3.1. Governing equations for microparticle dynamics

The motions of microparticles suspended in a liquid medium, when subjected to an external magnetic field, can be governed by Newton’s second law, written as

(3.1) \begin{align} m_i\dfrac {\partial ^2 \boldsymbol{x}_i}{\partial t^2}=\boldsymbol{F}^H_i+\boldsymbol{F}^D_i+\boldsymbol{F}^G_i+\boldsymbol{F}^B_i+\boldsymbol{ F}^{P-P}_i+\boldsymbol{F}^{P-W}_i, \end{align}

where subscript ‘ $i$ ’ denotes the $i$ th microparticle, $m_i$ represents the $i$ th microparticle mass, given by $m=V_p\rho _p = {4}/{3}\pi R^3\rho _p$ , and $R$ and $\rho _p$ are the radius and density of microparticle, respectively. Here, $\boldsymbol{F}^H_i$ is the hydrodynamic force from the carrier fluid, $\boldsymbol{F}^D_i$ is the magnetic dipole force induced by the external magnetic field and $\boldsymbol{F}^G_i$ and $\boldsymbol{F}^B_i$ represent the gravity and buoyancy, respectively. Since the microfluidic chip is usually characterised by an extremely thin thickness, resulting in a negligible sedimentation effect during the motion of microparticles, the buoyancy force is assumed to be counteracted by gravity in this work. Here, $\boldsymbol{F}^{P-P}_i$ and $\boldsymbol{F}^{P-W}_i$ are the contacting forces due to the particle–particle and particle–wall interactions, respectively.

In the present work, the hydrodynamic force $\boldsymbol{F}^H_i$ is fully resolved by the LBM, which has been explained in § 2. The magnetic dipole force acting on the $i$ th microparticle $\boldsymbol{F}^D_i$ is the resultant force from other microparticles, which is modelled by

(3.2) \begin{align} \boldsymbol{F}_i^{D}&=\sum _{j\neq i}\dfrac {3}{4\pi \mu _0{||\boldsymbol{r}_{\textit{ij}}||}^4}\left \{\left (\boldsymbol{m}_i\boldsymbol{\cdot }\boldsymbol{ m}_{\!j}\right )\dfrac {\boldsymbol{r}_{\textit{ij}}}{||\boldsymbol{r}_{\textit{ij}}||}-5\left (\boldsymbol{m}_i\boldsymbol{\cdot }\boldsymbol{r}_{\textit{ij}}\right )\left (\boldsymbol{ m}_{\!j}\boldsymbol{\cdot }\boldsymbol{r}_{\textit{ij}}\right )\dfrac {\boldsymbol{r}_{\textit{ij}}}{||\boldsymbol{r}_{\textit{ij}}||^3}\right. \notag\\&\quad \left.+\,\left [\left (\boldsymbol{m}_i\boldsymbol{\cdot }\boldsymbol{ r}_{\textit{ij}}\right )\boldsymbol{m}_{\!j}+\left (\boldsymbol{m}_{\!j}\boldsymbol{\cdot }\boldsymbol{r}_{\textit{ij}}\right )\boldsymbol{m}_i\right ]\dfrac {1}{||\boldsymbol{ r}_{\textit{ij}}||}\right \}, \end{align}

where $\boldsymbol{r}_{\textit{ij}}=\boldsymbol{r}_i-\boldsymbol{r}_j$ is the relative position vector of the $i$ th and $j$ th microparticles, and $\boldsymbol{m}_i$ represents the magnetic moment induced by the external magnetic field, given as

(3.3) \begin{align} \boldsymbol{m}_i=\dfrac {4}{3}\pi R^3 \mu _0 \chi \boldsymbol{H}=\left (\dfrac {4}{3}\pi R^3 \mu _0 \chi H_0\right ) \boldsymbol{e}_H=m_0 \boldsymbol{e}_H, \end{align}

where $\mu _0$ is the magnetic permeability in vacuum, $\chi$ denotes the effective magnetic susceptibility of the microparticle and $\boldsymbol{H}$ represents the external magnetic field strength with magnitude $H_0$ along the $\boldsymbol{e}_H$ direction.

Furthermore, the magnetic dipole force model, i.e. (3.2), can be rewritten as

(3.4) \begin{align} \boldsymbol{F}_i^{D}&=\dfrac {3m_0^2}{4\pi \mu _0R^4}\sum _{j\neq i}\left (\dfrac {R}{{||\boldsymbol{r}_{\textit{ij}}||}}\right )^4\bigg (\boldsymbol{ e}_{\textit{ij}}-5\left (\boldsymbol{e}_H\boldsymbol{\cdot }\boldsymbol{e}_{\textit{ij}}\right )^2\boldsymbol{e}_{\textit{ij}}+2\left (\boldsymbol{e}_H\boldsymbol{\cdot }\boldsymbol{e}_{\textit{ij}}\right )\boldsymbol{e}_H\bigg ). \end{align}

Here, we introduce a non-dimensional symbol $\boldsymbol{I}^D$ , formulated as

(3.5) \begin{align} \boldsymbol{I}^D=\sum _{j\neq i}\left (\dfrac {R}{{||\boldsymbol{r}_{\textit{ij}}||}}\right )^4\bigg (\boldsymbol{e}_{\textit{ij}}-5\left (\boldsymbol{e}_H\boldsymbol{\cdot }\boldsymbol{e}_{\textit{ij}}\right )^2\boldsymbol{e}_{\textit{ij}}+2\left (\boldsymbol{e}_H\boldsymbol{\cdot }\boldsymbol{e}_{\textit{ij}}\right )\boldsymbol{e}_H\bigg ), \end{align}

where $\boldsymbol{I}^D$ is a function of the relative separation distance ${R}/{\boldsymbol{r}_{\textit{ij}}}$ between the two microparticles, and the orientation of the line connecting their centres with respect to the direction of the external magnetic field $\boldsymbol{e}_H\boldsymbol{\cdot }\boldsymbol{e}_{\textit{ij}}$ . Thus, (3.4) can be further simplified to

(3.6) \begin{align} \boldsymbol{F}_i^{D}&=\dfrac {3m_0^2}{4\pi \mu _0R^4}\boldsymbol{I}^D. \end{align}

Here, $\boldsymbol{F}^{P-P}_i$ and $\boldsymbol{F}^{P-W}_i$ are modelled by a repulsive force $\boldsymbol{F}_i^{rep}$ exerted on the microparticles to prevent the unphysical overlap, given by

(3.7) \begin{align} \boldsymbol{F}_i^{rep}&=\dfrac {3m_0^2}{4\pi \mu _0D^4}\exp (-k\delta )\dfrac {\boldsymbol{r}_{\textit{ij}}}{||r_{\textit{ij}}||}, \end{align}

where $D=2R$ denotes the diameter of the microparticle, $k$ is the Debye–Hückel parameter, set to 40 according to previous studies (Ido et al. Reference Ido, Inagaki and Umehara2008, Reference Ido, Inagaki and Yamaguchi2010), and $\delta$ is the distance between the surfaces of two solid objects.

