \boldsymbol{\cdot} (\bar{\boldsymbol{\omega}} - \bar{\boldsymbol{o}}(\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}))] - k^{2} [|\bar{\boldsymbol{\omega}}|^{2} - (\bar{\boldsymbol{\omega}} \boldsymbol{\cdot} \bar{\boldsymbol{o}})^{2}] \}}{ 8 \mu_0 M (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})} \nonumber\\ &\quad + \frac{2 \bar{\phi} \mu_0 M^{2} (\boldsymbol{k} \boldsymbol{\cdot} \bar{\boldsymbol{o}})(\boldsymbol{k} \boldsymbol{\cdot} \bar{\boldsymbol{H}})}{{\rm \pi}^{2} d^{4} \eta_0 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})} + \frac{{\rm \pi} k^{2} d^{5} \bar{\phi} \eta_0 \eta^{\prime} (|\bar{\boldsymbol{\omega}}|^{2} - (\bar{\boldsymbol{\omega}} \boldsymbol{\cdot} \bar{\boldsymbol{o}})^{2})}{12 \mu_0 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})} \nonumber\\ & \quad - \frac{2 k^{2} \bar{\phi} \mu_0 M^{2}}{{\rm \pi}^{2} \eta_0 d^{4}}. \end{align}

Here, the substitution $\bar {n} = \bar {\phi } ({\rm \pi} d^{3}/6)^{-1}$ has been made, where $\bar {\phi }$ is the volume fraction in the base state. The diffusion coefficient $\boldsymbol{D}$ is extracted from (2.38). This is written in scaled form as the sum of a ‘hydrodynamic’ contribution due to the vorticity disturbance, a ‘magnetic’ contribution due to the magnetic field disturbance, and an isotropic contribution due to the variation in viscosity and magnetic permeability with particle volume fraction:

(2.39)\begin{equation} \boldsymbol{D} = \bar{\phi} d^{2} |\boldsymbol{\omega}| (\boldsymbol{D}^{h} + \boldsymbol{D}^{m} + {D_i^{\prime}} \boldsymbol{I}), \end{equation}

where the scaled diffusivities are

(2.40)\begin{gather} \boldsymbol{D}^{h} = \frac{\{ ({{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} - \bar{\boldsymbol{o}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}})) ({{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} - \bar{\boldsymbol{o}}(\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}})) - \boldsymbol{I} [1 - ({{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \boldsymbol{\cdot} \bar{\boldsymbol{o}})^{2}] \}}{ 8\varSigma (\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}})}, \end{gather}

(2.41)\begin{gather}\boldsymbol{D}^{m} = \frac{\varSigma {M^{{\ast}}}}{\rm \pi} \left( \frac{\bar{\boldsymbol{o}} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}} + {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}} \bar{\boldsymbol{o}}}{\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}}} - 2 \boldsymbol{I} \right), \end{gather}

(2.42)\begin{gather}{D_i^{\prime}} = \frac{\eta^{\prime} (1 - ({{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \boldsymbol{\cdot} \bar{\boldsymbol{o}})^{2})}{12 \varSigma (\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}})}. \end{gather}

Here, ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}$ and ${{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$ are the unit vectors along the applied vorticity and magnetic fields, and the substitutions $\bar {\boldsymbol{\omega }} = |\bar {\boldsymbol{\omega }}| {{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}$ and $\bar {\boldsymbol{H}} = |\bar {\boldsymbol{H}}| {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$ have been used. There are two dimensionless numbers in (2.40)–(2.42), the scaled ratio of the hydrodynamic and magnetic torques $\varSigma$, and the scaled magnetic moment ${M^{\ast }}$:

(2.43)\begin{gather} \varSigma = \frac{\mu_0 M |\bar{\boldsymbol{H}}|}{{\rm \pi} \eta_0 d^{3} |\bar{\boldsymbol{\omega}}|}, \end{gather}

(2.44)\begin{gather}{M^{{\ast}}} = \frac{M}{d^{3} |\bar{\boldsymbol{H}}|}. \end{gather}

In the diffusion tensor (2.39), the contribution $\boldsymbol{D}^{m}$ due to the magnetic torque is proportional to $\bar {\phi } |\bar {\boldsymbol{\omega }}| d^{2} \varSigma \sim \bar {\phi } \mu _0 M |\bar {\boldsymbol{H}}|/ d \eta _0$ and is independent of the vorticity, but it does depend on the viscosity, the particle magnetic moment and the magnetic field. The contribution $\boldsymbol{D}^{h}$ due to the hydrodynamic torque is proportional to $\bar {\phi } |\bar {\boldsymbol{\omega }}| d^{2} \varSigma ^{-1} \sim \bar {\phi } d^{5} \eta _0 |\bar {\boldsymbol{\omega }}|^{2}/\mu _0 M |\bar {\boldsymbol{H}}|$, and is proportional to the square of the vorticity and inversely proportional to the magnetic field. The isotropic component ${D_i} \boldsymbol{I}$ is due to the variation of the viscosity with the particle volume fraction. This is positive, and it has a damping effect on concentration fluctuations.

2.2. Steady solution

The rotating and steady solutions for the orientation vector for an isolated dipolar spheroid in a magnetic field were derived in Kumaran (Reference Kumaran2020a). Here, the results for a spherical particle are summarised briefly. The orientation of the particle depends on the dimensionless parameter $\varSigma$ in (2.43). The projections of the orientation vector along the directions of the magnetic field and the vorticity are shown as functions of the parameter $\varSigma$ in figure 2. The stable solutions are shown by the blue lines, unstable solutions by the red lines, and neutral solutions by the green lines.

Figure 2.

The variation of (a) $\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$ and (b) $\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}$ with $\varSigma$ for a particle with a permanent dipole. The solid lines are the results for a parallel magnetic field ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}} = 0$, and the dashed lines are the results for an oblique magnetic field with ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}} = 0.1$ ($\circ$), $\frac {1}{2}$ ($\triangle$), $\frac {1}{\sqrt {2}}$ ($\nabla$) and $\frac {\sqrt {3}}{2}$ ($\diamond$). The blue lines are the stable stationary nodes, the red lines are the unstable stationary nodes, and the green lines are the neutral stationary nodes around which there are periodic orbits.

In the limit $\varSigma \rightarrow \infty$, the orientation vector is parallel to the magnetic field; this corresponds to $\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}} \rightarrow 1$ for the stable solution. There is also an unstable solution where the orientation vector is anti-parallel to the magnetic field, in which case $\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}} \rightarrow -1$. As $\varSigma$ decreases, there is a gradual decrease in $\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$, while $\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}$ increases and tends to $1$ for the stable solution in this limit. This implies that the particle is aligned along the vorticity direction in the limit $\varSigma \rightarrow 0$.

The parallel magnetic field, ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}} = 0$, is a special case. In figure 2, it is observed that there is a transition with a slope discontinuity in this case for $\varSigma = \frac {1}{2}$. The stable and unstable branches merge and bifurcate into two centre nodes, around which there are periodic closed orbits. There are no steady solutions in this case, and the distribution of orientations in the different orbits depends on the initial distribution. The effect of interactions for a parallel magnetic field is examined in § 2.3 for the parameter regime $\varSigma > \frac {1}{2}$, where there is a steady solution. The same study for an oblique magnetic field is presented in § 2.4 for $\varSigma > 0$, since there is a steady solution for all values of $\varSigma$.

2.3. Parallel magnetic field

Here, we consider the special case where the imposed magnetic field is in the flow plane and perpendicular to the vorticity, $\bar {\boldsymbol{H}} \boldsymbol{\cdot } \bar {\boldsymbol{\omega }} = 0$. From (2.9), $\bar {\boldsymbol{o}} \boldsymbol{\cdot } \bar {\boldsymbol{\omega }} = 0$ for steady solutions, that is, the orientation vector and the vorticity are also orthogonal. The configuration and coordinate system are shown in figure 3(a). A linear shear flow, shown by grey arrows, is applied in the fluid far from the particle. The mean vorticity is in the direction perpendicular to the plane of shear, and the magnetic field vector is in the plane of shear. The orthogonal unit vectors ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}} = \bar {\boldsymbol{\omega }}/|\bar {\boldsymbol{\omega }}|$ and ${{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}} = \bar {\boldsymbol{H}} / |\bar {\boldsymbol{H}}|$ are defined along the vorticity and magnetic field directions, and the third orthogonal vector ${{\boldsymbol {e}}_\perp } = {{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}} \times {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$ is perpendicular to the ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}\unicode{x2013}{{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$ plane.

Figure 3.

The configuration and coordinate system for analysing the effect of interactions for (a) a parallel magnetic field, and (b) an oblique magnetic field. The vorticity vector, shown in blue, is perpendicular to the plane of flow. The applied magnetic field shown in brown and the orientation vector shown in red are in the plane of flow in (a), and at an angle to the plane of flow in (b). The unit vector ${{\boldsymbol {e}}_\parallel }$ is perpendicular to $\bar {\boldsymbol{\omega }}$ in the $\bar {\boldsymbol{\omega }}\unicode{x2013}\bar {\boldsymbol{H}}$ plane in (b). The unit vector ${{\boldsymbol {e}}_\perp }$ is perpendicular to the $\bar {\boldsymbol{\omega }}\unicode{x2013}\bar {\boldsymbol{H}}$ plane.

