2. Problem formulation and governing equations

Figure 2.

A neutrally buoyant spheroid with symmetry axis $\boldsymbol{p}$ in plane Poiseuille flow. The position of the spheroid relative to the laboratory frame ( $x_1,x_2,x_3$ ) is denoted by $\boldsymbol{y}$ ; $(r_1,r_2,r_3)$ represents the Cartesian frame with origin at the spheroid centre, and translating with it. Here $s= d/H$ denotes the dimensionless position of the spheroid with respect to the lower wall ( $s=0$ ). The zoomed-in image of the spheroid, on the right-hand side, shows the body-fixed coordinate system aligned with $\boldsymbol{p}$ , along with the polar ( $\theta _j$ ) and azimuthal ( ${\phi }_{\!j}$ ) angles that define the spheroid orientation. The dot–dashed line is the projection of the $\boldsymbol{r}_{b_1}$ -axis on the flow-gradient plane.

Figure 2. Long description

A neutrally buoyant spheroid with its symmetry axis in plane Poiseuille flow. The position of the spheroid relative to the laboratory frame is denoted by a vector. The Cartesian frame with origin at the spheroid center translates with it. The dimensionless position of the spheroid with respect to the lower wall is represented. The zoomed-in image of the spheroid shows the body-fixed coordinate system aligned with the axes, along with the polar and azimuthal angles that define the spheroid orientation. The dot-dashed line is the projection of the axis on the flow-gradient plane.

Figure 2 shows a neutrally buoyant spheroid of aspect ratio $\kappa =L/b$ ( $L$ and $b$ are the semimajor and minor axes) freely suspended in a wall-bounded plane Poiseuille flow at a distance $d$ from the lower wall; $\kappa \lt 0.14$ and ${\gt } 1$ for oblate and prolate spheroids, respectively. The spheroid motion is governed by the following equations:

(2.1a) \begin{align} {\nabla} ^2 \boldsymbol{u}-\boldsymbol{\nabla } p &={\textit{Re}}_{\!p}\Bigg (\frac {\partial \boldsymbol{u}}{\partial t}+\frac {{\rm d}\boldsymbol{U}_{\!p}}{{\rm d}t}+\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla u}+\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla u}^\infty +\boldsymbol{u}^\infty \boldsymbol{\cdot }\boldsymbol{\nabla u}\Bigg ), \end{align}

(2.1b) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u}&=0, \end{align}

for the disturbance velocity field, in a reference frame with its origin at the spheroid centre, and translating with the spheroid velocity $\boldsymbol{U}_{\!p}$ . Here, $\boldsymbol{u}$ satisfies

(2.2a) \begin{align} \boldsymbol{u}&=\boldsymbol{\varOmega }_{\!p}\wedge \boldsymbol{r}-\boldsymbol{u}^\infty \quad \text{ for } \quad \boldsymbol{r}\in S_{\!p}, \end{align}

(2.2b) \begin{align} \boldsymbol{u}&\rightarrow 0 \quad \text{ for } \quad r_1,r_3\rightarrow \infty \,(r_2\ \text{fixed}), \end{align}

(2.2c) \begin{align} \boldsymbol{u}&=0 \quad \text{ at } \quad r_2=-s\lambda ^{-1}, (1-s)\lambda ^{-1}, \end{align}

where $S_{\!p}$ denotes the spheroid surface. Here $\boldsymbol{U}_{\!p}$ and $\boldsymbol{\varOmega }_{\!p}$ in (2.1a ) and (2.2a ) are the spheroid translational and angular velocities. In (2.1a )–(2.2c ), all variables are non-dimensionalized using $L$ and the velocity scale $V_c=V_{\textit{max}}L/H$ , this being the order of the velocity change across the ends of the spheroid. The particle Reynolds number is therefore given by ${\textit{Re}}_{\!p}=V_c L/\nu = V_{\textit{max}}L^2/H\nu$ , $\nu$ being the kinematic viscosity of the fluid. The ambient plane Poiseuille flow, $\boldsymbol{u}^\infty$ in (2.1a ) and (2.2a ), is given by

(2.3) \begin{align} \boldsymbol{u}^\infty =\big(\alpha +\beta r_2 +\gamma \lambda {r_2}^2\big)\boldsymbol{1}_1-\boldsymbol{U}_{\!p}, \end{align}

where $\alpha \boldsymbol{1}_1 - \boldsymbol{U}_{\!p}$ , with $\alpha =4 \lambda ^{-1} s(1-s)$ , is the slip velocity at the spheroid centre, $\beta =4(1-2s)$ is the local shear rate that varies linearly across the channel and $\gamma =-4$ is the constant curvature of the plane Poiseuille profile. Here $s = d/H$ denotes the non-dimensional spheroid location, with $s=1/2$ corresponding to the channel centreline. Here $\lambda =L/H$ is the confinement ratio, with ${\textit{Re}}_{\!p}$ and $\lambda$ being assumed small, while allowing for the channel Reynolds number, ${\textit{Re}}_c={\textit{Re}}_{\!p}/\lambda ^2$ , to be arbitrary. The analysis below also assumes $s, 1-s \gg \lambda$ , and is therefore restricted to spheroid motion at distances greater than $O(L)$ from either wall.

For small ${\textit{Re}}_{\!p}$ , $\boldsymbol{U}_{\!p}$ and $\boldsymbol{\varOmega }_{\!p}$ may be determined using the Faxen laws for translation and rotation of a force-free and torque-free spheroid (Happel & Brenner Reference Happel and Brenner2012; Anand & Subramanian Reference Anand and Subramanian2023). This, at leading order in $\lambda$ , yields

(2.4) \begin{align} \boldsymbol{U}_{p}&= \left [\alpha +\frac {\gamma \lambda }{3\kappa ^2}\big[\cos ^2{\phi }_{\!j}+\big(\cos ^2\theta _j+\kappa ^2\sin ^2\theta _j\big)\sin ^2{\phi }_{\!j}\big] \right ]\boldsymbol{1}_1, \end{align}

(2.5) \begin{align} \boldsymbol{\varOmega }_{p}&=-\frac {\beta (\kappa ^2-1)}{4(\kappa ^2+1)}\cos {\phi }_{\!j}\sin 2\theta _j\boldsymbol{1}_1+\frac {\beta (\kappa ^2-1)}{4(\kappa ^2+1)}\sin {\phi }_{\!j}\sin 2\theta _j\boldsymbol{1}_2\nonumber \\ &\quad +\frac {\beta }{2}\left [-1+\frac {(\kappa ^2-1)}{(\kappa ^2+1)}\cos 2{\phi }_{\!j}\sin ^2 \theta _j\right ]\boldsymbol{1}_3, \end{align}

where the Cartesian unit vectors correspond to the space-fixed directions shown in figure 2.