It is noted that $\boldsymbol{F}^{P-P}_i$ and $\boldsymbol{F}^{P-W}_i$ are categorised as the short-range forces that are only exerted when an exact physical contact occurs. Such short-range forces have a significant effect exclusively in dense particulate flows where interparticle collisions occur frequently. The parameters $\boldsymbol{F}^H_i$ and $\boldsymbol{F}^D_i$ belong to the long-range forces, which are exerted throughout the entire motions of the microparticles and dominate their dynamic behaviours. Therefore, the dynamic equation of microparticles before physical contact occurs can be further rewritten as

(3.8) \begin{align} m_i\dfrac {\partial ^2 \boldsymbol{x}_i}{\partial t^2}=\boldsymbol{F}^D_i+\boldsymbol{F}^H_i. \end{align}

3.2. Dimensionless analysis of dynamic equations

A dimensionless analysis is conducted for the dynamic equation of microparticles, and several key characteristic parameters are identified in this subsection. The hydrodynamic force $\boldsymbol{F}^H_i$ in (3.8) represents the resultant force exerted on the microparticle by the surrounding fluid, consisting of the drag force $\boldsymbol{F}^{\textit{drag}}_i$ , the added mass force $\boldsymbol{F}^{added}_i$ and the Basset force along the direction of microparticle motion, as well as the Magnus force and the Saffman lift force perpendicular to the motion direction. The parameter $\boldsymbol{F}^H_i$ is usually approximated by the drag force models in previous studies (Banerjee et al. Reference Banerjee, Bit, Ganguly and Hardt2012; Gao et al. Reference Gao, Hulsen, Kang and den Toonder2012), while it is fully resolved in the LBM model, as given by (2.9a ). The contributions of individual force components are all absorbed into $\boldsymbol{F}^H_i$ and are not explicitly distinguished. In the following theoretical analysis, it is unnecessary to exhaustively list all force components acting on the microparticles. Instead, we only need to consider the leading forces and guarantee that all factors of interest are included in the dynamic governing equations.

For a rigid sphere moving in an unbounded quiescent flow, the drag force (Stokes Reference Stokes2009) acting on the $i$ th microparticle can be evaluated by

(3.9) \begin{align} \boldsymbol{F}^{\textit{drag}}_i=-C_d^{\textit{ub}}\mu _{\!f}R\left (\boldsymbol{u}_p^i-\boldsymbol{u}_{\!f}^i\right )\!, \end{align}

where $C_d^{\textit{ub}}=6\pi$ represents the drag coefficient in the unbounded flow, $\boldsymbol{u}_p^i$ denotes the $i$ th microparticle velocity, $\boldsymbol{u}_{\!f}^i$ refers to the undisturbed fluid velocity at the location of the $i$ th microparticle and $\mu _{\!f}$ is the dynamic viscosity of the carrier fluid, which follows a basic relation with the kinematic viscosity $\nu _{\!f}$ and density $\rho _{f}$ of the carrier fluid, i.e. $\mu _{\!f}=\rho _{f}\boldsymbol{\cdot }\nu _{\!f}$ .

To incorporate the wall effects, Happel & Brenner (Reference Happel and Brenner1983) proposed a modified drag force model for a sphere moving between two parallel walls, read as

(3.10) \begin{align} \boldsymbol{F}^{\textit{drag}}_i=-C_d\mu _{\!f}R\left (\boldsymbol{u}_p^i-\boldsymbol{u}_{\!f}^i\right )=-\left (C_d^{\textit{ub}}+C_d^{w\textit{b}}\right )\mu _{\!f}R\left (\boldsymbol{ u}_p^i-\boldsymbol{u}_{\!f}^i\right )\!, \end{align}

where the net drag coefficient $C_d$ is decomposed into the unbounded drag coefficient $C_d^{\textit{ub}}$ and the wall-bounded drag coefficient $C_d^{w\textit{b}}$ ; $C_d^{w\textit{b}}$ is a function of the particle–wall separation distance $(R/l)$ , as

(3.11) \begin{align} C_d^{w\textit{b}}\left (R/l\right )=6\pi \left \{\left [1-1.004\left (\!\dfrac {R}{l}\right )+0.418\left (\!\dfrac {R}{l}\right )^3\!+0.21\left (\!\dfrac {R}{l}\right )^4\!-0.169\left (\!\dfrac {R}{l}\right )^5\right ]^{-1}\!-\!1\!\right \}\!, \end{align}

where $l$ refers to the separation distance between the sphere and the microchannel wall, and the sphere is assumed to lie in the middle of two walls.

By approximating the hydrodynamic force with the drag force and substituting (3.6) and (3.10) into (3.8), we have

(3.12) \begin{align} \left (\dfrac {4}{3}\pi R^3\rho _p\right )\dfrac {\partial ^2 \boldsymbol{x}_i}{\partial t^2}+C_d\mu _{\!f}R\dfrac {\partial \boldsymbol{ x}_i}{\partial t}=C_d\mu _{\!f}R\boldsymbol{u}_{\!f}^i+\dfrac {4}{3}\pi R^2 \mu _0 \chi ^2 H_0^2 \boldsymbol{I}^D. \end{align}

The units and dimensions of the physical variables involved in (3.12) are listed in table 1. We take the radius of the microparticle and the density of the carrier fluid as the characteristic length and density, and then have $L^{\textit{ref}}=R$ , $M^{\textit{ref}}=\rho _{f} R^3$ . Now, a characteristic velocity is required to determine the characteristic time. However, previous studies, e.g. Klingenberg, Ulicny & Golden (Reference Klingenberg, Ulicny and Golden2007), Kang et al. (Reference Kang, Gao, Hulsen, den Toonder and Anderson2013), Seong et al. (Reference Seong, Kang, Hulsen, den Toonder and Anderson2016) and Zhang et al. (Reference Zhang, Zhou and Wang2019), exhibited a significant inconsistency in the selection of reference time scales, which hinders the quantitative comparison of the time evolution of particle dynamics and the comprehension of the underlying physics. Two widely used categories of reference time scales are introduced here to explore the essential relation between them, which are defined as

(3.13) \begin{align} T_1^{\textit{ref}}=\dfrac {R^2}{\nu _{\!f}},\quad T_2^{\textit{ref}}=\dfrac {\rho _{f} \nu _{\!f}}{\mu _0 \chi ^2{H_0}^2}, \end{align}

where $T_1^{\textit{ref}}$ reflects the relaxation time of the carrier fluid and $T_2^{\textit{ref}}$ represents the response time of the microparticle under the combined effect of hydrodynamic force and magnetic dipole interaction (Seong et al. Reference Seong, Kang, Hulsen, den Toonder and Anderson2016; Zhang et al. Reference Zhang, Zhou and Wang2019).