The solution for the orientation vector for the stable stationary node is

(2.45)\begin{gather} \bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}} = \frac{\sqrt{4 \varSigma^{2} - 1}}{2 \varSigma}, \end{gather}

(2.46)\begin{gather}\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_\perp} = \frac{1}{2 \varSigma}. \end{gather}

In (2.39) for the diffusion coefficient, the orientation vector is expressed as $\bar {\boldsymbol{o}} = \hat {o}_{\boldsymbol{H}} {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}} + (\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_\perp }) {{\boldsymbol {e}}_\perp }$, and (2.45)–(2.46) are substituted to obtain the diffusion tensor due to hydrodynamic and magnetic interactions:

(2.47)\begin{equation} \boldsymbol{D}^{h/m} = \left({{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}}\quad {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}}\quad {{\boldsymbol{e}}_\perp}\right) \left( \begin{array}{ccc} {D_{\bar{\boldsymbol{\omega}} \bar{\boldsymbol{\omega}}}^{h/m}} & 0 & 0 \\ 0 & {D_{\bar{\boldsymbol{H}} \bar{\boldsymbol{H}}}^{h/m}} & {D_{\bar{\boldsymbol{H}} \perp}^{h/m}} \\ 0 & {D_{\bar{\boldsymbol{H}} \perp}^{h/m}} & {D_{{\perp}{\perp}}^{h/m}} \end{array} \right) \left( \begin{array}{c} {{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \\ {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}} \\ {{\boldsymbol{e}}_\perp} \end{array} \right), \end{equation}

where

(2.48a,b)\begin{gather} {D_{\bar{\boldsymbol{\omega}} \bar{\boldsymbol{\omega}}}^{h}} = 0, \quad {D_{\bar{\boldsymbol{\omega}}\bar{\boldsymbol{\omega}}}^{m}} ={-} \frac{2 \varSigma {M^{{\ast}}}}{\rm \pi}, \end{gather}

(2.49a,b)\begin{gather}{D_{\bar{\boldsymbol{H}} \bar{\boldsymbol{H}}}^{h}} ={-} \frac{1}{4 \sqrt{4 \varSigma^{2} - 1}}, \quad {D_{\bar{\boldsymbol{H}} \bar{\boldsymbol{H}}}^{m}} = 0, \end{gather}

(2.50a,b)\begin{gather}{D_{\bar{\boldsymbol{H}} \perp}^{h}} = 0, \quad {D_{\bar{\boldsymbol{H}} \perp}^{m}} = \frac{{M^{{\ast}}} \varSigma}{{\rm \pi} \sqrt{4 \varSigma^{2} - 1}}, \end{gather}

(2.51a,b)\begin{gather}{D_{{\perp}{\perp}}^{h}} ={-} \frac{1}{4 \sqrt{4 \varSigma^{2} - 1}}, \quad {D_{{\perp}{\perp}}^{m}} ={-} \frac{2 \varSigma {M^{{\ast}}}}{\rm \pi}. \end{gather}

The isotropic contributions to the diffusion tensor due to the concentration dependence of the viscosity and the magnetic permeability are

(2.52)\begin{equation} {D_i^{\prime}} = \frac{\eta^{\prime}}{6 \sqrt{4 \varSigma^{2}-1}}. \end{equation}

The values of the coefficients in the limit $\varSigma \gg 1$ and $\varSigma - \frac {1}{2} \ll 1$ are listed in table 1. The eigenvalues and eigenvectors of the diffusion matrix are also provided. Since the diffusion matrix is symmetric, the eigenvalues are real, the eigenvectors are orthogonal, and the eigenvectors are the principal directions of amplification (along the directions with negative eigenvalues) or damping (along the directions with positive eigenvalues). If all three principal eigenvalues are negative, then there is amplification of concentration fluctuations in all three directions and the instability is expected to be in the form of particle clusters. If two eigenvalues are negative and one is positive, then there is amplification in two directions and damping in one direction, resulting in rod-like aggregates aligned along the direction with positive eigenvalue. If one eigenvalue is negative and two are positive, then there is amplification along one direction and damping along the other two directions, resulting in the formation of planar aggregates perpendicular to the unstable direction.

Table 1.

The asymptotic behaviour of the diffusion coefficients for $\varSigma, \varSigma _i \gg 1$ and close to the transition from steady to rotating states for a parallel magnetic field for permanent and induced dipoles.

The isotropic part of the diffusion tensor ${D_i}$ due to the dependence of viscosity on concentration, shown as a function of $\varSigma$ in figure 4, is not included in the calculation of the eigenvalues in table 1. When the isotropic part is included, the eigenvectors remain unchanged and the eigenvalues transform as $\lambda \rightarrow \lambda + {D_i}$.

Figure 4.

The scaled isotropic part of the diffusion tensor, ${D_i^{\prime }}/\eta ^{\prime }$ as a function of the parameter $\varSigma$ for particles with a permanent dipole. Here, ${D_i^{\prime }}$ is given in (2.42). The orientations of the vorticity and magnetic field are ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}} = 0.1$ ($\circ$), $\frac {1}{2}$ ($\triangle$), $\frac {1}{\sqrt {2}}$ ($\nabla$) and $\frac {\sqrt {3}}{2}$ ($\diamond$). The solid line is the result for a parallel magnetic field.

Equation (2.47) shows that one of the principal directions is always the vorticity direction, and the diffusion coefficient in this direction is negative, resulting in concentration amplification in this direction for all values of $\varSigma$. The magnitude of this diffusion coefficient increases proportional to $\varSigma {M^{\ast }}$ for $\varSigma \gg 1$. This diffusion is due to the modification of the magnetic field by fluctuations in the particle number density, which is the last term in the square brackets on the right in (2.11).

The diffusion in the flow plane is anisotropic, and the diffusion coefficients contain contributions due to hydrodynamic and magnetic interactions. The principal directions for diffusion in the plane, which are the eigenvectors of the $2 \times 2$ square sub-matrix in (2.47), are presented in table 1 for high magnetic field in the limit $\varSigma \gg 1$, and close to the transition between static and rotating states $\varSigma - \frac {1}{2} \ll 1$. For $\varSigma \gg 1$, the two principal axes are along the ${{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$ and ${{\boldsymbol {e}}_\perp }$ directions. The eigenvalue is negative along the ${{\boldsymbol {e}}_\perp }$ direction with magnitude $2 \varSigma {M^{\ast }}/{\rm \pi}$, which is the same as that in the ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}$ direction. The eigenvalue in the ${{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$ direction is positive for ${M^{\ast }} > {\rm \pi}$, and negative otherwise, and the magnitude is proportional to $\varSigma ^{-1}$ in this limit. This implies a weak damping of fluctuations for ${M^{\ast }} > {\rm \pi}$, and a weak amplification for ${M^{\ast }} < {\rm \pi}$. Thus there is anisotropic clustering with a strong amplification of fluctuations in the two directions perpendicular to the magnetic field, and weak damping or amplification along the magnetic field for $\varSigma \gg 1$.

For $\varSigma - \frac {1}{2} \ll 1$, the diffusion coefficient in the vorticity direction tends to a finite value, $- ({M^{\ast }} / {\rm \pi})$. The diffusion coefficients in the flow plane diverge proportional to $(\varSigma - \frac {1}{2})^{-1/2}$. The eigenvectors in the flow plane are along the directions rotated by angles $- {\rm \pi}/4$ and $+ {\rm \pi}/4$, respectively, from the magnetic field direction. One of the eigenvalues is always negative, while the other is positive for ${M^{\ast }} > {\rm \pi}/2$ and negative otherwise. In both cases, the eigenvalues diverge proportional to $(\varSigma - \frac {1}{2})^{-1/2}$. Thus there is strong concentration amplification along one principal direction in the flow plane, and strong damping in the perpendicular direction for ${M^{\ast }} > {\rm \pi}/2$, and strong amplification in both principal directions for ${M^{\ast }} < {\rm \pi}/2$.

It should be noted that the divergence proportional to $(4 \varSigma ^{2} - 1)^{-1/2}$ is the result of a mean-field calculation; a more complex renormalisation group calculation is required to include the effect of fluctuations.

2.4. Oblique magnetic field

Analytical solutions for the steady orientation have been derived for a spherical particle in an oblique magnetic field, where the vorticity and magnetic field are not perpendicular. The orthogonal coordinate system shown in figure 3(b) is used, where the unit vector ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}$ is along the vorticity direction, ${{\boldsymbol {e}}_\parallel }$ is perpendicular to ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}$ in the ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}\unicode{x2013}{{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$ plane, and ${{\boldsymbol {e}}_\perp }$ is perpendicular to the ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}\unicode{x2013}{{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$ plane. The following notations are used for the dot products:

(2.53a,b)\begin{equation} \hat{\omega}_{\boldsymbol{H}} = {{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}}, \quad \hat{o}_{\boldsymbol{H}} = \bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}}. \end{equation}

It should be noted that $\hat {\omega }_{\boldsymbol{H}}$ is specified, since it depends on the relative orientation of the vorticity and magnetic field, whereas $\hat {o}_{\boldsymbol{H}}$ is determined from the solution for the orientation vector. From (2.9), the dot product $\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}$ is $\hat {\omega }_{\boldsymbol{H}}/\hat {o}_{\boldsymbol{H}}$. The unit vectors ${{\boldsymbol {e}}_\parallel }$ and ${{\boldsymbol {e}}_\perp }$ are

(2.54)\begin{gather} {{\boldsymbol{e}}_\parallel} = \frac{{{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}} - \hat{\omega}_{\boldsymbol{H}} {{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}}}{\sqrt{1 - \hat{\omega}_{\boldsymbol{H}}^{2}}}, \end{gather}

(2.55)\begin{gather}{{\boldsymbol{e}}_\perp} = \frac{{{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \times {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}}}{\sqrt{1 - \hat{\omega}_{\boldsymbol{H}}^{2}}}. \end{gather}

The orientation vector for the steady solution is specified by

(2.56)\begin{gather} \bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} = \sqrt{- \frac{4 \varSigma^{2}-1}{2} + \frac{\sqrt{(4 \varSigma^{2} - 1)^{2} + 16 \varSigma^{2} \hat{\omega}_{\boldsymbol{H}}^{2}}}{2}}, \end{gather}

(2.57)\begin{gather}\hat{o}_{\boldsymbol{H}} = \frac{1}{2 \varSigma} \sqrt{\frac{4 \varSigma^{2}-1}{2} + \frac{\sqrt{(4 \varSigma^{2} - 1)^{2} + 16 \varSigma^{2} \hat{\omega}_{\boldsymbol{H}}^{2}}}{2}}. \end{gather}