The local linear flow approximation of the plane Poiseuille profile is a simple shear flow, and the inertial screening length for a particle in simple shear is $L {\textit{Re}}_{\!p}^{-1/2}$ or $H {\textit{Re}}_c^{-1/2}$ . The analysis for shear-induced migration depends on ${\textit{Re}}_c$ . When ${\textit{Re}}_c\ll 1$ , the inertial screening length is much larger than the channel width, implying that the inner region, characterized by scales of $O(H)$ , gives the dominant contribution to the lift velocity; inertia in this case is a regular perturbation. When ${\textit{Re}}_c\gtrsim O(1)$ , the inertial screening length is of order the channel width or smaller, and contributions from scales of $O(H)$ and $O(H {\textit{Re}}_c^{-1/2})$ must be accounted for via a matched asymptotic expansions approach (Schonberg & Hinch Reference Schonberg and Hinch1989; Anand & Subramanian Reference Anand and Subramanian2023). Scaling arguments towards the end of this section show that, for the rotation-arrested scenario which turns out to be most relevant to inertial migration, the assumption of a small ${\textit{Re}}_c$ imposes a severe restriction on spheroid aspect ratios, and we therefore focus on the case ${\textit{Re}} \gtrsim O(1)$ below.

The Stokes velocity disturbance due to a freely rotating spheroid only decays as $1/r^2$ at distances large compared with $L$ , with the algebraic decay implying that distinct inner ( $r \sim O(L)$ ) and outer-region expansions ( $r \sim O(H{\textit{Re}}_c^{-1/2})$ ) for the velocity and pressure fields are necessary to account for inertial effects for ${\textit{Re}}_c \gtrsim O(1)$ ; the leading terms in the latter expansion satisfy the linearized Navier–Stokes equations. To capture the dominant outer-region contribution, one transforms to outer coordinates using $\boldsymbol{r}={\textit{Re}}_{\!p}^{-1/2}\boldsymbol{R}$ , with the Stokesian rates of decay in the inner region implying the scalings $\boldsymbol{u}={\textit{Re}}_{\!p}\boldsymbol{U}$ and $p={\textit{Re}}_{\!p}^{3/2} P$ for the leading-order terms. When used in (2.1a )–(2.2c ), $\boldsymbol{U}$ and $P$ satisfy

(2.6a) \begin{align} \frac {\partial ^2 U_i}{\partial R_j\partial R_j}&-\frac {\partial P}{\partial R_i}-\frac {\partial U_i}{\partial t}-\frac {1}{{\textit{Re}}_{\!p}}\frac {dU_{p,i}}{dt}-U_2\big(\beta +2\gamma R_2 {\textit{Re}}_c^{-1/2}\big)\delta _{i1}\nonumber \\ &\quad -\big(\beta R_2+\gamma R_2^2 {\textit{Re}}_c^{-1/2}\big)\frac {\partial U_i}{\partial R_1}=\beta S_{\textit{ij}}\frac {\partial \delta (\boldsymbol{R})}{\partial R_j}, \end{align}

(2.6b) \begin{align} &\qquad\qquad\qquad\qquad\frac {\partial {U}_i}{\partial x_i}=0, \end{align}

where $U_i$ satisfies

(2.7a) \begin{align} U_i &\sim \frac {3\beta R_i S_{jm}R_j R_m}{4\pi R^5} \quad \text{ for } \quad \boldsymbol{R}\rightarrow 0, \end{align}

(2.7b) \begin{align} U_i &\rightarrow 0 \quad \text{ for } \quad R_1,R_3\rightarrow \infty \,(R_2\ \text{fixed}), \end{align}

(2.7c) \begin{align} U_i&=0 \quad \text{ at } \quad R_2=-s {\textit{Re}}_c^{1/2}, (1-s){\textit{Re}}_c^{1/2}. \\[10pt] \nonumber \end{align}

The Faxen correction in (2.4) contributes at a higher order in $\lambda$ (Anand & Subramanian Reference Anand and Subramanian2023) and therefore, $u^\infty _i\approx {\textit{Re}}_{\!p}^{-1/2}(\beta R_2+\gamma \,{\textit{Re}}_c^{-1/2} R_2^2) \delta _{i1}$ in (2.6a ). Note also that the no slip condition (2.2a ) has been replaced by (2.7a ), the requirement of matching to the far-field limit of the inner stresslet field, since the spheroid on scales of the inertial screening length is equivalent to a point force-dipole. The tensorial stresslet amplitude is given by (Marath & Subramanian Reference Marath and Subramanian2017)

(2.8) \begin{align} \boldsymbol{S}(\boldsymbol{p})&= A_1 \frac {3}{2}(\boldsymbol{E}:\boldsymbol{pp})\left (\boldsymbol{pp}-\frac {\boldsymbol{I}}{3}\right ) + A_2 ((\boldsymbol{I}-\boldsymbol{pp})\boldsymbol{\cdot }\boldsymbol{E}\boldsymbol{\cdot }\boldsymbol{pp}+\boldsymbol{pp}\boldsymbol{\cdot }\boldsymbol{E}\boldsymbol{\cdot }(\boldsymbol{I}-\boldsymbol{pp}))\nonumber \\&\quad +A_3\left ((\boldsymbol{I}-\boldsymbol{pp})\boldsymbol{\cdot }\boldsymbol{E}\boldsymbol{\cdot }(\boldsymbol{I}-\boldsymbol{pp})+(\boldsymbol{I}-\boldsymbol{pp})\frac {\boldsymbol{E}:\boldsymbol{pp}}{2}\right ), \end{align}

in terms of the unit vector $\boldsymbol{p}$ denoting the spheroid orientation; here, $E_{\textit{ij}}=({\beta }/{2})(\delta _{i1}\delta _{j2}+\delta _{i2}\delta _{j1})$ is the rate of strain tensor associated with the local simple shear. The above expression arises from resolving simple shear flow into an axisymmetric extension aligned with $\boldsymbol{p}$ , longitudinal planar extensions in a pair of orthogonal planes containing $\boldsymbol{p}$ , and a pair of transverse planar extensions in the plane perpendicular to $\boldsymbol{p}$ (Subramanian & Koch Reference Subramanian and Koch2006; Dabade et al. Reference Dabade, Marath and Subramanian2016), with the $A_i$ in (2.8) being the $\kappa$ -dependent scalar amplitudes corresponding to these component flows (Kim & Karrila Reference Kim and Karrila2005; Marath & Subramanian Reference Marath and Subramanian2017). For $\kappa \gt 1$ , these amplitudes are given by