Table 1.

The units and dimensions of the physical parameters in the dynamic equation of the microparticle.

Utilising the characteristic length $R$ and the reference time scale $T_1^{\textit{ref}}$ to normalise (3.12) yields

(3.14) \begin{align} \dfrac {\partial ^2 \boldsymbol{x}^*_i}{\partial t_1^{*2}}+\dfrac {3}{4\pi }C_d \left (\dfrac {\rho _{f}}{\rho _p}\right ) \dfrac {\partial \boldsymbol{x}^*_i}{\partial t_1^*} =\dfrac {3}{4\pi }C_d \left (\dfrac {\rho _{f}}{\rho _p}\right ) \dfrac {R u_{\!f}}{\nu _{\!f}} \boldsymbol{n}^f+ \left (\dfrac {\rho _{f}}{\rho _p}\right )\dfrac {\mu _0 \chi ^2 H_0^2}{\rho _{f}\left (\nu _{\!f}^2/R^2\right )} \boldsymbol{I}^D, \end{align}

where superscript ‘*’ denotes the normalised variables and subscript $t_1^*$ denotes normalisation using $T_1^{\textit{ref}}$ . Here, $u_{\!f}$ is the magnitude of the fluid velocity $\boldsymbol{u}_{\!f}^i$ , and the unit direction vector is denoted as $\boldsymbol{ n}^f=\boldsymbol{u}_{\!f}^i/u_{\!f}^i$ . It can be found that (3.14) involves four dimensionless characteristic parameters. The first one is the density ratio between the fluid and the microparticle $\rho _{f} /\rho _p$ . Since the sedimentation effect of the microparticle is not considered in this work, these microparticles are assumed to be neutrally buoyant, i.e. $\rho _{f} /\rho _p=1$ . The second parameter is the net drag coefficient $C_d$ , which is a function of the particle–wall separation distance $R/l$ , as demonstrated in (3.11). It also should be noted, however, that $C_d$ presented in (3.11) is derived under the assumption of a single particle moving between two parallel plates, and thus may not be fully applicable to the multi-particle system analysed in this work. Therefore, the particle–wall separation distance $R/l$ is employed as the characteristic parameter to represent the effect of microchannel wall interaction on the microparticle dynamics in the subsequent sections. The third one is widely known as the Reynolds number, given by $Re=u_{\!f} R/\nu _{\!f}$ . The last one reflects the ratio of the magnetic dipole interaction to the hydrodynamic viscous effect acting on the microparticle. We refer to this characteristic parameter as the magnetic number ( $N_{\textit{mag}}$ ), which is defined by

(3.15) \begin{align} {N_{\textit{mag}}}=\dfrac {\mu _0 \chi ^2 H_0^2}{\rho _{f}\left (\nu _{\!f}^2/R^2\right )}. \end{align}

Noted that some previous studies on magnetorheological Melle et al. (Reference Melle, Calderón, Rubio and Fuller2003), Sherman, Becnel & Wereley (Reference Sherman, Becnel and Wereley2015) and electrorheological Marshall, Zukoski & Goodwin (Reference Marshall, Zukoski and Goodwin1989) fluids have also introduced the Mason number to characterise the dynamic response of microparticles under external magnetic and electric fields, which denotes the ratio of the hydrodynamic effect to the magnetic or electric dipole effect. The flows addressed in these studies are generally shear flows with a well-defined shear rate. Consequently, the hydrodynamic viscous effect can be characterised by the dynamic viscosity $\mu _{\!f}$ and the shear rate $\dot {\gamma }$ of the carrier fluid. However, the problem discussed in this work is a quasi-stationary flow, and it is challenging to identify such a shear rate. Although the drag force is independent of the fluid density, the non-zero added mass force induced by the dynamic variable-speed motion of microparticles is associated with the fluid density. Therefore, this work employs the fluid density $\rho _{f}$ , kinematic viscosity $\nu _{\!f}$ , as well as the radius $R$ of microparticles to characterise the hydrodynamic viscous effect acting on microparticles. The parameter $N_{\textit{mag}}$ proposed in this work is conceptually the reciprocal of the Mason number.

Then, (3.14) can be further expressed as

(3.16) \begin{align} \dfrac {\partial ^2 \boldsymbol{x}^*_i}{\partial t_1^{*2}}+\dfrac {3}{4\pi }C_d \left (\dfrac {\rho _{f}}{\rho _p}\right )\dfrac {\partial \boldsymbol{x}^*_i}{\partial t_1^*} =\dfrac {3}{4\pi }C_d \left (\dfrac {\rho _{f}}{\rho _p}\right ) Re \boldsymbol{n}^f+ \left (\dfrac {\rho _{f}}{\rho _p}\right ){N_{\textit{mag}}}\boldsymbol{ I}^D. \end{align}

Similarly, we utilise the reference time scale $T_2^{\textit{ref}}$ to normalise (3.12), and then obtain the dynamic equation of the microparticle in a dimensionless form as

(3.17) \begin{align} \dfrac {\partial ^2 \boldsymbol{x}^*_i}{\partial t_2^{*2}}+\dfrac {3}{4\pi }C_d\left (\dfrac {\rho _{f}}{\rho _p}\right )\dfrac {1}{N_{\textit{mag}}}\dfrac {\partial \boldsymbol{x}^*_i}{\partial t_2^*}=\dfrac {3}{4\pi }C_d\left (\dfrac {\rho _{f}}{\rho _p}\right ) \dfrac {Re}{N_{\textit{mag}}^2} \boldsymbol{ n}^f+\left (\dfrac {\rho _{f}}{\rho _p}\right )\dfrac {1}{N_{\textit{mag}}} \boldsymbol{I}^D. \end{align}

Given that the dispersed particles discussed in this work are of micron size, it follows that the mass of these microparticles is sufficiently small, thereby rendering the contribution of microparticle acceleration in (3.16) and (3.17) negligible, as reported in the literature, e.g. Klingenberg et al. (Reference Klingenberg, van Swol and Zukoski1989) and Sherman et al. (Reference Sherman, Becnel and Wereley2015). Thus, (3.16) and (3.17) can be further simplified into a unified first-order differential equation, which is expressed as