The projection of the orientation vector on the unit vector ${{\boldsymbol {e}}_\parallel }$ is

(2.58)\begin{equation} \bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_\parallel} = \frac{\hat{\omega}_{\boldsymbol{H}}(1 - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}})^{2})}{ (\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}}) \sqrt{1 - \hat{\omega}_{\boldsymbol{H}}^{2}}}. \end{equation}

The dot product $\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_\perp }$ is determined from the torque balance equation:

(2.59)\begin{align} \bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_\perp} & = \frac{\bar{\boldsymbol{o}} \boldsymbol{\cdot} ({{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \times {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}})}{\sqrt{1 - ({{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}})^{2}}} ={-} \frac{{{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}})}{\sqrt{1 - ({{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}})^{2}}} \nonumber\\ & = \frac{{{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \boldsymbol{\cdot} (\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}}}{2 \varSigma \sqrt{1 - ({{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}})^{2}}} = \frac{1 - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}})^{2}}{2 \varSigma \sqrt{1 - \hat{\omega}_{\boldsymbol{H}}^{2}}}. \end{align}

The torque balance equation (2.8) and the definition (2.43) of $\varSigma$ have been used in the third step in the above equation. Equations (2.57)–(2.59) are substituted into (2.39) to obtain (2.47) for the diffusion coefficient, where $\boldsymbol{D}^{h}$ and $\boldsymbol{D}^{m}$ are symmetric tensors of the form

(2.60)\begin{equation} \boldsymbol{D}^{h/m} = \left({{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \quad {{\boldsymbol{e}}_\parallel} \quad {{\boldsymbol{e}}_\perp}\right) \left( \begin{array}{ccc} {D_{\bar{\boldsymbol{\omega}} \bar{\boldsymbol{\omega}}}^{h/m}} & {D_{\bar{\boldsymbol{\omega}} \parallel}^{h/m}} & {D_{\bar{\boldsymbol{\omega}} \perp}^{h/m}} \\[5pt] {D_{\bar{\boldsymbol{\omega}} \parallel}^{h/m}} & {D_{{\parallel}{\parallel}}^{h/m}} & {D_{{\parallel}{\perp}}^{h/m}} \\[5pt] {D_{\bar{\boldsymbol{\omega}} \perp}^{h/m}} & {D_{{\parallel}{\perp}}^{h/m}} & {D_{{\perp}{\perp}}^{h/m}} \end{array} \right) \left( \begin{array}{c} {{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \\ {{\boldsymbol{e}}_\parallel} \\ {{\boldsymbol{e}}_\perp} \end{array} \right). \end{equation}

The solutions for the elements of the tensors $\boldsymbol{D}^{h}$ and $\boldsymbol{D}^{m}$, and the asymptotic expansion of these elements for small and large $\varSigma$, are provided in table 2. The elements of $\boldsymbol{D}^{h}$ and $\boldsymbol{D}^{m}$ do depend on the solution (2.57) for $\hat {o}_{\boldsymbol{H}}$, and the asymptotic expansions employ the small and large $\varSigma$ approximations for the $\hat {o}_{\boldsymbol{H}}$ solution:

(2.61)\begin{gather} \hat{o}_{\boldsymbol{H}} = \hat{\omega}_{\boldsymbol{H}} + 2 \varSigma^{2} \hat{\omega}_{\boldsymbol{H}} (1 - \hat{\omega}_{\boldsymbol{H}}^{2}) \quad \mbox{for} \ \varSigma \ll 1, \end{gather}

(2.62)\begin{gather}\hat{o}_{\boldsymbol{H}} = 1 - \frac{1 - \hat{\omega}_{\boldsymbol{H}}^{2}}{4 \varSigma^{2}} \quad \mbox{for} \ \varSigma \gg 1. \end{gather}

It is verified easily that the elements of $\boldsymbol{D}^{h}$ and $\boldsymbol{D}^{m}$ reduce to (2.39) for a parallel magnetic field with $\hat {\omega }_{\boldsymbol{H}} = 0$.

Table 2.

The asymptotic behaviour of the diffusion coefficients for $\varSigma \gg 1$ and $\varSigma \ll 1$ for an oblique magnetic field for particles with permanent dipoles.

For an oblique magnetic field, all the elements of $\boldsymbol{D}^{h}$ and $\boldsymbol{D}^{m}$ are non-zero. This is in contrast to the diffusion matrix (2.39) for a parallel magnetic field, where there are only five independent non-zero components. For a parallel magnetic field, there is a transition from a static to a rotating state for the orientation vector for a parallel magnetic field at $\varSigma = \frac {1}{2}$. In contrast, there is no transition for an oblique magnetic field. Therefore, the diffusion coefficients for a parallel magnetic field are defined only in the range $\frac {1}{2} < \varSigma < \infty$, whereas those for an oblique magnetic field are defined for $0 < \varSigma < \infty$ in figure 5.

Figure 5.

The components of the diffusion tensor: (a) $- {D_{\bar {\boldsymbol{\omega }} \bar {\boldsymbol{\omega }}}^{h}}$ and $- {D_{\bar {\boldsymbol{\omega }}\bar {\boldsymbol{\omega }}}^{m}}/{M^{\ast }}$, (b) $- {D_{\bar {\boldsymbol{\omega }} \parallel }^{h}}$ and ${D_{\bar {\boldsymbol{\omega }} \parallel }^{m}}/{M^{\ast }}$, (c) $- {D_{\bar {\boldsymbol{\omega }} \perp }^{h}}$ and ${D_{\bar {\boldsymbol{\omega }} \perp }^{m}}/{M^{\ast }}$, (d) $- {D_{\parallel \parallel }^{h}}$ and $- {D_{\parallel \parallel }^{m}}/{M^{\ast }}$, (e) ${D_{\parallel \perp }^{h}}$ and ${D_{\parallel \perp }^{m}}/{M^{\ast }}$, and (f) $- {D_{\perp \perp }^{h}}$ and $- {D_{\perp \perp }^{m}}/{M^{\ast }}$, due to hydrodynamic interactions (blue lines) and magnetic interactions (red lines) as functions of the parameter $\varSigma$ for particles with a permanent dipole. The orientations of the vorticity and magnetic field are ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}} = 0.1$ ($\circ$), $\frac {1}{2}$ ($\triangle$), $\frac {1}{\sqrt {2}}$ ($\nabla$) and $\frac {\sqrt {3}}{2}$ ($\diamond$). The solid red and blue lines are the non-zero results for a parallel magnetic field.

Table 2 shows that all components of the hydrodynamic contribution to the diffusion tensor $\boldsymbol{D}^{h}$ are negative, with the exception of the component ${D_{\parallel \perp }^{h}}$. In contrast, all components of the magnetic contribution to the diffusion tensor $\boldsymbol{D}^{m}$ are positive. This implies that the magnetic interactions tend to dampen concentration fluctuations and stabilise the uniform state, whereas hydrodynamic interactions tend to destabilise the concentration fluctuations. The diffusion in the direction perpendicular to the vorticity and magnetic field is due entirely to hydrodynamic interactions, and the diffusion coefficient ${D_{\perp \perp }}$ is negative. Therefore, concentration fluctuations are amplified in the direction perpendicular to the vorticity and magnetic field. In other directions, the stability is determined by a balance between the contributions due to hydrodynamic and magnetic interactions.

The components of $\boldsymbol{D}$ are shown as functions of $\varSigma$ for different values of $\hat {\omega }_{\boldsymbol{H}}$ in figures 4 and 5. Also shown by solid lines in figures 5(a), 5(d), 5(e) and 5(f) are the results for a parallel magnetic field. The components of the diffusion tensor for a parallel magnetic field do not extend to $\varSigma < \frac {1}{2}$, due to the transition to a rotating state; the divergence in the diffusion coefficients ${D_{\bar {\boldsymbol{H}} \bar {\boldsymbol{H}}}^{h}}$, ${D_{\bar {\boldsymbol{H}} \perp }^{m}}$ and ${D_{\perp \perp }^{h}}$ predicted by (2.49a,b)–(2.51a,b) is apparent in figure 5. The results for $\hat {\omega }_{\boldsymbol{H}} = 0.1$ are in close agreement with those for a parallel magnetic field for $\varSigma \gtrsim \frac {1}{2}$, but the divergence in the coefficients ${D_{\bar {\boldsymbol{H}} \bar {\boldsymbol{H}}}^{h}}$, ${D_{\bar {\boldsymbol{H}} \perp }^{m}}$ and ${D_{\perp \perp }^{h}}$ is cut off, and the diffusion coefficients are finite, $\varSigma < \frac {1}{2}$.

The coefficients ${D_{\bar {\boldsymbol{\omega }} \bar {\boldsymbol{\omega }}}^{h}}, {D_{\bar {\boldsymbol{\omega }} \parallel }^{h}}, {D_{\bar {\boldsymbol{\omega }} \parallel }^{m}}, {D_{\bar {\boldsymbol{\omega }} \perp }^{h}}, {D_{\bar {\boldsymbol{\omega }} \perp }^{m}}, {D_{\parallel \parallel }^{m}}, {D_{\parallel \perp }^{h}}$ are all zero for a parallel magnetic field with $\hat {\omega }_{\boldsymbol{H}} = 0$. Figure 5 shows that these coefficients do decrease as $\hat {\omega }_{\boldsymbol{H}}$ decreases for $\varSigma > \frac {1}{2}$, in accordance with the $\varSigma \gg 1$ expressions in table 2. For $\varSigma < \frac {1}{2}$, these do not decrease as $\hat {\omega }_{\boldsymbol{H}}$ decreases. The latter regime is not accessible for a parallel magnetic field, since there is no steady orientation. Therefore, the diffusion due to interactions for a nearly parallel magnetic field is qualitatively different from that in a parallel magnetic field, due to the transition to rotating states in the latter.