(2.9) \begin{align} A_1&=-\frac {16\pi (\kappa ^2-1)^{5/2}}{9\kappa ^3[-3(\kappa ^2-1)^{1/2}\kappa +2\kappa ^2 \cosh ^{-1}(\kappa )+\cosh ^{-1}(\kappa )]}, \end{align}

(2.10) \begin{align} A_2&=-\frac {16\pi (\kappa ^2-1)^3}{3\kappa ^2(\kappa ^2+1)(\kappa ^4+\kappa ^2-3(\kappa ^2-1)^{1/2}\kappa \cosh ^{-1}(\kappa )-2)}, \end{align}

(2.11) \begin{align} A_3&=-\frac {32\pi (\kappa ^2-1)^3}{3\kappa ^3(2\kappa ^5-7\kappa ^3+3(\kappa ^2-1)^{1/2}\cosh ^{-1}(\kappa )+5\kappa )}. \end{align}

The corresponding expressions for an oblate spheroid ( $\kappa \lt 1$ ) may be obtained by first substituting $\kappa =\xi _0/(\xi _0^2-1)^{1/2}$ in terms of the coordinate label ( $\xi _0\gt 1$ ) for the spheroid, and then using the transformation $d\to -\iota d, \xi _0\to \iota (\xi _0^2-1)^{1/2}$ in the dimensional stresslet, obtained from multiplying $\boldsymbol{S}$ above by $\mu L^3 V_{\textit{max}}/H$ with $L= d\xi _0$ (Marath & Subramanian Reference Marath and Subramanian2017).

The stresslet given by (2.8) is a function of time owing to $\boldsymbol{p}$ in general being time dependent. The nature of this time dependence for small ${\textit{Re}}_{\!p}$ depends on the Jeffery orbit in which the spheroid rotates. For ${\textit{Re}}_{\!p} = 0$ , a spheroid can rotate in any of an infinite number of these orbits parameterized by an orbit constant $C \in [0,\infty )$ , with the limiting values corresponding to spinning ( $C=0$ ) and tumbling ( $C= \infty$ ) modes. Rotation along a Jeffery orbit generates an unsteady disturbance field for any $C \neq 0$ , with the associated fluid inertial forces, in a region of $O(L^3)$ around the rotating spheroid, leading to an $O({\textit{Re}}_{\!p})$ drift across orbits. This orbital drift stabilizes the tumbling mode for a prolate spheroid, and either the spinning or tumbling mode for oblate spheroids with $\kappa \lesssim 0.14$ (Einarsson et al. Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015; Dabade et al. Reference Dabade, Marath and Subramanian2016). For such oblate spheroids, rotational Brownian motion has been shown to play a crucial role in determining the partitioning between spinning and tumbling modes, via a barrier-hopping process acting over sufficiently long times (Dabade et al. Reference Dabade, Marath and Subramanian2016; Marath et al. Reference Marath, Dwivedi and Subramanian2017; Marath & Subramanian Reference Marath and Subramanian2018). In the present work, however, we are interested in the migration of spheroids with a major axis that is at least several microns in length, flowing through channels that are not excessively long. So, the role of thermal fluctuations, either direct or indirect, is assumed negligible.

The Jeffery time period remains unchanged to $O({\textit{Re}}_{\!p})$ for spheroids with $\kappa \sim O(1)$ , with the first correction only occurring at $O({\textit{Re}}_{\!p}^{3/2})$ (Marath & Subramanian Reference Marath and Subramanian2017). The angular velocity of a spheroid, tumbling stably in the flow-gradient plane ( $\theta _j = {\pi }/{2}$ ), can thus be written in the form (Dabade et al. Reference Dabade, Marath and Subramanian2016)

(2.12) \begin{align} \dot {\phi }_{\!j}=\dot {\phi }_{\!j}^{(0)}+ {\textit{Re}}_s \dot {\phi }_{\!j}^{(1)} + O\big({\textit{Re}}_{\!p}^{3/2}\big), \end{align}

for small ${\textit{Re}}_{\!p}$ , where ${\textit{Re}}_s=|\beta | {\textit{Re}}_{\!p}$ is the particle Reynolds number based on the magnitude of the local shear rate. The Jeffery angular velocity $\dot {\phi }_{\!j}^{(0)}$ , and the first inertial correction $\dot {\phi }_{\!j}^{(1)}$ , are given by

(2.13) \begin{align} \dot {\phi }_{\!j}^{(0)} &=\beta \left (-\frac {1}{2}+\frac {\kappa ^2-1}{2 (\kappa ^2+1)}\cos 2{\phi }_{\!j}\right ), \end{align}

(2.14) \begin{align} \dot {\phi }_{\!j}^{(1)}&=\beta \sin {\phi }_{\!j}\cos {\phi }_{\!j}\big[G_1(\kappa )\!+\!G_2(\kappa )\cos 2\theta _j\!+\!G_3(\kappa )\cos 2{\phi }_{\!j}\!+\!G_4(\kappa )\cos 2\theta _j \cos 2{\phi }_{\!j}\big], \end{align}

where the expression for $\dot {\phi }_{\!j}^{(0)}$ may be inferred from (2.5). In (2.13) and (2.14), $\theta _j$ and ${\phi }_{\!j}$ are the polar and azimuthal angles depicted in figure 2, and that define the spheroid orientation $\boldsymbol{p} = \sin \theta _j \cos {\phi }_{\!j} \boldsymbol{1}_1 + \sin \theta _j \sin {\phi }_{\!j} \boldsymbol{1}_2 + \cos \theta _j \boldsymbol{1}_3$ , with the orbit constant $C$ mentioned above being given by $C=\kappa ^{-1}\tan \theta _j(\kappa ^2\sin ^2{\phi }_{\!j}+\cos ^2{\phi }_{\!j})^{1/2}$ . The aspect-ratio-dependent functions $G_1-G_4$ in $\dot {\phi }_{\!j}^{(1)}$ above include additive contributions due to particle and fluid inertia; detailed expressions are given in Appendix C. In the general case, these contributions are multiplied by $St$ and ${\textit{Re}}_{\!p}$ , respectively. Here, $St= \rho _{\!p} V_{max}L^2/H\mu$ , with $\rho _{\!p}$ being the particle density, is the Stokes number, and a dimensionless measure of particle inertia. For the neutrally buoyant particles of interest with $\rho _{\!p} = \rho _f$ , $St = {\textit{Re}}_{\!p}$ , which leads to the fluid inertial contributions being dominant for all $\kappa$ ; the dominance of fluid inertia being analogous to what was originally found in the context of the orientation dynamics problem (Dabade et al. Reference Dabade, Marath and Subramanian2016).