(3.18) \begin{align} \dfrac {3}{4\pi }\dfrac {\partial \boldsymbol{x}^*_i}{\partial t^*}=\dfrac {3}{4\pi }C_d \dfrac {Re}{N_{\textit{mag}}} \boldsymbol{n}^f+\boldsymbol{ I}^D, \end{align}

where

(3.19) \begin{align} t^*&=\dfrac {N_{\textit{mag}}}{C_d}{t_1^*}=\left (\dfrac {N_{\textit{mag}}}{C_d}\right )\dfrac {t}{T_1^{\textit{ref}}}\notag\\ &=\dfrac {1}{C_d}{t_2^*}=\left (\dfrac {1}{C_d}\right )\dfrac {t}{T_2^{\textit{ref}}}. \end{align}

Here, $t^*$ is a normalised time, whose role is to facilitate the convergence of the microparticle dynamics equation with varying $N_{\textit{mag}}$ and $C_d$ under different reference time scales, i.e. (3.18). Conversely, a mapping relation between the physical time $t$ under different reference time scales and the normalised time $t^*$ can also be established from (3.19), i.e.

(3.20) \begin{align} \dfrac {t}{T_1^{\textit{ref}}}=\left (\dfrac {C_d}{N_{\textit{mag}}}\right )\boldsymbol{\cdot }t^*,\quad \quad \dfrac {t}{T_2^{\textit{ref}}}={C_d}\boldsymbol{\cdot }t^*. \end{align}

For a given $t^*$ , the physical time $t$ under $T_1^{\textit{ref}}$ will be shortened as $N_{\textit{mag}}$ increases but prolonged as $C_d$ increases. When $T_2^{\textit{ref}}$ is adopted as the reference time scale, the physical time required to reach $t^*$ remains independent of $N_{\textit{mag}}$ but is proportional to $C_d$ .

Figure 2.

Flow patterns induced by the relative motion between the two microparticles: (a) an approaching movement caused by the attractive magnetic dipole interaction, and (b) a separating movement resulting from the repulsive magnetic dipole interaction.

Additionally, the two microparticles will approach or separate from each other under the attractive or repulsive magnetic dipole interaction, leading to the carrier fluid being squeezed or refilled within the interparticle gap, as demonstrated in figure 2. When the global carrier flow field is quasi-stationary, it can be assumed that the local fluid velocity ${u}_{\!f}$ is proportional to the microparticle velocity ${u}_p$ , owing to the hydrodynamic viscous effect. Establishing a precise mapping relation between ${u}_{\!f}$ and ${u}_p$ is significantly challenging, but can be generally expressed as

(3.21) \begin{align} u_{\!f}\!\propto u_p=\mathscr{M}\boldsymbol{\cdot }\left (\dfrac {R}{T_2^{\textit{ref}}}\right )=\mathscr{M}\boldsymbol{\cdot }\left (\dfrac {\mu _0 \chi ^2{H_0}^2}{\rho _{{\kern-1pt}f}\kern-1pt \nu _{\!f}{\kern-1pt}/{\kern-1pt}R}\right )\!, \end{align}

where the microparticle velocity ${u}_p$ is characterised by the radius $R$ and the response time $T_2^{\textit{ref}}$ of the microparticle, and $\mathscr{M}$ denotes the mapping coefficient. Substituting (3.21) into (3.18) yields

(3.22) \begin{align} \dfrac {3}{4\pi }\dfrac {\partial \boldsymbol{x}^*_i}{\partial t^*}=\dfrac {3}{4\pi }C_d \boldsymbol{\cdot }\mathscr{M}\boldsymbol{\cdot }\boldsymbol{n}^f+\boldsymbol{ I}^D. \end{align}

Then, the spatial trajectory of the microparticle suspended in a quasi-stationary liquid can be described by

(3.23) \begin{align} \boldsymbol{x}^*_i=\int _{0}^{\tau ^*}\left (C_d \boldsymbol{\cdot }\mathscr{M}\boldsymbol{\cdot }\boldsymbol{n}^f+\dfrac {4\pi }{3}\boldsymbol{I}^D \right ){\rm d} t^*, \end{align}

where $\tau ^*$ denotes the dimensionless aggregation time at which the two microparticles come into physical contact. Although the effect of external magnetic fields is not explicitly presented in (3.23), it is important to note that this effect has been implicitly incorporated into the dimensionless time $t^*$ , as shown in (3.13) and (3.19). The absence of the external magnetic field results in $t^*$ or ${\rm d}t^*$ equalling zero. It indicates that the spatial trajectory $\boldsymbol{x}^*_i$ becomes independent of $t^*$ , and these microparticles will not undergo any motion in microchannels.

Figure 3.

Schematic of microparticle dynamics in the Eulerian and Lagrangian frameworks as characterised by different reference time scales.

Several key points concerning the magnetic field-induced dynamics of microparticles suspended in a quasi-stationary liquid within a microchannel can be concluded from (3.18)–(3.23). First, since the solution of the first-order differential equation (i.e. (3.22)) exhibits self-similarity, the spatial trajectories of microparticles mainly depend on the microchannel wall interaction reflected by $C_d$ , as well as their initial positions $\boldsymbol{ I}^D(t^*=0)$ , including the initial inter-particle distance $\boldsymbol{r}_{\textit{ij}}(t^*=0)$ and the initial orientation relative to the external magnetic field $\boldsymbol{e}_H\boldsymbol{\cdot }\boldsymbol{e}_{\textit{ij}}(t^*=0)$ . The strength of the external magnetic field, characterised by $N_{\textit{mag}}$ , has little effect on the spatial trajectories of microparticles under fixed initial positions and microchannel wall interactions.

Second, when $T_1^{\textit{ref}}$ serves as the reference time scale, the dynamics of microparticles is analysed in the framework of the Eulerian (fixed) coordinates. The corresponding dimensionless governing equation reflects the effects of the external magnetic field strength $N_{\textit{mag}}$ and the microchannel wall interaction $C_d$ on the spatial trajectories and the aggregation time $\tau$ of microparticles. The aggregation time $\tau$ exhibits an inverse linear relationship with $N_{\textit{mag}}$ , as demonstrated in (3.20). In other words, a $K_m$ -fold increase in the value of $N_{\textit{mag}}$ results in a corresponding decrease by a factor of $K_m$ in the aggregation time $\tau$ .

Third, when $T_2^{\textit{ref}}$ serves as the reference time scale, the dynamics of microparticles is discussed in the framework of the Lagrangian (moving) coordinates. The associated dimensionless governing equation remains unaffected by the external magnetic field strength $N_{\textit{mag}}$ . This property enables a straightforward comparison between the numerical results of microparticle trajectories presented in this work and the benchmark data from the existing literature, even when the values of $N_{\textit{mag}}$ are not identical.