The magnetic contributions to the diffusion tensor are larger than the hydrodynamic contributions for $\varSigma \gg 1$. For $\varSigma \gg 1$, the largest contributions to the diffusion tensor are ${D_{\bar {\boldsymbol{\omega }}\bar {\boldsymbol{\omega }}}^{m}}$, ${D_{\bar {\boldsymbol{\omega }} \parallel }^{m}}$, ${D_{\parallel \parallel }^{m}}$ and ${D_{\perp \perp }^{m}}$, which diverge proportional to $\varSigma$. The coefficient ${D_{\perp \perp }^{m}} = - (2 {M^{\ast }} \varSigma /{\rm \pi} )$ is negative for all $\varSigma$, indicating that concentration fluctuations are amplified in the ${{\boldsymbol {e}}_\perp }$ direction. The diffusion in the ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}\unicode{x2013}{{\boldsymbol {e}}_\parallel }$ plane is along two principal directions. The eigenvalue $- (2 {M^{\ast }} \varSigma /{\rm \pi} )$ along $\boldsymbol{e}_2$ in table 2 also increases proportional to $\varSigma$ for $\varSigma \gg 1$, indicating strong amplification of concentration fluctuations in this direction. The eigenvalue along the $\boldsymbol{e}_1$ direction decreases proportional to $\varSigma ^{-1}$ for $\varSigma \gg 1$, and the concentration fluctuations are dampened/amplified for ${M^{\ast }} \gtrless {\rm \pi}$. Thus the behaviour of concentration fluctuations for $\varSigma \gg 1$ is very similar to that for a parallel magnetic field, with strong concentration amplifications in two directions ${{\boldsymbol {e}}_\perp }$ and $\boldsymbol{e}_2$, and weak amplification/damping in the third perpendicular direction.

In the limit $\varSigma \ll 1$, the diffusion coefficient along the ${{\boldsymbol {e}}_\perp }$ direction perpendicular to the vorticity and magnetic field is ${D_{\perp \perp }^{m}} = - (2 {M^{\ast }} \varSigma /{\rm \pi} )$. The $O(\varSigma )$ contributions to the diffusion tensor are ${D_{\bar {\boldsymbol{\omega }} \bar {\boldsymbol{\omega }}}^{h}}$, ${D_{\bar {\boldsymbol{\omega }}\bar {\boldsymbol{\omega }}}^{m}}$, ${D_{\parallel \parallel }^{h}}$, ${D_{\bar {\boldsymbol{\omega }} \parallel }^{m}}$ and ${D_{\perp \perp }^{m}}$. For diffusion in the ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}\unicode{x2013}{{\boldsymbol {e}}_\parallel }$ plane, concentration fluctuations are unstable along the $\boldsymbol{e}_2$ direction where the eigenvalue $\lambda _2$ is negative, while they are stable in the perpendicular direction $\boldsymbol{e}_1$ for $2 {M^{\ast }} > {\rm \pi}(1 + \hat {\omega }_{\boldsymbol{H}})$ and unstable otherwise.

3.1. Diffusion due to interactions

For an induced dipole, the particle magnetic moment is modelled as

(3.1)\begin{equation} \boldsymbol{M} = \chi \boldsymbol{o} [\boldsymbol{o} \boldsymbol{\cdot} (\boldsymbol{H} + n \boldsymbol{M})], \end{equation}

that is, the magnetic moment is proportional to the component of the magnetic field along the particle orientation. Here, $\chi$ is the magnetic susceptibility of the particle. Substituting $\boldsymbol{M} = M \boldsymbol{o}$ in (3.1), the magnitude of the particle magnetic moment is

(3.2)\begin{equation} M = \frac{\chi \boldsymbol{o} \boldsymbol{\cdot} \boldsymbol{H}}{1 - \chi n}. \end{equation}

The torque balance equation in the absence of interactions in the direction perpendicular to the orientation vector, analogous to (2.8), is

(3.3)\begin{equation} \frac{1}{2} {\rm \pi}\eta d^{3} (\boldsymbol{I} - \boldsymbol{o} \boldsymbol{o}) \boldsymbol{\cdot} \boldsymbol{\omega} + \frac{\mu_0 \chi (\boldsymbol{o} \boldsymbol{\cdot} \boldsymbol{H}) (\boldsymbol{o} \times \boldsymbol{H})}{ 1 - \chi n} = 0. \end{equation}

The substitution (3.2) is made in (2.1) to obtain the expression for the force,

(3.4)\begin{align} \boldsymbol{F} & = \boldsymbol{\nabla} \left[\frac{\chi (\boldsymbol{o} \boldsymbol{\cdot} \boldsymbol{H})}{1 - \chi n}\,\boldsymbol{o} \boldsymbol{\cdot} \left(\boldsymbol{H} + \frac{\chi n \boldsymbol{o} (\boldsymbol{o} \boldsymbol{\cdot} \boldsymbol{H})}{1 - \chi n} \right) \right] \nonumber\\ & = \boldsymbol{\nabla} \left[ \frac{\chi (\boldsymbol{o} \boldsymbol{\cdot} \boldsymbol{H})^{2}}{(1 - \chi n)^{2}} \right]. \end{align}

When this expression is linearised in the perturbations and transformed into Fourier space, the equivalent of (2.11) for the interaction force is

(3.5)\begin{equation} \hat{\boldsymbol{F}}_{\boldsymbol{k}} ={-} 2 {{\rm{i}}} \boldsymbol{k} \mu_0 \chi (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) \left[\frac{ \bar{\boldsymbol{o}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}} + \bar{\boldsymbol{H}} \boldsymbol{\cdot} \hat{\boldsymbol{o}}_{\boldsymbol{k}}}{(1 - \chi \bar{n})^{2}} + \frac{\chi \hat{n}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})}{(1 - \chi \bar{n})^{3}} \right]. \end{equation}

Since we are considering the limit where the disturbance is small compared to the applied magnetic field, the approximation $\bar {n} \chi \ll 1$ is made in (3.5). With this approximation, the equivalent of (2.16) for the particle concentration field is

(3.6)\begin{equation} \frac{\partial \hat{n}_{\boldsymbol{k}}}{\partial t} + {{\rm{i}}} \boldsymbol{k} \boldsymbol{\cdot} (\bar{\boldsymbol{v}} \hat{n}_{\boldsymbol{k}}) + D_B k^{2} \hat{n}_{\boldsymbol{k}} - \left( \frac{2 k^{2} \bar{n} \mu_0 \chi (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}} + \bar{\boldsymbol{H}} \boldsymbol{\cdot} \hat{\boldsymbol{o}}_{\boldsymbol{k}}+\chi \hat{n}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}))}{3 {\rm \pi}\eta_0 d} \right) = 0. \end{equation}

Comparing (3.6) with (2.17), the magnetophoretic diffusion term in the particle number density equation is

(3.7)\begin{equation} \boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{D} \boldsymbol{\cdot} \boldsymbol{k} ={-} \frac{2 k^{2} \bar{n} \mu_0 \chi (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}} + \hat{\boldsymbol{o}}_{\boldsymbol{k}} \boldsymbol{\cdot} \bar{\boldsymbol{H}} + \chi \hat{n}_{\boldsymbol{k}} \bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})}{3 {\rm \pi}\eta_0 d \hat{n}_{\boldsymbol{k}}}. \end{equation}

The equivalent of (2.19) for the disturbance to the magnetic field due to interactions is

(3.8)\begin{equation} \hat{\boldsymbol{H}}_{\boldsymbol{k}} ={-} \frac{\boldsymbol{k} \boldsymbol{k}}{k^{2}} \boldsymbol{\cdot} \left[ \frac{\hat{n}_{\boldsymbol{k}} \chi \bar{\boldsymbol{o}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})}{ (1 - \chi \bar{n})^{2}} + \frac{\bar{n} \hat{\boldsymbol{o}}_{\boldsymbol{k}} \boldsymbol{\cdot} (\boldsymbol{I} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) + \bar{\boldsymbol{H}} \bar{\boldsymbol{o}})}{1 - \chi \bar{n}} \right]. \end{equation}

The second term in the square brackets on the right, proportional to $\hat {\boldsymbol{o}}_{\boldsymbol{k}}$, equivalent to the term proportional to $\hat {\boldsymbol{H}}''$ in (2.19), is neglected. The reason for this is discussed in the paragraph following (2.28). In the denominator of the first term in the square brackets on the right of (3.8), $\chi \bar {n}$ is neglected in comparison to $1$, because the disturbance to the magnetic field is small compared to the applied magnetic field. The disturbance to the vorticity field is given in (2.24). In this expression, the term proportional to $\bar {n} d^{3}$ on the left-hand side is neglected because the volume fraction is small, and the term proportional to $\hat {\boldsymbol{\omega }}''$ on the right-hand side is neglected for the reason discussed in the paragraph after (2.28). With these approximations, the torque balance equation, equivalent to (2.28), is

(3.9)\begin{align} &\frac{{\rm \pi} d^{3} \eta_0}{2 \mu_0 \chi}\,[(\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \hat{\boldsymbol{\omega}}_{\boldsymbol{k}} - \bar{\boldsymbol{o}} (\hat{\boldsymbol{o}}_{\boldsymbol{k}} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}) - \hat{\boldsymbol{o}}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}) + \eta^{\prime} \hat{\phi}_{\boldsymbol{k}} (\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}] \nonumber\\ &\quad + \{(\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})(\bar{\boldsymbol{o}} \times \hat{\boldsymbol{H}}_{\boldsymbol{k}}) + (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}}) (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) + (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) (\hat{\boldsymbol{o}}_{\boldsymbol{k}} \times \bar{\boldsymbol{H}}) \nonumber\\ &\quad + (\hat{\boldsymbol{o}}_{\boldsymbol{k}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) + \chi \hat{n}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})(\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) \} = 0. \end{align}