While $\dot {\phi }_{\!j}^{(0)} \sim O(1)\,\forall \,{\phi }_{\!j}$ for spheroids with $\kappa \sim O(1)$ , this is no longer true for spheroids of extreme-aspect-ratios ( $\kappa ,\kappa ^{-1}\gg 1$ ), where one may differentiate between aligned and non-aligned phases of rotation. $\dot {\phi }_{\!j}^{(0)}$ remains order unity only during the latter phase, but becomes asymptotically small during periods of near-alignment – it is $O(\kappa ^{-2})$ ( $O(\kappa ^2))$ for a prolate (oblate) spheroid nearly aligned with the flow (gradient) direction. To see this and the resulting amplification of inertial effects, consider the limiting forms of (2.12) for small ${\phi }_{\!j}$ ( $\pi /2-{\phi }_{\!j}$ ) for a prolate (oblate) spheroid,

(2.15a) \begin{align} \dot {\phi }_{\!j}\beta ^{-1}&= -\left ({\phi }_{\!j}^2+\frac {1}{\kappa ^2}\right )+ \frac {2{\phi }_{\!j} {\textit{Re}}_s}{30\ln (2\kappa )-45}+ O\left (\frac {{\phi }_{\!j}^2}{\kappa ^2}\right )\,\,\,\text{for}\,\,\,\, \kappa \gg 1, \end{align}

(2.15b) \begin{align} \dot {\phi }_{\!j}\beta ^{-1}&=-\left [\left (\frac {\pi }{2}-{\phi }_{\!j}\right )^2+\kappa ^2\right ]-\frac {2 {\textit{Re}}_s}{15}\left (\frac {\pi }{2}-{\phi }_{\!j}\right )+O\left (\!\left (\frac {\pi }{2}-{\phi }_{\!j}\right )^2\kappa ^2\right )\,\,\,\text{for}\,\,\,\, \kappa ^{-1}\gg 1, \end{align}

where the asymptotic forms of the aspect-ratio functions in (2.13) and (2.14) for large and small $\kappa$ have been used in (2.15a ) and (2.15b ), respectively. The Jeffery contribution, corresponding to the first bracketed term in the above equations, is seen to be asymptotically small for extreme aspect ratios. It is $O(\kappa ^{-2})$ for ${\phi }_{\!j} \sim O(\kappa ^{-1})$ in (2.15a ), and $O(\kappa ^2)$ for $ {\pi }/{2}-{\phi }_{\!j} \sim O(\kappa )$ in (2.15b ), leading to a prolate (oblate) spheroid spending a time of $O(\kappa H/V_{max})$ ( $O(\kappa ^{-1}H/V_{max}$ )) in the near-aligned phase, which then controls the Jeffery time period. The inertial terms are $O({\textit{Re}}_s/\kappa \ln \kappa )$ and $O({\textit{Re}}_s\kappa )$ for the prolate and oblate cases, and therefore, become comparable to the leading Jeffery contributions during the aligned phase, leading to a threshold ${\textit{Re}}_s$ value beyond which the spheroid adopts a stationary orientation. The rotation arrest occurs at ${\textit{Re}}_s={\textit{Re}}^{\textit{arrest}}=(30\ln (2\kappa )-45)/\kappa$ for slender prolate spheroids – this threshold, to leading logarithmic order, was predicted by Subramanian & Koch (Reference Subramanian and Koch2005). An analogous arrest occurs for flat disks at ${\textit{Re}}_s={\textit{Re}}^{\textit{arrest}}=15\kappa$ (Marath & Subramanian Reference Marath and Subramanian2017). Note that both thresholds fall within the purview of a small- ${\textit{Re}}_{\!p}$ theory in the respective limits. Figure 3 plots all three asymptotes mentioned above as a function of $\kappa$ ; the region below the horizontal dashed line (corresponding to ${\textit{Re}}_s = 1$ ) may be regarded as the regime of validity of the theory.

Figure 3.

(a) The analytical predictions for ${\textit{Re}}^{\textit{arrest}}$ plotted as a function of $\kappa$ ; these include (i) $(30\ln (2\kappa )-45)/\kappa$ , and its leading logarithmic-order approximation $30\ln \kappa /\kappa$ , for $\kappa \gg 1$ ; and (ii) $15\kappa$ for $\kappa \ll 1$ . The region below the horizontal dashed line, ${\textit{Re}}^{\textit{arrest}}=1$ , is a rough indication of the range of validity of a small- ${\textit{Re}}_{\!p}$ theory. Also included are results for ${\textit{Re}}^{\textit{arrest}}$ , obtained from the lattice Boltzmann simulations of Ding & Aidun (Reference Ding and Aidun2000) and Liu, Huang & Lu (Reference Liu, Huang and Lu2022), and shown as purple stars and pink triangles, respectively. The lone result for $\kappa \lt 1$ is for an oblate spheroid with $\kappa = 0.5$ , while the remaining ones pertain to two-dimensional simulations involving elliptic cylinders in plane Couette flow; note that results from the two simulations coincide for the elliptic cylinder with $\kappa = 2$ . (b) The various components of the inertially averaged stresslet, $\langle S_{\textit{ij}}\rangle$ , plotted as a function of ${\textit{Re}}_c$ , for $s =0.3$ , for $\kappa =10$ . The diagonal components $\langle S_{11} \rangle$ and $\langle S_{22} \rangle$ exhibit a linear scaling, while the inertial correction to the Jeffery-averaged stresslet, $\langle S_{12} \rangle _J$ exhibits a quadratic scaling over the entire range of ${\textit{Re}}_c$ .

Figure 3. Long description

The line graph consists of two subplots labeled (a) and (b). Subplot (a) presents analytical predictions for a function of kappa, including various approximations and results from simulations. The x-axis represents kappa on a logarithmic scale ranging from 10^−3 to 10^3, and the y-axis represents the arrest parameter on a logarithmic scale ranging from 0.1 to 10. The graph includes data points from different studies, marked by different symbols. Subplot (b) shows the various components of the inertially averaged stresslet as a function of Re_c. The x-axis represents Re_c on a logarithmic scale ranging from 10^−2 to 10^2, and the y-axis represents the magnitude of the stresslet components on a logarithmic scale ranging from 10^−10 to 10^−2. The diagonal components exhibit a linear scaling, while the inertial correction to the Jeffery-averaged stresslet exhibits a quadratic scaling over the entire range of Re_c. All values are approximated.