The theoretical prediction inferred from the second and third points can be visualised in figure 3. Since the same characteristic length and different reference time scales are adopted in the dimensionless analysis, it can be considered that there is no transformation between the spatial dimensions (i.e. the $x$ - and $y$ -axes) in the two frameworks, but there exists a scaling in the temporal dimension (i.e. the $t$ -axis). Relative to the fixed Eulerian framework, the Lagrangian framework moves at the velocity of the microparticle (i.e. $R/T_2^{\textit{ref}}$ ) along the $t$ -axis. In the Eulerian framework, the time required for a microparticle to travel a characteristic length is $T_1^{\textit{ref}}$ and the time for the microparticle to cover the same distance in the Lagrangian framework is $T_2^{\textit{ref}}$ . In others, the time taken by the microparticle to move from the initial position to the final position in the Lagrangian framework is $T_2^{\textit{ref}}/T_1^{\textit{ref}}=1/N_{\textit{mag}}$ times that in the Eulerian framework. Therefore, the magnetic number $N_{\textit{mag}}$ for the problem under investigation can also represent the scaling in the temporal dimension.

Finally, it can be observed from (3.20) that the aggregation time $\tau$ is proportional to the net drag coefficient $C_d$ , which is also a function of the particle–wall separation distance $R/l$ . As the microparticle approaches the microchannel wall, the particle–wall separation distance $R/l$ increases and the net drag coefficient $C_d$ also increases, consequently leading to an extended aggregation time.

These theoretical predictions offer valuable insight into the spatial trajectory and aggregation time of microparticles within a microchannel when exposed to an external magnetic field, and are validated by a series of 3-D LBM simulations. Although some of them are comparable to those reported in prior studies on 2-D circular particles, it is crucial to note that several critical assumptions incorporated in the theoretical analysis and their underlying physical mechanisms display substantial differences. For example, previous studies (Klingenberg et al. Reference Klingenberg, van Swol and Zukoski1989; Hass Reference Hass1993) directly overlooked the effect of the carrier flow on the dynamic behaviour of microparticles by setting $Re$ to 0. But the present work argues that the local flow becomes particularly significant in a confined microchannel, resulting in a non-zero local Reynolds number. Even though the Reynolds number vanished in (3.22), this does not imply that its contributions are directly neglected. Instead, it is implicitly absorbed into the term of $C_d\boldsymbol{\cdot }\mathscr{M}$ , and exerts a modulating effect on the spatial trajectories of microparticles, which can be verified by the numerical results presented in § 5.2.

Figure 4.

The sketch of two magnetic microparticles suspended in a quasi-stationary flow under a uniform external magnetic field applied along the $x$ -direction.

5.1. Effect of external magnetic field strength on microparticle dynamics

The effect of an external magnetic field on the dynamic behaviour of microparticles within a microchannel is explored in this subsection, with a focus on the effect of external magnetic field strength. Once the physical properties of the carrier fluid are specified, the value of $N_{\textit{mag}}$ given by (3.15) can serve as an indicator for the strength of the applied magnetic field. In the benchmark problem of § 4, the density $\rho _{f}$ or kinematic viscosity $\nu _{\!f}$ of the carrier fluid is not clarified, making it impossible to determine the value of $N_{\textit{mag}}$ . Here, we assume the carrier fluid is a non-magnetic and Newtonian liquid with a constant density of $2.13\times 10^3\,\rm kg\,m^{-3}$ , resulting in $N_{\textit{mag}}$ for this problem being estimated as $N_{\textit{mag}}^0=0.0015$ . This assumption is intended exclusively to serve as a reference for parameter settings and to assist readers in understanding the physical problem, without affecting the subsequent analyses and discussions.

Figure 7.

Spatial trajectories of two microparticles inside a microchannel remain unaffected by the applied magnetic field strengths, characterised by $N_{\textit{mag}}$ . The initial orientations of two microparticles: (a) $\theta =0^\circ$ , (b) $\theta =20^\circ$ , (c) $\theta =45^\circ$ , (d) $\theta =54.7^\circ$ and (e) $\theta =80^\circ$ . The dashed and solid circles represent the initial and final positions, respectively.

Taking $N_{\textit{mag}}^0$ as the base, three external magnetic fields with various strengths are applied along the $x$ -direction in the LBM simulations of this subsection. The corresponding $N_{\textit{mag}}$ are set to $200 N_{\textit{mag}}^0$ , $400 N_{\textit{mag}}^0$ and $800 N_{\textit{mag}}^0$ , with the amplification factors $K_m$ of 200, 400 and 800, respectively. Except for the magnetic field strength $H_0$ serving as a variable, all other parameters involved in $N_{\textit{mag}}$ are retained from § 4. These microparticles are initially located at the central horizontal plane, with a particle–wall separation distance $(R/l)$ of $1/5$ , thereby maintaining a constant effect of the microchannel wall interaction on the microparticle dynamics.

Figure 7 illustrates that the spatial trajectories of the two microparticles initially align along different orientations $\theta$ under a uniform external magnetic field with varying $N_{\textit{mag}}$ . It can be observed from figures 7(a) to 7(e) that the motions of microparticles exhibit distinct differences depending on their initial orientations relative to the external magnetic field. However, the spatial trajectories of the microparticles originating from the same initial position tend to converge when the strength of the external magnetic field varies with $N_{\textit{mag}}$ Klingenberg et al. (Reference Klingenberg, Ulicny and Golden2007). This phenomenon may be attributed to the fact that these microparticles suspended in a stationary liquid are set in motion when driven by the magnetic dipole force and subsequently undergo relative motion with the surrounding fluid. It causes the microparticles to experience a hydrodynamic force, whose direction is always opposite to both the direction of relative motion and the magnetic dipole force. The active driving–passive relationship generated between the magnetic dipole force and hydrodynamic force is clearly illustrated in figure 12(c). Owing to this relationship, the spatial trajectory of microparticles within stationary liquids is dominated by the magnetic dipole force, whose acting path is dependent on the initial position of microparticles. This is the reason why microparticles originating from the same initial positions still exhibit convergent spatial trajectories under external magnetic fields of varying strengths.

Figure 8.

Aggregation time of two microparticles is inversely proportional to the strength of the external magnetic field, characterised by $N_{\textit{mag}}$ . Time histories of the inter-particle distance normalised by (a) $R^2/\nu _{\!f}$ , (b) $(R^2/\nu _{\!f})/K_m$ and (c) $\rho _{f}\nu _{\!f} /(\mu _0\chi ^2H_0^2)$ .