The expression (2.29), is substituted into (3.9) to obtain

(3.10)\begin{align} &\frac{{\rm \pi} d^{3} \eta_0}{2 \mu_0 \chi}\,\{(\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \hat{\boldsymbol{\omega}}_{\boldsymbol{k}} - \bar{\boldsymbol{o}} \hat{o}^{\ddagger}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) \boldsymbol{\cdot} \bar{\boldsymbol{\omega}} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}) [\hat{o}^{{{\dagger}}}_{\boldsymbol{k}} (\bar{\boldsymbol{H}} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) \bar{\boldsymbol{o}}) \nonumber\\ &\quad + \hat{o}^{\ddagger}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}})] + \eta^{\prime} \hat{\phi}_{\boldsymbol{k}} (\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} \bar{\boldsymbol{\omega}} \} + \{(\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})(\bar{\boldsymbol{o}} \times \hat{\boldsymbol{H}}_{\boldsymbol{k}}) \nonumber\\ &\quad + (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}}) (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) + \hat{o}^{{{\dagger}}}_{\boldsymbol{k}} [|\bar{\boldsymbol{H}}|^{2} - 2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2}] (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) \nonumber\\ &\quad + \hat{o}^{\ddagger}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) [\bar{\boldsymbol{H}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) - \bar{\boldsymbol{o}} |\bar{\boldsymbol{H}}|^{2}] + \chi \hat{n}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})(\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) \} = 0. \end{align}

The functions $\hat {o}^{{{\dagger}} }_{\boldsymbol{k}}$ and $\hat {o}^{\ddagger }_{\boldsymbol{k}}$ are determined by taking the dot product of (3.10) with $\bar {\boldsymbol{o}} \times \bar {\boldsymbol{H}}$ and $\bar {\boldsymbol{o}}$, respectively:

(3.11)\begin{align} &\frac{{\rm \pi} d^{3} \eta_0}{2 \mu_0 \chi}\,[\hat{\boldsymbol{\omega}}_{\boldsymbol{k}} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) - \hat{o}^{\ddagger}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}})(|\bar{\boldsymbol{H}}|^{2} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2}) + \eta^{\prime} \hat{\phi}_{\boldsymbol{k}} \bar{\boldsymbol{\omega}} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}})] \nonumber\\ &\quad + \{(\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})[\bar{\boldsymbol{H}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}} - 2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})(\hat{\boldsymbol{H}}_{\boldsymbol{k}} \boldsymbol{\cdot} \bar{\boldsymbol{o}})] + (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}}) |\bar{\boldsymbol{H}}|^{2} \nonumber\\ &\quad + [|\bar{\boldsymbol{H}}|^{2} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2}][\hat{o}^{{{\dagger}}}_{\boldsymbol{k}} (|\bar{\boldsymbol{H}}|^{2} - 2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2}) + \chi \hat{n}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})] \} = 0, \end{align}

(3.12)\begin{align} &- \frac{{\rm \pi} d^{3} \eta_0}{2 \mu_0 \chi}\,[\hat{o}^{\ddagger}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) \boldsymbol{\cdot} \bar{\boldsymbol{\omega}}] - \{ \hat{o}^{\ddagger}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) [|\bar{\boldsymbol{H}}|^{2} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2}] \} = 0. \end{align}

These are solved to obtain

(3.13)\begin{align} \hat{o}^{{{\dagger}}}_{\boldsymbol{k}} & = \frac{{\rm \pi} \eta_0 d^{3}}{2 \mu_0 \chi}\,\frac{\hat{\boldsymbol{\omega}}_{\boldsymbol{k}} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) + \eta^{\prime} \hat{\phi}_{\boldsymbol{k}} \bar{\boldsymbol{\omega}} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}})}{[2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2} - |\bar{\boldsymbol{H}}|^{2}] [|\bar{\boldsymbol{H}}|^{2} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2}]} + \frac{\chi \hat{n}_{\boldsymbol{k}} \bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}}{2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2} - |\bar{\boldsymbol{H}}|^{2}} \nonumber\\ & \quad + \frac{\{(\bar{\boldsymbol{o}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}})[|\bar{\boldsymbol{H}}|^{2} - 2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2}] + (\bar{\boldsymbol{H}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}})(\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})\}}{[2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2} - |\bar{\boldsymbol{H}}|^{2}] [|\bar{\boldsymbol{H}}|^{2} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2}]}, \end{align}

(3.14)\begin{gather} \hat{o}^{\ddagger}_{\boldsymbol{k}} = 0. \end{gather}

The expression for the magnetophoretic diffusion in (3.7) is

(3.15)\begin{align} \boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{D} \boldsymbol{\cdot} \boldsymbol{k} & ={-} \frac{2 k^{2} \bar{n} \mu_0 \chi (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})}{3 {\rm \pi}\eta_0 d\hat{n}_{\boldsymbol{k}}} \left[ \bar{\boldsymbol{o}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}} + \hat{o}^{{{\dagger}}}_{\boldsymbol{k}} [|\bar{\boldsymbol{H}}|^{2} - (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2}] + \chi \hat{n}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) \right] \nonumber\\ & ={-} \frac{2 k^{2} \bar{n} \mu_0 \chi (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})}{3 {\rm \pi}\eta_0 d \hat{n}_{\boldsymbol{k}}} \left[ \frac{{\rm \pi} d^{3} \eta_0 [\hat{\boldsymbol{\omega}}_{\boldsymbol{k}} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) + \eta^{\prime} \hat{\phi}_{\boldsymbol{k}} \bar{\boldsymbol{\omega}} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}})]}{2 \mu_0 \chi [2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2} - |\bar{\boldsymbol{H}}|^{2}]} \right. \nonumber\\ &\quad \left. {}+ \frac{(\bar{\boldsymbol{H}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}}) (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})}{2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2} - |\bar{\boldsymbol{H}}|^{2}} + \frac{\chi \hat{n}_{\boldsymbol{k}} \bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}} [|\bar{\boldsymbol{H}}|^{2}-(\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})]}{2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2} - |\bar{\boldsymbol{H}}|^{2}} + \chi \hat{n}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) \right]. \nonumber\\ & ={-} \frac{2 k^{2} \bar{n} \mu_0 \chi (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})}{3 {\rm \pi}\eta_0 d \hat{n}_{\boldsymbol{k}}} \left[ \frac{{\rm \pi} d^{3} \eta_0 [\hat{\boldsymbol{\omega}}_{\boldsymbol{k}} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) + \eta^{\prime} \hat{\phi}_{\boldsymbol{k}} \bar{\boldsymbol{\omega}} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}})]}{2 \mu_0 \chi [2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2} - |\bar{\boldsymbol{H}}|^{2}]} \right. \nonumber\\ & \quad \left. {}+ \frac{(\bar{\boldsymbol{H}} \boldsymbol{\cdot} \hat{\boldsymbol{H}}_{\boldsymbol{k}}) (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})}{2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2} - |\bar{\boldsymbol{H}}|^{2}} + \frac{\chi \hat{n}_{\boldsymbol{k}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{3}}{2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2} - |\bar{\boldsymbol{H}}|^{2}} \right]. \end{align}

Substituting (3.8) and (2.24) for $\hat {\boldsymbol{H}}_{\boldsymbol{k}}$ and $\hat {\boldsymbol{\omega }}_{\boldsymbol{k}}$, the final expression for the diffusion coefficient is

(3.16)\begin{align} \boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{D} \boldsymbol{\cdot} \boldsymbol{k} & ={-} \frac{d^{2} \bar{\phi} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) [(\boldsymbol{k} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}} - (\boldsymbol{k} \boldsymbol{\cdot} \bar{\boldsymbol{o}})(\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}})) (\boldsymbol{k} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}})) - k^{2} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}}) \boldsymbol{\cdot} (\bar{\boldsymbol{\omega}} - (\bar{\boldsymbol{\omega}} \boldsymbol{\cdot} \bar{\boldsymbol{o}}) \bar{\boldsymbol{o}})]}{2 (2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2} - |\bar{\boldsymbol{H}}|^{2})} \nonumber\\ &\quad - \frac{k^{2} \bar{n} d^{2} \eta^{\prime} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}}) \hat{\phi}_{\boldsymbol{k}} \bar{\boldsymbol{\omega}} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times \bar{\boldsymbol{H}})}{6 \hat{n}_{\boldsymbol{k}} (2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2} - |\bar{\boldsymbol{H}}|^{2})} + \frac{4 \mu_0 \chi^{2} \bar{\phi} (\boldsymbol{k} \boldsymbol{\cdot} \bar{\boldsymbol{H}})(\boldsymbol{k} \boldsymbol{\cdot} \bar{\boldsymbol{o}}) (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{3}}{{\rm \pi}^{2} d^{4} \eta_0 [2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2} - |\bar{\boldsymbol{H}}|^{2}]} \nonumber\\ &\quad - \frac{4 k^{2} \bar{\phi} \mu_0 \chi^{2} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{4}}{{\rm \pi}^{2} \eta_0 d^{4} [2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2} - |\bar{\boldsymbol{H}}|^{2}]}. \end{align}

Here, the number density is expressed in terms of the volume fraction using the expression $\bar {n} = \bar {\phi } / ({\rm \pi} d^{3}/6)$. Equation (3.3) is used to express $\bar {\boldsymbol{o}} \times \bar {\boldsymbol{H}}$ in the first term on the right in (3.16):