There is numerical support for inertia-induced rotation arrest, although not in the specific case of three dimensions and with ${\textit{Re}}_{\!p}$ small, which precludes a quantitative comparison with the asymptotes above. Ding & Aidun (Reference Ding and Aidun2000) performed lattice Boltzmann simulations for vorticity-aligned circular and elliptic cylinders, the latter with a cross-sectional aspect ratio of $2$ , and an oblate spheroid with $\kappa =0.5$ , all of which were suspended freely in plane Couette flow with $0.05\leqslant \lambda \leqslant 0.25$ ; ${\textit{Re}}_s={\textit{Re}}_{\!p}$ for plane Couette flow. While the rotation rate of the circular cylinder approached zero only for ${\textit{Re}}_s \rightarrow \infty$ , rotation of the elliptic cylinder was arrested for ${\textit{Re}}^{\textit{arrest}}\simeq 7.25$ (defined using the semimajor axis), with the period of rotation diverging as $({\textit{Re}}^{\textit{arrest}}-{\textit{Re}}_s)^{-1/2}$ for ${\textit{Re}}_s \rightarrow {\textit{Re}}^{\textit{arrest}}$ . The tumbling oblate spheroid, which appears to be the only three-dimensional case examined in literature, attained a steady orientation at ${\textit{Re}}^{\textit{arrest}}\simeq 20.25$ . More recently, Liu et al. (Reference Liu, Huang and Lu2022) studied the motion of elliptic cylinders with cross-sectional aspect ratios in the interval $[2,10]$ , using the immersed boundary-lattice Boltzmann method, again in plane Couette flow, and found ${\textit{Re}}^{\textit{arrest}}$ to decrease from $7.2$ down to $0.64$ over the aspect-ratio interval above; this inverse dependence is qualitatively similar to that for a slender fibre, albeit without the additional logarithmic factor that comes from slender body theory in three dimensions. The ${\textit{Re}}^{\textit{arrest}}$ values of Liu et al. (Reference Liu, Huang and Lu2022), and those for the oblate spheroid in Ding & Aidun (Reference Ding and Aidun2000), are plotted alongside the analytical predictions in figure 3(a), with the comparison highlighting the amplified effect of inertia in two dimensions. It is worth reiterating that the increase in time period for $\kappa \sim O(1)$ is controlled by the $O({\textit{Re}}_s^{3/2})$ correction in (2.12) (Marath & Subramanian Reference Marath and Subramanian2017). For such $\kappa$ , one may expand the time period integral ( $\textstyle \int _{2\pi }^0 {\rm d}{\phi }_{\!j} \dot {\phi }_{\!j}^{-1}$ ) for small ${\textit{Re}}_s$ , and the $O({\textit{Re}}_s)$ correction involving $\dot {\phi }_{\!j}^{(1)}$ causes equal amounts of slowing down and speeding up in the extensional and compressional quadrants, respectively, leading to no net change in the tumble period. This is no longer true for extreme aspect ratios where, as seen above, $\dot {\phi }_{\!j}^{(1)}$ becomes comparable to $\dot {\phi }_{\!j}^{(0)}$ , and causes a divergence of the time period (rotation arrest). In our emphasis on extreme aspect ratios, we have neglected the $O({\textit{Re}}_s^{3/2})$ contribution since it remains smaller than $\dot {\phi }_{\!j}^{(0)}$ regardless of $\kappa$ .

We now analyse the implications of rotation arrest from the perspective of calculating the inertial lift velocity. This is important since, as will be seen in § 4 below, onset of rotation arrest heralds a significant departure of the actual lift profile from the Jeffery-averaged approximation obtained in Anand & Subramanian (Reference Anand and Subramanian2023). As indicated earlier in this section, the nature of the migration calculation depends on ${\textit{Re}}_c$ , with a semianalytical approach based on a reciprocal theorem formulation being possible when ${\textit{Re}}_c \ll 1$ . In contrast, one resorts to a numerical implementation of a matched asymptotic expansions approach for ${\textit{Re}}_c \gtrsim 1$ . The discussion above shows that arrest for slender prolate spheroids occurs at ${\textit{Re}}^{\textit{arrest}} \approx 30\ln \kappa /\kappa$ . Using the maximum value, ${\textit{Re}}_s = 4{\textit{Re}}_{\!p}$ , corresponding to the channel walls ( $s=0,1$ ), and ${\textit{Re}}_{\!p} = \lambda ^2 {\textit{Re}}_c$ , the arrest criterion above corresponds to ${\textit{Re}}_c \approx 15\ln \kappa /(2\lambda ^2\kappa )$ . The assumption of small ${\textit{Re}}_c$ would therefore imply $(15\ln \kappa /2\kappa )^{1/2}\ll \lambda \ll 1$ ; for thin oblate spheroids, one obtains $(15\kappa /4)^{1/2} \ll \lambda \ll 1$ . Assuming $\lambda = 0.1$ , $(15\ln \kappa /2\kappa )^{1/2}$ dips below $0.1$ only for $\kappa \gtrsim 5000$ . The oblate-side criterion yields $\kappa \lesssim 0.002$ . Clearly, the assumption of a small ${\textit{Re}}_c$ is very restrictive, and most migration scenarios involving spheroids of extreme aspect ratios correspond to ${\textit{Re}}_c \gtrsim O(1)$ . In light of this, rather than having separate sections devoted to the analyses for small and order unity ${\textit{Re}}_c$ , as in Anand & Subramanian (Reference Anand and Subramanian2023), we relegate the details of the small- ${\textit{Re}}_c$ analysis to Appendix B, and only focus on ${\textit{Re}}_c \gtrsim O(1)$ in the next section.

3. Inertial lift velocity for $\boldsymbol{{Re}_{{c}} \gtrsim O(1)}$

As discussed in detail in Anand & Subramanian (Reference Anand and Subramanian2023), for small $\lambda$ , the time scale over which the spheroid migrates to its equilibrium position, under the influence of inertia, is asymptotically larger than that required for it to complete a single tumble. This implies that the leading-order migration is the result of an averaged lift velocity, the average being over the orientations sampled during a tumble period. Therefore, defining the time averaging operator as

(3.1) \begin{align} \langle f({\phi }_{\!j})\rangle =\frac {1}{T}\int _{2\pi }^{0}f({\phi }_{\!j}) \dfrac {{\rm d}{\phi }_{\!j}}{\dot {\phi }_{\!j}}, \end{align}

with $\dot {\phi }_{\!j}$ given by (2.12), and with $T=\int _{2\pi }^{0}( {{\rm d}{\phi }_{\!j}}/{\dot {\phi }_{\!j}})$ being the tumble time period, we average equations ((2.6a ), (2.6b )) and the boundary conditions ((2.7a )–(2.7c )), for a fixed $\boldsymbol{R}$ . This gives