Figure 8 shows the inter-particle distance $||\boldsymbol{ X}_1-\boldsymbol{X}_2||$ between the two microparticles as a function of the time $t$ normalised by different reference time scales. As demonstrated in (3.20), the physical time $t$ required for microparticles to come into contact from an identical initial position is linearly proportional to $1/N_{\textit{mag}}$ , when $T_1^{\textit{ref}}$ works as the reference time scale. When $N_{\textit{mag}}$ is increased by 200, 400 and 800 times relative to the basic value $N_{\textit{mag}}^0$ (i.e. $K_m=200$ , $400$ and $800$ ), it can be observed from figure 8(a) that the aggregation time of the two microparticles is significantly shortened with the increase of the $N_{\textit{mag}}$ . As mentioned above, the motion of microparticles is governed by a resultant force composed of the magnetic dipole force and hydrodynamic force. Since the magnetic number $N_{\textit{mag}}$ functions as an index that characterises the ratio between the two forces, the magnitude of the resultant force is proportional to the value of $N_{\textit{mag}}$ . Thus, it can be logically inferred that when $N_{\textit{mag}}$ is relatively larger, the resultant force exerted on the microparticles is correspondingly greater, and then the time required for a microparticle to cover a specific distance is shorter.

To further quantify this reduction in aggregation time, the time histories of the inter-particle distance between the two microparticles are normalised by $T_1^{\textit{ref}}/K_m$ and compared in figure 8(b). It is evident that these time history curves associated with various $N_{\textit{mag}}$ converge into a unified consequence, suggesting a linear correlation between the aggregation time and the reciprocal of $N_{\textit{mag}}$ (i.e. $1/N_{\textit{mag}}$ ). Furthermore, no significant deviation can be observed among the time history curves of $||\boldsymbol{X}_1-\boldsymbol{X}_2||$ when normalised by $T_2^{\textit{ref}}$ across varying values of $N_{\textit{mag}}$ , as illustrated in figure 8(c).

In light of the above results, two key points need further clarification. On the one hand, the denominator of $N_{\textit{mag}}$ characterises the inherent properties of the microparticle and the carrier fluid, whereas its numerator represents the strength of the external magnetic field. Given that this section is intended to examine the effect of the magnetic field strength on the dynamic behaviour of microparticles, the effects of particle size and fluid viscosity are not explored in depth. Accordingly, the aforementioned conclusion should be more precisely stated as the fact that the aggregation time is inversely proportional to $\mu \chi ^2H_0^2$ . It should be noted, however, that the aggregation behaviour of the microparticles inside microchannels is not only governed by $N_{\textit{mag}}$ , but is also closely related to other dimensionless parameters, such as the particle–wall separation distance $R/l$ and initial particle position $\boldsymbol{I}^D$ (as a function of initial interparticle distance $r_{\textit{ij}}/R$ and initial orientation $\theta$ ). When assessing the effect of magnetic field strength on aggregation time, in principle, other dimensionless parameters should be maintained consistent. Otherwise, the results and associated conclusions will not be comparable. For this reason, the magnetic number $N_{\textit{mag}}$ is adopted to characterise the strength of the applied magnetic field in this work.

On the other hand, although the independence of the microparticle spatial trajectory from the magnetic field strength has been reported in previous work (Klingenberg et al. Reference Klingenberg, Ulicny and Golden2007), this work aims to emphasise that this conclusion is valid only under a specific prerequisite in the application of microfluidic chips. Most previous studies focused on dilute particulate flow in an unbounded domain, where only the interaction between the boundary layers of microparticles was considered. Such an interaction occurs only before particle collision, and its effect is negligible. Therefore, this result appears to hold unconditionally. But in microfluidic chip applications, an additional interaction between the microparticle boundary layer and the microchannel boundary layer needs to be considered, and its effect persists throughout the entire motion, as illustrated in figure 6(b). When the distance between microparticle and microchannel wall is larger than $5R$ (i.e. $R/l\le 1/5$ ), there is no interaction between the microparticle boundary layer and the microchannel boundary layer. In such a case, the dynamic behaviour of microparticles should be comparable to that in an unbounded flow. And when the distance between microparticle and microchannel wall is reduced to $3R$ (i.e. $R/l\ge 1/3$ ), a significant interaction occurs between boundary layers, which induces a non-negligible modulation on the microparticle dynamics. Therefore, for microfluidic chip applications, the independence of the spatial trajectory from the magnetic field, as well as the linear relationship between the aggregation time and the magnetic number $N_{\textit{mag}}$ , must be predicted under the assumption that the particle–wall interaction within the microchannel remains constant. The effect of microchannel wall interaction on microparticle dynamics will be discussed in § 5.2, and these conclusions will be verified further.

5.2. Effect of microchannel wall interaction on microparticle dynamics

The effect of the microchannel wall interaction on the dynamics of microparticles is explored here by varying the particle–wall separation distance ( $R/l$ ) in the LBM simulations. Specifically, the separation distance between the upper and lower microchannel walls ranges from $4R$ to $10R$ , and the microparticles remain initially located on the central horizontal plane of the microchannel, resulting in the particle–wall separation distance of $R/l\in [1/5,1/2]$ . The magnetic number $N_{\textit{mag}}$ in this subsection is fixed to $800N_{\textit{mag}}^0$ , and other parameters are consistent with those employed in § 4.

Figure 9.

Vortex migrates towards the surfaces of microparticles as the particle–wall separation distance $R/l$ increases: (a) $R/l=1/5$ , (b) $R/l=1/4$ , (c) $R/l=1/3$ and (d) $R/l=1/2$ . The hexagram symbols in (d) represent the centre of the vortex.

Figure 9. Long description

Panel A: A 3D diagram showing magnetic field lines around two interacting objects. The objects are represented as spheres, one orange and one blue. The field lines are depicted in various colors, with blue and red indicating different directions of the magnetic field. The top view shows the field lines from above, the right side view from the side, and the front view from the front. Panel B: Another 3D diagram similar to Panel A but with a different orientation of the objects. The field lines and views are similarly depicted, showing the interaction of the magnetic fields from different angles.