(3.17)\begin{align} \boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{D} \boldsymbol{\cdot} \boldsymbol{k} & = \frac{{\rm \pi} \bar{\phi} \eta_0 d^{5} [(\boldsymbol{k} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}} \!-\! (\boldsymbol{k} \boldsymbol{\cdot} \bar{\boldsymbol{o}})(\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}})) (\boldsymbol{k} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}} \!-\! (\boldsymbol{k} \boldsymbol{\cdot} \bar{\boldsymbol{o}}) (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{\omega}})) - k^{2} (|\bar{\boldsymbol{\omega}}|^{2} - (\bar{\boldsymbol{\omega}} \boldsymbol{\cdot} \bar{\boldsymbol{o}}))^{2}]}{4 \mu_0 \chi (2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2} - |\bar{\boldsymbol{H}}|^{2})} \nonumber\\ &\quad + \frac{{\rm \pi} k^{2} \bar{\phi} \eta_0 d^{5} \eta^{\prime} (|\bar{\boldsymbol{\omega}}|^{2}-(\bar{\boldsymbol{\omega}} \boldsymbol{\cdot} \bar{\boldsymbol{o}})^{2})}{6 \mu_0 \chi (2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2} - |\bar{\boldsymbol{H}}|^{2})} + \frac{4 \mu_0 \chi^{2} \phi (\boldsymbol{k} \boldsymbol{\cdot} \bar{\boldsymbol{H}})(\boldsymbol{k} \boldsymbol{\cdot} \bar{\boldsymbol{o}}) (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{3}}{{\rm \pi}^{2} d^{4} \eta_0 (2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2} - |\bar{\boldsymbol{H}}|^{2})} \nonumber\\ &\quad - \frac{4 k^{2} \bar{\phi} \mu_0 \chi^{2} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{4}}{{\rm \pi}^{2} \eta_0 d^{4} [2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} \bar{\boldsymbol{H}})^{2} - |\bar{\boldsymbol{H}}|^{2}]}. \end{align}

The diffusion coefficient extracted from this equation is of the form (2.39), where $\boldsymbol{D}^{h}$ and $\boldsymbol{D}^{m}$ are

(3.18)\begin{equation} \left.\begin{gathered} \boldsymbol{D}^{h} = \frac{ ({{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} - \bar{\boldsymbol{o}}(\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}})) ({{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} - \bar{\boldsymbol{o}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}})) - \boldsymbol{I} (1 - ({{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \boldsymbol{\cdot} \bar{\boldsymbol{o}}))^{2}}{4 \varSigma_i [2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}})^{2} - 1]},\\ \boldsymbol{D}^{m} = \frac{2 \varSigma_i \chi^{{\ast}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}})^{3} ({{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}} \bar{\boldsymbol{o}} + \bar{\boldsymbol{o}} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}})}{{\rm \pi}[2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}})^{2} - 1]} - \frac{4 \varSigma_i \chi^{{\ast}} (\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}})^{4} \boldsymbol{I}}{{\rm \pi} [2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}})^{2}- 1]}, \\ {D_i^{\prime}} = \frac{\eta^{\prime} (1 - ({{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \boldsymbol{\cdot} \bar{\boldsymbol{o}})^{2})}{6 \varSigma_i [2 (\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}})^{2}-1]}. \end{gathered}\right\}\end{equation}

Here, the dimensionless ratio of the magnetic and hydrodynamic torques is

(3.19)\begin{equation} \varSigma_i = \frac{\mu_0 \chi |\bar{\boldsymbol{H}}|^{2}}{{\rm \pi} \eta_0 d^{3} |\bar{\boldsymbol{\omega}}|}, \end{equation}

and the scaled susceptibility per unit volume is

(3.20)\begin{equation} \chi^{{\ast}} = \frac{\chi}{d^{3}}. \end{equation}

In going from (3.5) to (3.6), we had made the approximation $\bar {n} \chi \ll 1$. If this approximation is not made, then there is only one change in the diffusion coefficients: $\chi ^{\ast } (1 + \bar {n} \chi )$ should be substituted for $\chi ^{\ast }$ in the first term on the right in the expression for $\boldsymbol{D}^{m}$.

3.2. Steady state

The orthogonal basis vectors $({{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}, {{\boldsymbol {e}}_\parallel }, {{\boldsymbol {e}}_\perp })$ in figure 3(b) are used for an oblique magnetic field, where ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}$ is along the vorticity, ${{\boldsymbol {e}}_\parallel }$ is perpendicular to the vorticity in the $\bar {\boldsymbol{\omega }}- \bar {\boldsymbol{H}}$ plane, and ${{\boldsymbol {e}}_\perp }$ is perpendicular to the $\bar {\boldsymbol{\omega }}- \bar {\boldsymbol{H}}$ plane. The unit vectors ${{\boldsymbol {e}}_\parallel }$ and ${{\boldsymbol {e}}_\perp }$ are defined in (2.54) and (2.55). The solution $\hat {o}_{\boldsymbol{H}}$ (see (2.53a,b)) has to be determined numerically in this case; this is in contrast to the permanent dipole in an oblique magnetic field where it is possible to obtain an analytical solution, (2.57). The solutions have been derived in Kumaran (Reference Kumaran2021a,Reference Kumaranb) for a spheroid. The solution $\hat {o}_{\boldsymbol{H}}^{2}$ for a spherical particle satisfies the cubic equation

(3.21)\begin{equation} 4 \varSigma_i^{2} \hat{o}_{\boldsymbol{H}}^{4} [1 - \hat{o}_{\boldsymbol{H}}^{2}] + [\hat{\omega}_{\boldsymbol{H}}^{2} - \hat{o}_{\boldsymbol{H}}^{2}] = 0, \end{equation}

where $\hat {\omega }_{\boldsymbol{H}}$ and $\hat {o}_{\boldsymbol{H}}$ are defined in (2.53a,b).

In the limit $\varSigma _i \gg 1$, the particle aligns along the magnetic field and $\hat {o}_{\boldsymbol{H}} \rightarrow 1$. As $\varSigma _i$ is decreased, there are steady stable solutions for the orientation vector for the parameter regimes

(3.22)\begin{align} \varSigma_i^{2} & > \frac{1 + 18 \hat{\omega}_{\boldsymbol{H}}^{2} - 27 \hat{\omega}_{\boldsymbol{H}}^{4} - \sqrt{(1 - \hat{\omega}_{\boldsymbol{H}}^{2})(1 - 9 \hat{\omega}_{\boldsymbol{H}}^{2})^{3}}}{32 \hat{\omega}_{\boldsymbol{H}}^{2}} \quad \mbox{for} \ 0 \leq \hat{\omega}_{\boldsymbol{H}}^{2} \leq \frac{1}{9}, \end{align}

(3.23)\begin{align} & > \frac{9 (1 - 3 \hat{\omega}_{\boldsymbol{H}}^{2})}{8} \quad \mbox{for} \ \frac{1}{9} \leq \hat{\omega}_{\boldsymbol{H}}^{2} \leq \frac{1}{3}, \end{align}

(3.24)\begin{align} & > 0 \quad \mbox{for} \ \hat{\omega}_{\boldsymbol{H}}^{2} \geq \frac{1}{3}. \end{align}

When the conditions (3.22)–(3.24) are not satisfied, there are stable limit cycles and possibly an unstable steady solution. The boundary between the steady and rotating solutions in the $\varSigma _i- \hat {\omega }_{\boldsymbol{H}}$ parameter space is shown by the blue line in figure 6.

Figure 6.

The boundary between the stable stationary solutions (above) and rotating states (below) for the single-particle dynamics, (3.24), is shown by the blue line, and the boundary for the dynamical transition at $\varSigma _i = \sqrt {1 - 2 \hat {\omega }_{\boldsymbol{H}}^{2}}$ is shown by the red line, in the $\hat {\omega }_{\boldsymbol{H}}- \varSigma _i$ plane.

The solutions of (3.21) for $\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$ and the corresponding solutions of $\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}$ are shown as functions of $\varSigma$ in figure 7. These are qualitatively different from those for particles with a permanent dipole shown in figure 2 because $\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$ is necessarily positive for an induced dipole; the parameter space $\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}} < 0$ does not exist in this case. The solutions for the orientation of a particle acted upon by a shear flow and a magnetic field are derived in orientation space consisting of the azimuthal and meridional angles of the particle orientation (Kumaran Reference Kumaran2021a,Reference Kumaranb). In this orientation space, the red lines in figure 7 are the unstable steady solutions, the blue lines are the stable steady solutions, and the brown lines are saddle points. The evolution of the fixed points for $0 < \hat {\omega }_{\boldsymbol{H}}^{2} < \tfrac {1}{9}$ is illustrated by the curve for $\hat {\omega }_{\boldsymbol{H}} = 0.1$ in figure 7. There is one stable fixed point for $\varSigma _i \gg 1$. As $\varSigma _i$ is decreased, one unstable fixed point and one saddle point appear in the phase diagram. When there is a further decrease in $\varSigma _i$, the saddle and stable fixed point merge, and there remain one unstable fixed point and one stable limit cycle. For $\frac {1}{9} < \hat {\omega }_{\boldsymbol{H}}^{2} < \frac {1}{3}$, the phase plot contains a stable fixed point and an unstable limit cycle for $\varSigma _i \gg 1$, and there is an exchange of stability to an unstable fixed point and a stable limit cycle for $\varSigma _i \ll 1$, as shown by the curve for $\hat {\omega }_{\boldsymbol{H}} = 0.5$ in figure 7. For $\frac {1}{3} < \hat {\omega }_{\boldsymbol{H}}^{2} < 1$, there is a stable fixed point that is aligned with the magnetic field for $\varSigma _i \gg 1$, and with the vorticity for $\varSigma _i \ll 1$.

Figure 7.

The variation of (a) $\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$ and (b) $\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}$ with $\varSigma _i$, for a particle with an induced dipole. The solid lines are the results for a parallel magnetic field ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}} = 0$, and the dashed lines are the results for an oblique magnetic field with ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}} = 0.1$ ($\circ$), $\frac {1}{2}$ ($\triangle$), $\frac {1}{\sqrt {2}}$ ($\nabla$) and $\frac {\sqrt {3}}{2}$ ($\diamond$). The blue lines are the stable stationary nodes, the red lines are the unstable stationary nodes, and the brown lines are the saddle nodes. The value of ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$ for a parallel magnetic field is not shown in (b) because it is zero.