(3.2a) \begin{align} \frac {\partial ^2 \langle U_i\rangle }{\partial R_j\partial R_j}&-\frac {\partial \langle P\rangle }{\partial R_i}-\langle U_2\rangle \big(\beta +2\gamma R_2 {\textit{Re}}_c^{-1/2}\big)\delta _{i1}\nonumber\\&-\big(\beta R_2+\gamma R_2^2 {\textit{Re}}_c^{-1/2}\big)\frac {\partial \langle U_i\rangle }{\partial R_1} =\beta \langle S_{\textit{ij}}\rangle \frac {\partial \delta (\boldsymbol{R})}{\partial R_j}, \end{align}

(3.2b) \begin{align} &\qquad\qquad\qquad \frac {\partial {\langle U}_i\rangle }{\partial x_i}=0, \end{align}

where $\langle U_i\rangle$ satisfies

(3.3a) \begin{align} \langle U_i\rangle &\sim \frac {3\beta R_i \langle S_{jm}\rangle R_j R_m}{4\pi R^5} \quad \text{ for } \quad \boldsymbol{R}\rightarrow 0, \end{align}

(3.3b) \begin{align} \langle U_i\rangle &\rightarrow 0 \quad \text{ for } \quad R_1,R_3\rightarrow \infty \,(R_2\ \text{fixed}), \end{align}

(3.3c) \begin{align} \langle U_i\rangle &=0 \quad \text{ at } \quad R_2=-s {\textit{Re}}_c^{1/2}, (1-s){\textit{Re}}_c^{1/2}, \end{align}

with the translational ( ${\rm d}\boldsymbol{U}_{\!p}/{\rm d}t$ ) and unsteady ( $ {\partial U_i}/{\partial t}$ ) acceleration terms in (2.6a ) integrating to zero over a single period. Note that the assumption of a fixed $\boldsymbol{R}$ also neglects the variation in spheroid position due to interactions with the walls, which occurs on the tumbling time scale. As mentioned in the introduction, this variation is only $O(\lambda ^2)$ for ${\textit{Re}}_c \rightarrow 0$ , and will be even smaller for finite ${\textit{Re}}_c$ owing to inertia-induced screening of the spheroid disturbance field.

Approximating $\dot {\phi }_{\!j}$ by $\dot {\phi }_{\!j}^{(0)}$ in the time-averaging operation leads to the Jeffery-averaged analysis in Anand & Subramanian (Reference Anand and Subramanian2023), with $T=T_{\textit{jeff}}=\int _{2\pi }^{0}{\rm d}{\phi }_{\!j}/\dot {\phi }_{\!j}^{(0)}= 2\pi \beta ^{-1}(\kappa +\kappa ^{-1})$ being the non-dimensional Jeffery time period. Note that the Jeffery time-averaging operator is independent of the ambient flow owing to $\dot {\phi }_{\!j}^{(0)}$ being only a function of $\kappa$ , and the Jeffery-averaged stresslet must therefore be linear in $\boldsymbol{E}$ , as for a sphere. This implies that, for the plane Poiseuille profile, the forcing in (3.2a ) must only involve $\langle S_{12} \rangle _J$ , and is of the form $\beta \langle S_{12} \rangle _J \delta (R_3)[\delta (R_1)\delta^\prime(R_2)\boldsymbol{1}_1 + \delta^\prime(R_1)\delta (R_2)\boldsymbol{1}_2]$ (the subscript $J$ denoting a Jeffery-averaged quantity), with $\langle S_{12}\rangle _J$ defined by

(3.4) \begin{align} \langle S_{12}\rangle _J &= \frac { (3 A_1+A_3)\kappa +2 A_2 (\kappa ^2+1)}{4(\kappa +1)^2}. \end{align}

The above led Anand & Subramanian (Reference Anand and Subramanian2023) to conclude that the Jeffery-averaged lift velocity of a tumbling spheroid, which is a linear functional of $\langle U_2\rangle$ (see (3.11) below), can be written as

(3.5) \begin{align} \langle V_{\!p} \rangle _J(s;{\textit{Re}}_c,\kappa ) = \frac {\langle S_{12}\rangle _J(\kappa )}{(-10\pi /3)} V_{\!p}^{\textit{sphere}}(s;{\textit{Re}}_c), \end{align}

and differs from that for a sphere only by a multiplicative function of $\kappa$ ; the complete expression for this function is given in Appendix B – see (B12) therein. The relation (3.5) implies that the Jeffery-averaged equilibria, corresponding to $\langle V_{\!p} \rangle _J = 0$ , are independent of $\kappa$ , being identical to those for a sphere (Schonberg & Hinch Reference Schonberg and Hinch1989) at the same ${\textit{Re}}_c$ . Note that the proportionality relation in (3.5) is always true for a spinning spheroid within the small- ${\textit{Re}}_{\!p}$ framework used here. This is due to the steady velocity disturbance, even though the rate of spin is altered by inertia (at $O({\textit{Re}}_{\!p}^{3/2}$ )). Since weak inertia stabilizes the spinning mode for oblate spheroids with $\kappa \gtrsim 0.14$ , our focus is on oblate spheroids with aspect ratios less than this threshold value.

Figure 4.

(a) Comparison of lift velocity profiles, obtained using the inertial and Jeffery orientation distributions, for tumbling prolate spheroids with $\kappa =2$ ; $\lambda =0.05$ and ${\textit{Re}}_c = 1$ and $100$ . The two profiles exhibit excellent agreement over the entire range of $s$ examined. (b) A comparison of actual (solid curves) and Jeffery-averaged (dashed curves) lift velocity profiles for $\lambda = 0.05, {\textit{Re}}_c=100$ , for prolate spheroids of different aspect ratios: $\kappa =2$ (circles), $5$ (triangles), $10$ (squares) and $25$ (stars). With increasing $\kappa$ , the actual profiles start to deviate from the Jeffery-averaged ones; the deviation begins in the near-wall region, and for $\kappa = 25$ , leads to a small shift of the equilibrium position towards the centreline.

Figure 4. Long description

The image contains two line graphs labeled (a) and (b). Graph (a) compares lift velocity profiles obtained using inertial and Jeffery orientation distributions for tumbling prolate spheroids with an aspect ratio of = 2 and varying Reynolds numbers. The profiles show excellent agreement across the entire range of s values. Graph (b) compares actual and Jeffery-averaged lift velocity profiles for a Reynolds number of 100, for prolate spheroids with different aspect ratios: 2, 3, 5, and 10. The actual profiles start to deviate from the Jeffery-averaged ones as the aspect ratio increases, with the deviation beginning in the near-wall region and leading to a small shift of the equilibrium position towards the centerline for an aspect ratio of 10. The x-axis represents the variable s, and the y-axis represents the normalized lift velocity. All values are approximated.