Taking the two microparticles initially aligned along the orientation of $\theta = 54.7^\circ$ as an example, a visualisation analysis of the flow field within the microchannel is performed to demonstrate the effect of microchannel wall interaction on the hydrodynamic behaviours of the carrier fluid. Figure 9( a–d ) illustrates the streamline distributions during the relative motion of the two microparticles in microchannels with the thicknesses of $2l=10R$ , $8R$ , $6R$ and $4R$ , respectively, corresponding to the particle–wall separation distances of $1/5$ , $1/4$ , $1/3$ and $1/2$ . Meanwhile, the cross-sectional views of the flow field on the $x{-}y$ (top view), $x{-}z$ (front view) and $y{-}z$ (right side view) central planes are presented, offering a comprehensive depiction of fluid behaviour in 3-D space. Due to the motion of microparticles between the two parallel walls and the viscous effects of the carrier fluid, it can be observed from the front and right side views of figures 9(a) and 9(b) that some vortices in the wall-normal direction are induced when the spacing between the upper and lower walls is sufficiently large (i.e. $R/l=1/5$ and $1/4$ ). As the thickness of the microchannel $2l$ further decreases to $6R$ and $4R$ (i.e. $R/l=1/3$ and $1/2$ ), these vortices in the wall-normal direction are significantly suppressed and ultimately eliminated, which can be observed from the front and right side views of figures 9(c) and 9(d).

Additionally, due to the magnetic dipole interaction, the two microparticles move toward each other and displace the intervening liquid, thereby resulting in upward and downward flows and forming two small vortices in the gap between the two microparticles. When encountering the obstructions from the upper and lower microchannel walls, the upward and downward flows alter their directions and propagate horizontally along the microchannel walls, consequently generating two larger vortices on the left-hand side of the microparticle heading direction. Such hydrodynamic characteristics can be observed in the top view of each subfigure in figure 9. By comparing the four top views presented in figures 9(a) to 9(d), it can be found that the centres of vortices on the horizontal plane move closer to the microparticles as the microchannel thickness decreases. It indicates that as the microparticle moves closer to the microchannel wall, the flow field in the vicinity of the microparticle surface becomes increasingly complex, and the hydrodynamic viscous effects acting on the microparticles become more significant.

Figure 10.

Modulation effect of microchannel wall interaction on the microparticle spatial trajectories due to vortex migration with $R/l$ . The initial orientations of two microparticles: (a) $\theta =0^\circ$ , (b) $\theta =20^\circ$ , (c) $\theta =45^\circ$ , (d) $\theta =54.7^\circ$ and (e) $\theta =80^\circ$ .

In figure 10, the spatial trajectories of the two microparticles in microchannels with different particle–wall separation distances $R/l$ are compared under an external magnetic field of $N_{\textit{mag}}=800N_{\textit{mag}}^0$ . For $\theta =0^\circ$ , the two microparticles experience a centre-to-centre attractive dipole force along the external magnetic field direction, and the flow pattern of the carrier fluid is also symmetric with respect to their central connecting line, rendering the effect of the microchannel wall interaction on the spatial trajectories of the two microparticles negligible, as shown in figure 10(a). When the initial orientation $\theta$ of the two microparticles varies in the range of $0^\circ \lt \theta \le 54.7^\circ$ , the magnetic dipole force between them remains attractive throughout the entire motion. As shown in figures 10(b) to 10(d), variations in the particle–wall separation distance $R/l$ lead to slight deviations on the spatial trajectories of the microparticles, with the magnitude of the order of $10^{-3}\sim 10^{-2}R$ . The microparticles suspended in a microchannel with a larger thickness experience a weaker hydrodynamic viscous effect, which easily alters their motion when driven by the magnetic dipole force. Consequently, they can exhibit a more sensitive dynamic response to the attractive magnetic dipole force, thereby leading to their spatial trajectories with a smaller radius of curvature. This can also be interpreted as follows. The hydrodynamic viscous effect, which is enhanced as $R/l$ increases, is equivalent to exerting a damping effect on the microparticle motion. As a result, the particles tend to preserve their original motion trajectories and are relatively resistant to alteration by the magnetic dipole force.

For the initial orientation $\theta =80^\circ$ , the magnetic dipole force between the two microparticles is repulsive at the onset of their motion. Subsequently, as the relative orientation of the two microparticles with respect to the external magnetic field direction evolves during their motion, the magnetic dipole force shifts from a repulsive to an attractive interaction. It can be found from figure 10(e) that the microparticles in a microchannel with a larger thickness tend to be transported to a more distant position when the repulsive dipole force dominates their motions. When the magnetic dipole force turns into an attractive force, the spatial trajectory of these microparticles exhibits a similar trend as that illustrated in figure 10( b–d ). In some engineering applications, such as biomedical particle sorting, it is expected that microparticles move along a predefined track in microchannels. But such a wall-induced vortex effect may lead to deviations in the spatial trajectories of microparticles, thereby reducing the sorting accuracy. To minimise this negative effect, two potentially feasible suggestions are provided here. One potential approach is to implement a specialised microstructure in a microfluidic chip, which allows microparticles to pass through microchannels in a sparse, linear and uniform manner, thereby reducing the interparticle interactions and suppressing the generation of complex local flows. Another possibility is to design microchannels with a larger cross-sectional area, thereby achieving a larger separation distance between the particles and the microchannel wall with a smaller $R/l$ , and weakening their hydrodynamic interaction. This helps keep these vortices away from microparticles, thereby reducing their effects on the spatial trajectories of microparticles.

Figure 11.

(a) Aggregation time $\tau$ increases with particle–wall separation distances $R/l$ due to enhanced viscous drag from microchannel wall interactions, and (b) the curve fitting results of the aggregation time $\tau$ .

Figure 11(a) compares the time histories of the inter-particle distance $||\boldsymbol{X}_1-\boldsymbol{X}_2||$ between the two microparticles suspended in microchannels with varying particle–wall separation distances $R/l$ . When the spacing between the microparticles and the microchannel wall is relatively large (i.e. $R/l=1/5$ and $1/4$ ), the weaker hydrodynamic viscous effect from the carrier fluid enables the two microparticles departing from the same initial position to aggregate together within a shorter time. When $R/l$ is increased to $1/3$ and $1/2$ , the two microparticles experience a stronger hydrodynamic viscous effect due to the enhanced microchannel wall interaction, resulting in an extended aggregation time.

To further quantitatively evaluate the aggregation behaviour of microparticles inside microchannels, the time required for the two microparticles to move from their initial positions until they reach physical contact is defined as the aggregation time and denoted by $\tau$ . It is clearly illustrated that (3.23), the dynamics of the microparticle in a microchannel is governed by the initial position $\boldsymbol{I}^D$ , the strength of the applied magnetic field $N_{\textit{mag}}$ and the microchannel wall interaction $C_d$ . Furthermore, $\boldsymbol{I}^D$ is a function of initial interparticle distance $r_0/R$ and the initial orientation $\theta$ , while $C_d$ is related to the particle–wall separation distance $R/l$ . Therefore, this work tends to establish a predictive model for the aggregation time $\tau$ , which can be characterised by $r_0/R$ , $\theta$ , $N_{\textit{mag}}$ and $R/l$ .