3.3. Parallel magnetic field

Equation (3.21) has analytical solutions when the magnetic field is parallel to the flow plane, $\hat {\omega }_{\boldsymbol{H}} = 0$. The stable solution for the orientation vector exists only for $\varSigma _i > 1$, and the orientation vector is given by

(3.25a,b)\begin{gather} \hat{o}_{\boldsymbol{H}} = \sqrt{\frac{1}{2} + \frac{\sqrt{\varSigma_i^{2} - 1}}{2 \varSigma_i}}, \quad \bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_\perp} = \sqrt{\frac{1}{2} - \frac{\sqrt{\varSigma_i^{2} - 1}}{2 \varSigma_i}},\end{gather}

(3.26a,b)\begin{gather} 2 \hat{o}_{\boldsymbol{H}}^{2} - 1 = \frac{\sqrt{\varSigma_i^{2}-1}}{\varSigma_i}, \quad \hat{o}_{\boldsymbol{H}}(\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_\perp}) = \frac{1}{2 \varSigma_i}. \end{gather}

The orthogonal basis unit vectors $({{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}, {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}, {{\boldsymbol {e}}_\perp })$ shown in figure 3(a) are used, where ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}$ is along the vorticity direction, ${{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$ is the along the direction of the magnetic field, and ${{\boldsymbol {e}}_\perp } = {{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}} \times {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$. The diffusion coefficient due to hydrodynamic and magnetic interactions, (3.18), is reduced to the form in (2.47), where

(3.27a,b)\begin{gather} {D_{\bar{\boldsymbol{\omega}} \bar{\boldsymbol{\omega}}}^{h}} = 0, \quad {D_{\bar{\boldsymbol{\omega}}\bar{\boldsymbol{\omega}}}^{m}} ={-} \frac{\chi^{{\ast}} \left(\varSigma_i + \sqrt{\varSigma_i^{2}-1}\right)^{2}}{{\rm \pi} \sqrt{\varSigma_i^{2}-1}}, \end{gather}

(3.28a,b)\begin{gather}{D_{\bar{\boldsymbol{H}} \bar{\boldsymbol{H}}}^{h}} ={-} \frac{1}{4 \sqrt{\varSigma_i^{2}-1}}, \quad {D_{\bar{\boldsymbol{H}} \bar{\boldsymbol{H}}}^{m}} = 0, \end{gather}

(3.29a,b)\begin{gather}{D_{\bar{\boldsymbol{H}} \perp}^{h}} = 0, \quad {D_{\bar{\boldsymbol{H}} \perp}^{m}} = \frac{\chi^{{\ast}} \left(\varSigma_i + \sqrt{\varSigma_i^{2}-1}\right)}{2 {\rm \pi}\sqrt{\varSigma_i^{2}-1}}, \end{gather}

(3.30a,b)\begin{gather}{D_{{\perp}{\perp}}^{h}} ={-} \frac{1}{4 \sqrt{\varSigma_i^{2}-1}}, \quad {D_{{\perp}{\perp}}^{m}} ={-} \frac{\chi^{{\ast}} \left(\varSigma_i + \sqrt{\varSigma_i^{2}-1}\right)^{2}}{{\rm \pi} \sqrt{\varSigma_i^{2}-1}}. \end{gather}

The isotropic part of the diffusion tensor due to the concentration dependence of the viscosity and magnetic permeability is

(3.31)\begin{equation} {D_i^{\prime}} = \frac{\eta^{\prime}}{6 \sqrt{\varSigma_i^{2}-1}}. \end{equation}

The asymptotic values of the diffusion matrix for $\varSigma _i \gg 1$ and $\varSigma _i - 1 \ll 1$ are reported in table 1. The qualitative characteristics of the diffusion matrix are very similar to those for permanent dipoles in a parallel magnetic field discussed at the end of § 2.3. An important difference is that ${D_{\bar {\boldsymbol{\omega }}\bar {\boldsymbol{\omega }}}^{m}}$ and ${D_{\perp \perp }^{m}}$ diverge for $\varSigma _i \rightarrow 1$ where a transition occurs between static and rotating states.

The eigenvalues and eigenvectors of the diffusion matrix are provided in table 1. For the symmetric diffusion tensor, the eigenvalues are real, and the orthogonal eigenvectors are the principal axes of extension/compression. There is extension along directions with positive eigenvalues, indicating that concentration fluctuations are damped, and compression along directions with negative eigenvalues, resulting in amplification of concentration fluctuations. Planar aggregates perpendicular to the unstable direction are expected if two eigenvalues are positive and one is negative. If one eigenvalue is positive and two are negative, then cylindrical aggregates are expected to form aligned along the direction with positive eigenvalue. Spheroidal clusters are expected if all three directions are unstable, that is, all eigenvalues are negative. The eigenvalues in table 1 are calculated without including the isotropic component ${D_i} \boldsymbol{I}$ in the diffusion tensor; that is, these do not include the effect of variations of viscosity with concentration. The eigenvectors are unchanged when the isotropic diffusion tensor with the viscosity correction is included, and the eigenvalues are transformed as $\lambda \rightarrow \lambda + {D_i}$.

The component ${D_{\bar {\boldsymbol{\omega }}\bar {\boldsymbol{\omega }}}^{m}}$ of the diffusion tensor is always negative, and it diverges as $- 4 \chi ^{\ast } \varSigma _i/{\rm \pi}$ for $\varSigma _i \gg 1$. Therefore, perturbations are always unstable in the vorticity direction perpendicular to the flow, and there is clustering in this direction for a parallel magnetic field. For $\varSigma _i \gg 1$, the principal eigenvalues are aligned along the ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}, {{\boldsymbol {e}}_\parallel }, {{\boldsymbol {e}}_\perp }$ directions. The principal eigenvalue along the ${{\boldsymbol {e}}_\perp }$ direction diverges as $- 4 \chi ^{\ast } \varSigma _i / {\rm \pi}$. The principal eigenvalue along the ${{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$ direction is positive/negative for $\chi ^{\ast } \gtrless {\rm \pi}$, and its magnitude decreases proportional $\varSigma _i^{-1}$ in this limit. Thus, depending on the value of $\chi ^{\ast }$, there is weak amplification/damping of fluctuations in the direction of the magnetic field, and strong amplification of fluctuations in the other two directions. When the magnetic field is perpendicular to the fluid velocity, this would result in the formation of long and narrow clusters aligned along the magnetic field.

For $\varSigma _i - 1 \ll 1$ near the transition between rotating and steady states, all three eigenvalues diverge proportional to $(\varSigma _i - 1)^{-1/2}$. Two of the eigenvalues – $\lambda _3$ in the ${{\boldsymbol {e}}_\perp }$ direction, and $\lambda _2$ in the ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}- {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$ plane – are negative, therefore strong amplification of perturbations is predicted in these two directions. The other eigenvalue – $\lambda _1$ in the $\boldsymbol{e}_1$ direction – is positive/negative for $2 (\sqrt {2} - 1) \chi ^{\ast } \gtrless {\rm \pi}$, indicating damping of fluctuations if $\chi ^{\ast }$ exceeds a threshold. It should be noted that the divergence exponent $-\frac {1}{2}$ is a mean-field exponent, which could be altered if fluctuations are included.

3.4. Oblique magnetic field

The orthogonal coordinate system shown in figure 3(b) is used for an oblique magnetic field, where ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}$ is along the direction of the vorticity perpendicular to the flow plane, the unit vector ${{\boldsymbol {e}}_\parallel }$ is perpendicular to ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}$ in the $\bar {\boldsymbol{\omega }}- \bar {\boldsymbol{H}}$ plane, and ${{\boldsymbol {e}}_\perp }$ is perpendicular to the $\bar {\boldsymbol{\omega }}- \bar {\boldsymbol{H}}$ plane. After the solution of (3.21) for $\hat {o}_{\boldsymbol{H}}$ is determined, $\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}$ is calculated using (2.9). This specifies completely the orientation vector $\bar {\boldsymbol{o}}$. It is easily verified that for a parallel magnetic field with ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}} = 0$, the non-trivial solution reduces to (3.25a,b). The product $\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_\parallel }$ is given by (2.58), and the product $\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_\perp }$ is

(3.32)\begin{align} \bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_\perp} & = \frac{\bar{\boldsymbol{o}} \boldsymbol{\cdot} ({{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \times {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}})}{\sqrt{1 - ({{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}})^{2}}} ={-} \frac{{{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \boldsymbol{\cdot} (\bar{\boldsymbol{o}} \times {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}})}{\sqrt{1 - ({{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}})^{2}}} \nonumber\\ & = \frac{{{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}} \boldsymbol{\cdot} (\boldsymbol{I} - \bar{\boldsymbol{o}} \bar{\boldsymbol{o}}) \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{\omega}}}}}{2 \varSigma_i (\bar{\boldsymbol{o}} \boldsymbol{\cdot} {{\boldsymbol{e}}_{\bar{\boldsymbol{H}}}}) \sqrt{1 - \hat{\omega}_{\boldsymbol{H}}^{2}}} = \frac{\hat{o}_{\boldsymbol{H}}^{2} - \hat{\omega}_{\boldsymbol{H}}^{2}}{2 \varSigma_i \hat{o}_{\boldsymbol{H}}^{3} \sqrt{1 - \hat{\omega}_{\boldsymbol{H}}^{2}}}. \end{align}

Here, the torque balance equation (3.3) has been used to substitute for $\bar {\boldsymbol{o}} \times {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$, and (2.9) is used to substitute $\hat {o}_{\boldsymbol{H}} = \hat {\omega }_{\boldsymbol{H}}/(\bar {\boldsymbol{o}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}})$. Equations (2.58) and (3.32) are used to substitute for $\bar {\boldsymbol{o}}$ in (3.18) to obtain a diffusion matrix of the form (2.60). The elements of the matrix are listed in table 3, and the coefficients are plotted as a function of $\varSigma _i$ in figures 8 and 9.

Table 3.

The asymptotic behaviour of the diffusion coefficients for $\varSigma \gg 1$ and $\varSigma \ll 1$ for an oblique magnetic field for particles with induced dipoles.

Figure 8.