Intuitively speaking, one expects a Jeffery-averaged analysis to give the inertial lift for sufficiently small ${\textit{Re}}_c$ , for spheroids of order unity aspect ratios, since the inertial correction to the spheroid angular velocity is asymptotically small in this limit. The Jeffery-averaged lift is $O({\textit{Re}}_{\!p})$ for ${\textit{Re}}_c \ll 1$ , owing to $V_{\!p}^{\textit{sphere}}$ in (3.5) being of this order. It turns out, however, that there is a second contribution at the same order, and that crucially relies on the alteration of spheroid rotation by weak inertia. This latter $O({\textit{Re}}_{\!p})$ contribution was missed out in Anand & Subramanian (Reference Anand and Subramanian2023), owing to the authors writing down the lift velocity in terms of a Jeffery-averaged volume integral involving the $O({\textit{Re}}_{\!p})$ inertial acceleration. The correction to the Jeffery-averaged acceleration, arising from $\dot {\phi }_{\!j}^{(1)}$ in the time-averaging operator defined in (3.1), would only contribute to the lift velocity at $O({\textit{Re}}_{\!p}^2)$ , and this smaller contribution was accordingly neglected in the said analysis. While the neglect is justified for the inertial acceleration, the second $O({\textit{Re}}_{\!p})$ contribution to the inertial lift comes from a different source. As discussed in more detail in Appendix B, the origin of this contribution is the $O({\textit{Re}}_{\!p})$ term in the stresslet forcing in (3.2a ) that arises on using an inertial, rather than Jeffery, orientation average. The explicit form of this forcing is given below, and involves both the diagonal components of the time-averaged stresslet and an inertial correction to $\langle S_{12} \rangle _J$ that appears in (3.5) – see discussion in the next paragraph. Despite being formally of the same order, this second contribution turns out to involve an extremely small numerical prefactor for $\kappa \sim O(1)$ , as a result of which the Jeffery-averaged lift obtained in Anand & Subramanian (Reference Anand and Subramanian2023) remains a very accurate approximation of the inertial lift, for all ${\textit{Re}}_c$ , for $\kappa \sim O(1)$ ; for instance, see figure 4(a) in § 4 below. Notwithstanding its smallness, the calculation in Appendix B does show that a $\kappa$ -dependence of the equilibrium position arises even for $\kappa \sim O(1)$ and ${\textit{Re}}_c \ll 1$ . Interestingly, the equilibrium positions for prolate spheroids with $\kappa \in (1,1.33]$ exhibit a wallward shift (compared with those for spheres), while those for spheroids with $\kappa \gt 1.33$ are marginally closer to the centreline. The direction of shift (towards the centreline) for the latter range of aspect ratios remains consistent with that for spheroids of extreme aspect ratios analysed in § 4. Further, and importantly, the analysis shows that the essential mechanism underlying this shift is that of a repulsive interaction with an image stresslet induced by the wall boundary conditions, and this is considered in more detail in § 5.

We now move beyond the Jeffery-averaged analysis, considering ${\textit{Re}}_c \gtrsim O(1)$ . With the inertial angular velocity included in the time-averaging operator, the forcing function in (3.2a ) also involves the diagonal elements in addition to $\langle S_{12}\rangle$ , being of the form

(3.6) \begin{align} \langle S_{\textit{ij}}\rangle \frac {\partial \delta (\boldsymbol{R})}{\partial R_j} &= \langle S_{12}\rangle \delta (R_3)\big[\delta (R_1)\delta^\prime(R_2)\boldsymbol{1}_1+\delta ^\prime(R_1)\delta (R_2)\boldsymbol{1}_2\big]+\langle S_{11}\rangle \delta ^\prime(R_1)\delta (R_2)\delta (R_3)\boldsymbol{1}_1 \nonumber \\ &\quad +\langle S_{22}\rangle \delta (R_1)\delta ^\prime(R_2)\delta (R_3)\boldsymbol{1}_2+\langle S_{33}\rangle \delta (R_1)\delta (R_2)\delta ^\prime(R_3)\boldsymbol{1}_3. \end{align}

On account of the ${\textit{Re}}_{\!p}$ -dependent orientation dynamics, the averaged stresslets in the above expression are functions of ${\textit{Re}}_c$ (for $\lambda$ fixed), and the ${\textit{Re}}_c$ -dependence for the different components is shown in figure 3(b) for $\kappa = 10$ . For this $\kappa$ , one is always far from rotation arrest, and the quadratic scaling of $\langle S_{12}\rangle - \langle S_{12}\rangle _J$ and the linear scalings of $\langle S_{11}\rangle$ and $\langle S_{22}\rangle$ , over the entire range of ${\textit{Re}}_c$ in figure 3(b), merely reflect the corresponding small- ${\textit{Re}}_{\!p}$ scalings for $\lambda$ fixed. Since $\langle S_{12} \rangle$ contributes to the suspension viscosity (Batchelor Reference Batchelor1970), the $O({\textit{Re}}_c^2)$ scaling of $\langle S_{12}\rangle - \langle S_{12}\rangle _J$ in figure 3 implies a regular $O({\textit{Re}}_{\!p}^2)$ correction to the shear viscosity of a suspension of spheroids, arising from an inertial alteration of the orientation distribution. Earlier calculations for a non-interacting inertial suspension of spherical particles (drops) have shown that the leading-order correction to $S_{12}$ is $O({\textit{Re}}_{\!p}^{3/2})$ , and originates in the outer region (Lin et al. Reference Lin, Peery and Schowalter1970; Subramanian et al. Reference Subramanian, Koch, Zhang and Yang2011); this singular outer-region correction is not relevant for small ${\textit{Re}}_c$ , owing to the channel width being smaller than the inertial screening length in this limit.