A similar predictive model for the aggregation time $\tau$ of 2-D circular particles has been proposed by Banerjee et al. (Reference Banerjee, Bit, Ganguly and Hardt2012), in which the hydrodynamic viscous effect is evaluated by the Stokes law and the microchannel wall interaction is not considered. For actual microparticles in a microfluidic chip, the prediction from their model could maintain a qualitative consistency with the motion of microparticles. But it is challenging to quantitatively describe their dynamic behaviours in 3-D space. Meanwhile, Banerjee et al. (Reference Banerjee, Bit, Ganguly and Hardt2012) pointed out that the aggregation time $\tau$ follows a power relation with the normalised initial interparticle distance $r_0/R$ , i.e. $\tau \sim (r_0/R)^5$ . This result is preserved in the present model and will not be further studied in the subsequent discussion.

The mapping relation between the aggregation time $\tau$ and the particle–wall separation distance $R/l$ , as well as the initial orientation $\theta$ , is illustrated in figure 11(b)- $\unicode {x2460}$ , where the y-axis is presented on a logarithmic scale. The result suggests that the aggregation time $\tau$ exhibits a significant dependence on the initial orientations $\theta$ of the two microparticles for any given $R/l$ . And the aggregation time $\tau$ follows a consistent trend as a function of the particle–wall separation distance $R/l$ , regardless of the initial orientation $\theta$ . Thus, it can be inferred that the effects of the initial orientation $\theta$ and the particle–wall separation distance $R/l$ on the aggregation time $\tau$ are independent. Additionally, a linear inverse proportionality between the aggregation time and the external magnetic field strength characterised by $N_{\textit{mag}}$ has already been confirmed in § 5.1.

Given all that, a reasonable assumption is proposed that the aggregation time $\tau$ of the microparticles within a microchannel is independently governed by $N_{\textit{mag}}$ , $\theta$ and $R/l$ . Their contributions to the aggregation time $\tau$ are denoted as $\tau _H$ , $\tau _\theta$ and $\tau _l$ , respectively. Consequently, the aggregation time $\tau$ can be modelled as

(5.1) \begin{align} \tau /\left (r_0/R\right )^5\sim \tau _\theta \boldsymbol{\cdot }\tau _l \boldsymbol{\cdot }\tau _H. \end{align}

If both sides of (5.1) are divided by $\tau _H$ , the aggregation time $\tau$ becomes independent of the external magnetic field strength. It is already known that the dynamic behaviour of microparticles remains unaffected by the external magnetic field strength when $T_2^{\textit{ref}}$ is adopted as the reference time scale, as illustrated in figure 8(c). Thus, the contribution of the external magnetic field strength $\tau _H$ can be represented by $T_2^{\textit{ref}}$ as

(5.2) \begin{align} \tau _H\sim T_2^{\textit{ref}}=\dfrac {\rho _{f} \nu _{\!f}}{\mu _0 \chi ^2H_0^2}. \end{align}

Furthermore, the contributions of the initial orientation $\tau _\theta$ and the microchannel wall interaction $\tau _l$ can be analytically expressed via the curve fittings as

(5.3) \begin{align} \tau _\theta =0.3433\times \varOmega ^{-2}+0.2787, \end{align}

and

(5.4) \begin{align} \tau _l=6\pi \left [19.89-1.83\left (\dfrac {R}{l}\right )-86.05\left (\dfrac {R}{l}\right )^3+101\left (\dfrac {R}{l}\right )^4\right ]^{-1}, \end{align}

where $\varOmega$ and $R/l$ serve as the fitting model variables for (5.3) and (5.4), respectively. Here, $\varOmega$ is the radian of the orientation $\theta$ , following a relation as $\varOmega =(90^\circ -\theta )\pi /180^\circ$ . The curve fitting results for $\tau _\theta$ and $\tau _l$ are presented in figure 11(b)- $\unicode {x2461}$ and $\unicode {x2462}$ , which are in good agreement with the LBM numerical simulation results. By substituting (5.2)–(5.4) into (5.1), a predictive model for the aggregation time $\tau$ can be developed as a function of the initial orientation $\theta$ , the microchannel wall interaction characterised by $R/l$ and the external magnetic field strength associated with $T_2^{\textit{ref}}$ or $N_{\textit{mag}}$ , which can be formulated by

(5.5) \begin{align} \dfrac {\tau }{\left (r_0/R\right )^5}= &{\dfrac {\rho _{f} \nu _{\!f}}{\mu _0 \chi ^2H_0^2}\boldsymbol{\cdot }\dfrac {6\pi \left (0.3433\times \varOmega ^{-2}+0.2787\right )}{\left [19.89-1.83\left (\dfrac {R}{l}\right )-86.05\left (\dfrac {R}{l}\right )^3+101\left (\dfrac {R}{l}\right )^4\right ]}}\notag\\ =&{\dfrac {R^2}{\nu _{\!f} N_{\textit{mag}}}\boldsymbol{\cdot }\dfrac {6.4711\boldsymbol{\cdot }\varOmega ^{-2}+5.2534}{\left [19.89-1.83\left (\dfrac {R}{l}\right )-86.05\left (\dfrac {R}{l}\right )^3+101\left (\dfrac {R}{l}\right )^4\right ]}}. \end{align}

It must be recognised that, even though the predictive model exhibits a good consistency with the LBM simulation results, there remains a significant gap to bridge before it becomes a mature model for practical engineering applications. Some limitations of the present model should be realised and addressed in future work. First, the problem investigated in this work is a two-particle system, and the relevant results can be extended to dilute particulate flows. However, for multi-particle systems or dense particulate flows, particles will experience more active dipole–dipole interaction, more complex local flows induced by particle motions and frequent interparticle collisions. The applicability of the present predictive model requires a systematic verification. Secondly, two microparticles in this work are symmetrically distributed in microchannels, indicating that the net force along the wall-normal direction is zero. Nevertheless, if they are asymmetrically located in microchannels, a lateral force will be induced and act upon these microparticles, leading to an alteration in their spatial trajectories and aggregation times. Thirdly, the carrier fluid considered in this work is assumed to be a Newtonian liquid with constant rheological properties. For some non-Newtonian fluids, their rheological properties may be altered in response to the applied magnetic field, which undoubtedly poses a challenge for predicting the dynamic behaviour of particles in these liquids. Finally, the issue considered in this work is isothermal. For certain engineering applications involving heat transfer phenomena, the effect of thermal fluctuations needs to be taken into account.