The scaled isotropic part of the diffusion tensor ${D_i^{\prime }}/\eta ^{\prime }$ as a function of the parameter $\varSigma _i$ for particles with a induced dipole. Here, ${D_i^{\prime }}$ is given in (3.18). The orientations of the vorticity and magnetic field are ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}} = 0.1$ ($\circ$), $\frac {1}{2}$ ($\triangle$), $\frac {1}{\sqrt {2}}$ ($\nabla$) and $\frac {\sqrt {3}}{2}$ ($\diamond$). The solid line is the result for a parallel magnetic field.

Figure 9.

The components of the diffusion tensor: (a) $- {D_{\bar {\boldsymbol{\omega }} \bar {\boldsymbol{\omega }}}^{h}}$ and $- {D_{\bar {\boldsymbol{\omega }}\bar {\boldsymbol{\omega }}}^{m}}/\chi ^{\ast }$, (b) $- {D_{\bar {\boldsymbol{\omega }} \parallel }^{h}}$ and ${D_{\bar {\boldsymbol{\omega }} \parallel }^{m}}/\chi ^{\ast }$, (c) $- {D_{\bar {\boldsymbol{\omega }} \perp }^{h}}$ and ${D_{\bar {\boldsymbol{\omega }} \perp }^{m}}/\chi ^{\ast }$, (d) $- {D_{\parallel \parallel }^{h}}$ and $- {D_{\parallel \parallel }^{m}}/\chi ^{\ast }$, (e) ${D_{\parallel \perp }^{h}}$ and ${D_{\parallel \perp }^{m}}/\chi ^{\ast }$, and (f ) $- {D_{\perp \perp }^{h}}$ and $- {D_{\perp \perp }^{m}}/\chi ^{\ast }$, due to hydrodynamic interactions (blue lines) and magnetic interactions (red lines) as functions of the parameter $\varSigma _i$ for particles with an induced dipole moment. The orientations of the vorticity and magnetic field are ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}} \boldsymbol{\cdot } {{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}} = 0.1$ ($\circ$), $\frac {1}{2}$ ($\triangle$), $\frac {1}{\sqrt {2}}$ ($\nabla$) and $\frac {\sqrt {3}}{2}$ ($\diamond$). The solid blue and red lines are the non-zero results for a parallel magnetic field.

Also presented in table 3 are the expansions for the components of $\boldsymbol{D}^{h}$ and $\boldsymbol{D}^{m}$ for the limits of small and large $\varSigma _i$, respectively. These are determined using the expansion for the solutions of (3.21):

(3.33)\begin{gather} \hat{o}_{\boldsymbol{H}} = \hat{\omega}_{\boldsymbol{H}} + 2 \varSigma_i^{2} \hat{\omega}_{\boldsymbol{H}}^{3} (1 - \hat{\omega}_{\boldsymbol{H}}^{2}) \quad \mbox{for} \ \varSigma_i \ll 1, \end{gather}

(3.34)\begin{gather}\hat{o}_{\boldsymbol{H}} = 1 - \frac{1 - \hat{\omega}_{\boldsymbol{H}}^{2}}{8 \varSigma_i^{2}} \quad \mbox{for} \ \varSigma_i \gg 1. \end{gather}

The denominators of the diffusion coefficients in the third column of table 3 decrease to zero for $\varSigma _i \ll 1$ and $\hat {\omega }_{\boldsymbol{H}} = \frac {1}{\sqrt {2}}$. In this case, it is necessary to first substitute (3.33) into the expressions in the second column of table 3, and then substitute $\hat {\omega }_{\boldsymbol{H}} = \frac {1}{\sqrt {2}}$ and take the limit $\varSigma _i \ll 1$. The results obtained in this manner are presented in the fourth column of table 3, and these are shown for different values of $\hat {\omega }_{\boldsymbol{H}}$ in figure 9.

The striking feature of the diffusion tensor elements in table 3 is the singularity at $\hat {o}_{\boldsymbol{H}}^{2} = \frac {1}{2}$. From (3.21), this corresponds to $\varSigma _i = \sqrt {1 - 2 \hat {\omega }_{\boldsymbol{H}}^{2}}$, shown by the red line in figure 6, and this singularity exists only for $\hat {\omega }_{\boldsymbol{H}} < \frac {1}{\sqrt {2}}$. This boundary is different from the blue boundary between stationary and rotating states for the single-particle dynamics, and this represents a dynamical transition due to inter-particle interactions. At this boundary, the ‘susceptibility’ multiplying $\hat {o}^{{{\dagger}} }_{\boldsymbol{k}}$ in (3.11) decreases to zero, therefore the perturbation to the orientation vector $\hat {o}^{{{\dagger}} }_{\boldsymbol{k}}$ in (3.13) diverges. The divergence at $\hat {o}_{\boldsymbol{H}}^{2} = \frac {1}{2}$ is observed in all components of the diffusion tensor in figure 9, with the exception of ${D_{\perp \perp }^{m}}$. Of course, the analysis is not accurate at this point because it is carried out assuming $\hat {o}^{{{\dagger}} }_{\boldsymbol{k}} \ll 1$, and it is necessary to include nonlinear fluctuation effects in the vicinity of this point to extract the nature of the divergence. This is a subject for future study.

For $\varSigma _i \gg 1$, the characteristics of the diffusion tensor are similar to those for dipolar particles in figure 5. The largest contributions to the diffusion tensor are ${D_{\bar {\boldsymbol{\omega }}\bar {\boldsymbol{\omega }}}^{m}}$, ${D_{\bar {\boldsymbol{\omega }} \parallel }^{m}}$, ${D_{\parallel \parallel }^{m}}$ and ${D_{\parallel \perp }^{m}}$; all of these diverge proportional to $\varSigma _i$. The eigenvalue $\lambda _3$ corresponding to the ${{\boldsymbol {e}}_\perp }$ direction is negative, and diverges proportional to $\varSigma _i$, therefore concentration fluctuations are amplified in the direction perpendicular to the plane containing the vorticity and magnetic field. In the ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}- {{\boldsymbol {e}}_\parallel }$ plane, one of the eigenvalues ($\lambda _2$) diverges proportional to $- (4 \chi \varSigma _i/{\rm \pi} )$, indicating strong amplification of concentration fluctuations in the $\boldsymbol{e}_2$ direction. The third eigenvalue ($\lambda _1$) decreases proportional to $\varSigma _i^{-1}$, and this is positive/negative for $\chi ^{\ast } \gtrless {\rm \pi}$, indicating weak amplification/damping of fluctuations. The two directions $\boldsymbol{e}_1$ and $\boldsymbol{e}_2$ align with ${{\boldsymbol {e}}_{\bar {\boldsymbol{H}}}}$ and ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}}$ for a parallel magnetic field $\hat {\omega }_{\boldsymbol{H}} = 0$.

The characteristics of the diffusion tensor are similar to those for dipolar particles for $\varSigma _i \ll 1$ and $\hat {\omega }_{\boldsymbol{H}} > \frac {1}{\sqrt {2}}$. The diffusion coefficients ${D_{\bar {\boldsymbol{\omega }} \bar {\boldsymbol{\omega }}}^{h}}$, ${D_{\bar {\boldsymbol{\omega }} \parallel }^{m}}$, ${D_{\parallel \parallel }^{h}}$, ${D_{\parallel \parallel }^{m}}$ and ${D_{\perp \perp }^{m}}$, and all three eigenvalues, decrease proportional to $\varSigma _i$. The eigenvalue of the diffusion tensor in the ${{\boldsymbol {e}}_\perp }$ direction is negative for $\hat {\omega }_{\boldsymbol{H}} > \frac {1}{\sqrt {2}}$, $\lambda _3 = - 4 \chi ^{\ast } \varSigma _i \hat {\omega }_{\boldsymbol{H}}^{2}/{\rm \pi} (2 \hat {\omega }_{\boldsymbol{H}}^{2}-1)$, and this is an unstable direction. One of the eigenvalues in the $\boldsymbol{e}_2$ direction is also always negative, while the third eigenvalue is positive/negative for $2 \chi ^{\ast } \hat {\omega }_{\boldsymbol{H}} \gtrless {\rm \pi}(1 + \hat {\omega }_{\boldsymbol{H}})$. Thus concentration fluctuations are unstable in the ${{\boldsymbol {e}}_\perp }$ and $\boldsymbol{e}_2$ directions, and they could be stable/unstable in the $\boldsymbol{e}_1$ direction depending on the value of $\chi ^{\ast }$.

For $\hat {\omega }_{\boldsymbol{H}} > \frac {1}{\sqrt {2}}$, there is a dynamical transition at $\hat {o}_{\boldsymbol{H}} = \frac {1}{2}$ along the red line in figure 6. This is a transition in the collective dynamics of the particles, and is distinct from the single-particle transition between stationary and rotating states along the blue line in figure 6. All the coefficients of the diffusion matrix diverge proportional to $(\hat {o}_{\boldsymbol{H}} - \frac {1}{2})^{-1/2}$ at this transition. For $\hat {\omega }_{\boldsymbol{H}} = \frac {1}{\sqrt {2}}$, the elements of the diffusion tensor have a scaling different to that for $\hat {\omega }_{\boldsymbol{H}} > \frac {1}{\sqrt {2}}$. The elements ${D_{\bar {\boldsymbol{\omega }} \bar {\boldsymbol{\omega }}}^{h}}$, ${D_{\bar {\boldsymbol{\omega }} \parallel }^{m}}$, ${D_{\parallel \parallel }^{h}}$, ${D_{\parallel \parallel }^{m}}$ and ${D_{\perp \perp }^{m}}$ actually increase proportional to $\varSigma _i^{-1}$ in this limit. The eigenvalues of the diffusion matrix increase proportional to $\varSigma _i^{-1}$. Two of these are negative, therefore concentration fluctuations are amplified in one direction in the ${{\boldsymbol {e}}_{\bar {\boldsymbol{\omega }}}} - {{\boldsymbol {e}}_\parallel }$ plane and in the ${{\boldsymbol {e}}_\perp }$ direction. The third eigenvalue is positive for $2 \chi ^{\ast } (\sqrt {2} - 1) > {\rm \pi}$, and negative otherwise.