Having clarified the functional form of the stresslet forcing, we proceed to solve (3.2a )–(3.3c ) for the disturbance fields, which involves Fourier transforming along the spatially homogeneous flow and vorticity directions (Schonberg & Hinch Reference Schonberg and Hinch1989; Anand & Subramanian Reference Anand and Subramanian2023). Defining the partial Fourier transform,

(3.7) \begin{align} \hat {f}(k_1,R_2,k_3)=\int _{-\infty }^\infty \int _{-\infty }^\infty {\rm d}R_1 {\rm d}R_3 \,f(R_1,R_2,R_3)\,e^{\iota (k_1 R_1+k_3 R_3)}, \end{align}

one obtains, after some manipulations, the following coupled ordinary differential equations for $\langle \hat {T}\rangle$ and $\langle \hat {U}_2\rangle$ :

(3.8a) \begin{align} \frac {d^2 \hat {\langle T\rangle }}{{\rm d}R_2^2}-k_\perp ^2\hat {\langle T\rangle }&=2\iota k_1 \langle \hat {U}_2\rangle \big(\beta +2\gamma R_2 {\textit{Re}}_c^{-1/2}\big), \end{align}

(3.8b) \begin{align} \frac {d^2\langle \hat {U}_2\rangle }{{\rm d}R_2^2}-k_\perp ^2\langle \hat {U}_2\rangle &=\frac {{\rm d}\hat {\langle T\rangle }}{{\rm d}R_2}-\iota k_1\langle \hat {U}_2\rangle \big(\beta R_2+\gamma R_2^2 {\textit{Re}}_c^{-1/2}\big), \end{align}

in the intervals $-s{\textit{Re}}_c^{1/2} \lt R_2 \lt 0$ and $0\lt R_2 \lt (1-s){\textit{Re}}_c^{1/2}$ . Here, $\langle \hat {T}\rangle =\langle \hat {P}\rangle + \beta \langle S_{22}\rangle \delta (R_2)$ is a modified (transformed) pressure field. Equations (3.8a ) and (3.8b ) are subject to the following conditions:

(3.9a) \begin{align} \langle \hat {U}_2\rangle &\sim \frac {\iota \beta \langle S_{12}\rangle e^{-k_\perp R_2} R_2\,\,k_1}{2}+\frac {\beta R_2 e^{-k_\perp R_2} \left (\left(k_1^2+2 k_3^2\right)\langle S_{22}\rangle -\left(k_1^2-k_3^2\right)\langle S_{11}\rangle \right )}{4 k_\perp }\nonumber \\ &\quad \text{ for } k_1,k_3\to \infty \text{ and } R_2\rightarrow 0, \end{align}

(3.9b) \begin{align} \langle \hat {U}_2\rangle &=\frac {{\rm d}\langle \hat {U}_2\rangle }{{\rm d}R_2}= 0 \text{ at } R_2=-s {\textit{Re}}_c^{1/2},(1-s) {\textit{Re}}_c^{1/2}, \end{align}

where $k_\perp ^2=k_1^2+k_3^2$ . Equation (3.9a ) is the matching requirement to the transformed stresslet field in the inner region, that now accounts for the modified stresslet tensor resulting from the inertial orientation dynamics, while (3.9b ) is the no-slip boundary condition at the channel walls. The stresslet forcing in (3.2a ) enters the problem in two ways. The second term in $\langle \hat {T} \rangle$ above serves to absorb the localized contribution induced by the most singular component ( $\propto \langle S_{22} \rangle \delta^{\prime\prime}(R_2)$ ) of the aforementioned stresslet forcing that enters (3.8a ). The remaining part of the stresslet forcing enters via jump conditions at $R_2 = 0$ , that serve to connect the $\hat {\langle T\rangle }$ and $\hat {\langle U_2 \rangle }$ -fields for positive and negative $R_2$ . The detailed derivation in Appendix A leads to the following jump conditions:

(3.10a) \begin{align} \hat {\langle T\rangle }^{\!+} (k_1,0,k_3) -\hat {\langle T\rangle }^{\!-}(k_1,0,k_3) &=2\iota k_1 \beta \langle S_{12}\rangle , \end{align}

(3.10b) \begin{align} \frac {{\rm d}\hat {\langle T\rangle }^{\!+}}{{\rm d}R_2}(k_1,0,k_3)-\frac {{\rm d}\hat {\langle T\rangle }^{\!-}}{{\rm d}R_2}(k_1,0,k_3) &= \beta \big [\big(k_1^2-k_3^2\big)\langle S_{11}\rangle -\big(k_1^2+2 k_3^2\big)\langle S_{22}\rangle \big ], \end{align}

(3.10c) \begin{align} \langle \hat {U}_2\rangle ^{\!+}(k_1,0,k_3)&=\langle \hat {U}_2\rangle ^{\!-} (k_1,0,k_3), \end{align}

(3.10d) \begin{align} \frac {{\rm d}\langle \hat {U}_2\rangle ^{\!+}}{{\rm d}R_2}(k_1,0,k_3)-\frac {{\rm d}\langle \hat {U}_2\rangle ^{\!-}}{{\rm d}R_2}(k_1,0,k_3) &=\iota k_1 \beta \langle S_{12}\rangle , \end{align}

where the superscripts ‘+’ and ‘−’ denote the transformed fields for $R_2\gt 0$ and $R_2\lt 0$ , respectively. One solves the system (3.8a ), (3.8b ), along with the boundary conditions ((3.9a )–(3.9b )) and jump conditions ((3.10a )–(3.10d )), using a shooting method originally used by Schonberg & Hinch (Reference Schonberg and Hinch1989) for a spherical particle, and described in more detail by Anand & Subramanian (Reference Anand and Subramanian2023) as part of their Jeffery-averaged analysis for spheroids.

The limiting form of $\langle U_2 \rangle$ in the matching region $\boldsymbol{R}\ll 1$ , obtained from the above procedure, is the sum of the (inverse transformed) singular stresslet contribution given in (3.9a ), and a uniform flow along the gradient (cross-stream) direction that is a consequence of fluid inertia. The neutrally buoyant spheroid being force-free is convected by the latter uniform flow which therefore equals the inertial lift velocity. The lift velocity may thus be determined as the limit of the inverse transform for $\boldsymbol{R}\to 0$ ,

(3.11) \begin{align} \langle V_{\!p}\rangle &= \frac {{\textit{Re}}_{\!p}}{4\pi ^2}\,\,\Re \left \{\int _{-\infty }^\infty \int _{-\infty }^\infty \,\langle \hat {U}_2\rangle ^\pm (k_1,0,k_3)\,{\rm d}k_1\, {\rm d}k_3\right \}\!. \end{align}

Here, $\Re \{.\}$ denotes taking the real part, which eliminates the singular, albeit purely imaginary, stresslet contribution (although the stresslet as a whole vanishes anyway in the limit $R_2 \rightarrow 0$ , which may be seen from inverse transforming (3.9a )). Note that one may use either $\langle \hat {U}_2\rangle ^{\!+}$ or $\langle \hat {U}_2\rangle ^{\!-}$ in (3.11) on account of the continuity across $R_2=0$ (see (3.10c )).