2.1. Problem specification

A rigid sphere of radius $a$ is suspended in a linear shear flow with the origin of the Cartesian coordinate system located at the centre of the sphere (figure 2a). The coordinate unit vectors are $\boldsymbol {e}_{x}$, $\boldsymbol {e}_{{y}}$ and $\boldsymbol {e}_{z}$. A no-slip wall is placed at distance $(0, -l, 0)$ away from sphere centre. Outer boundaries are located at large distances $L(\gg l)$ away from the sphere centre to minimise any secondary boundary effects. For this study ${l/a}$ is varied from 1.2 to 9.5 to obtain the lift and drag force variation as a function of particle distance from the wall.

Figure 2. $(a)$ Schematic of a translating sphere of radius $a$ moving at velocity $u_{p}$ in a wall-bounded linear shear flow. $(b)$ The domain mesh for a particle located at $l/a=4$.

To obtain the forces generated due to wall and shear effects in the absence of relative motion between the particle and the fluid, the particle slip velocity ${\boldsymbol {u}_{{slip}}}$ is explicitly set to zero: That is,

(2.1)\begin{equation} \boldsymbol{u}_{{slip}} =\boldsymbol{u}_{{p}}-\boldsymbol{u}_{{f}}(y=0) = \boldsymbol{0}, \end{equation}

where $\boldsymbol {u}_{{p}}=u_{p}\boldsymbol {e}_{{x}}$ is the particle velocity and $\boldsymbol {u}_{{f}}$ is the undisturbed fluid velocity defined as

(2.2)\begin{equation} \boldsymbol{u}_{{f}} =\dot{\gamma}(y+l) \boldsymbol{e}_{{x}}. \end{equation}

Note that under this formulation the particle is constrained to translate only in the $x$ direction with particle velocity $u_{p}=\dot {\gamma }l$.

To determine the forces acting on the particle, we solve the steady-state Navier–Stokes equations in a frame of reference that moves with the particle (Batchelor Reference Batchelor1967). Specifically, we solve

(2.3)\begin{gather} \boldsymbol{\nabla} \boldsymbol{\cdot} \rho\boldsymbol{u'} = 0, \end{gather}

(2.4)\begin{gather} \boldsymbol{\nabla} \boldsymbol{\cdot} (\rho\boldsymbol{u'}\boldsymbol{u'}+\boldsymbol{\sigma}) = 0, \end{gather}

where $\boldsymbol {u'} = \boldsymbol {u} - \boldsymbol {u}_{{p}}$ and $\boldsymbol {u}$ is the local fluid velocity. The boundary conditions used in the moving frame of reference are

(2.5)\begin{align} \boldsymbol{u'}=\begin{cases} \displaystyle \dot{\gamma}(y+l)\boldsymbol{e}_{{x}}-\boldsymbol{u}_{p}, & y=+\infty;\ y=-l;\ x,z={\pm}\infty,\\ \displaystyle \boldsymbol{\omega}\times\boldsymbol{r}, & |\boldsymbol{r}|=a, \end{cases}\end{align}

where $\boldsymbol {r}$ is a radial displacement vector pointing from the sphere centre to the particle surface and $\boldsymbol {\omega }$ is the angular rotation of the particle.

The fluid is assumed to be Newtonian with a dynamic viscosity $\mu$ and density $\rho$. The total stress tensor ($\boldsymbol {\sigma }= p\boldsymbol {I}+\boldsymbol {\tau }$) (Bird, Stewart & Lightfoot (Reference Bird, Stewart and Lightfoot2002) sign convention), is computed using the fluid pressure, $p$ and viscous stress tensor, $\boldsymbol {\tau } = -\mu (\boldsymbol {\nabla } \boldsymbol {u'} + \boldsymbol {\nabla } \boldsymbol {u'}^{\text {T}})$. The forces acting on the particle are evaluated by integrating the total stress contributions around the particle surface $S$

(2.6)\begin{equation} \boldsymbol{F}_{{p}} = -\int_S \boldsymbol{n}\boldsymbol{\cdot}{\boldsymbol{\sigma}}\,\textrm{d}S. \end{equation}

Here, $\boldsymbol {n} (=\boldsymbol {\hat {r}})$ is a unit normal vector directed out of the particle. The drag $({F_{D}} = {\boldsymbol {F}_{{p}}}\boldsymbol {\cdot }\boldsymbol {e}_{{x}})$ and lift $(F_{L}=\boldsymbol {F}_{{p}}\boldsymbol {\cdot }\boldsymbol {e}_{{y}})$ are defined as the fluid forces acting on the sphere in ${+}x$ and ${+}y$ directions, respectively. Similarly the net torque $\boldsymbol {T}_{{p}}$ acting on the particle is evaluated using

(2.7)\begin{equation} \boldsymbol{T}_{{p}} = -\int_S \boldsymbol{r} \times \boldsymbol{\sigma}\boldsymbol{\cdot}\boldsymbol{n}\,\textrm{d}S. \end{equation}

In this study, two cases are considered: in the first, the particle is constrained from rotating, whereas, in the second, the particle is allowed to freely rotate about the $z$ axis, at an angular velocity $\omega _{p}$. For the non-rotating (first) case all components of the rotation $\boldsymbol {\omega }$ are explicitly set to zero, whereas for the (second) freely rotating case the $z$ component of the net torque $\boldsymbol {T}_{{p}}$ is explicitly set to zero and the $z$ component of $\boldsymbol {\omega }$ ($\boldsymbol {\omega } \boldsymbol {\cdot } \boldsymbol {e}_{{z}} = \omega _{p}$) is solved for as an unknown (with other components of $\boldsymbol {\omega }$ set to zero).

Unless stated otherwise, the results in the remainder of this study are presented in non-dimensional form (indicated by an asterisk) using a length scale $a$, velocity scale $\gamma a$ and force scale $\mu ^2/\rho$.

2.2. Numerical approach

The system of equations is solved using the finite volume package arb (Harvie Reference Harvie2010) over a non-uniform body-fitted structured mesh that is generated with gmsh (Geuzaine & Remacle Reference Geuzaine and Remacle2009) (figure 2b). The sphere surface is resolved with $N_{{p}}$ mesh points on each curved side length of a cubed sphere. This results in $6(N_{{p}} - 1)^2$ cells on the sphere surface; An $(N_{{p}}-1)$ number of inflation layers with an $\alpha$ geometric progression ratio around the sphere are used to capture gradients in velocity occurring near the particle wall. Outer boundaries, except the bottom wall located at $y^{\ast }=-l^{\ast }$, are placed at a distance $L^{\ast }$ from the sphere centre in all directions. To resolve the far field of the domain (excluding the inflation layers), $N_{{d}}$ points are used with a geometric progression $\beta$ expanding towards the domain boundaries. For larger separation distances, ${l^{\ast }} > 1.2$, ($N_{{w}}-1$) layers are introduced in the gap between the bounding box and the bottom wall, with a non-uniform progression of $\beta$ producing more refinement near the sphere and the bottom wall. The total cell count in the mesh is $N_{{t}}$.

The non-dimensional lift force magnitudes measured in this study are significantly low, $|F_{L}^{\ast }| \sim 10^{-6}{-}10^{-1}$, and hence require a high resolution mesh around the sphere to accurately determine the lift coefficient. Further, the lift force is highly sensitive to the cell distribution around the particle, requiring perfect symmetry of the mesh in the $y$ direction to correctly ensure that the lift reduces to zero as inertial effects are reduced. For this reason the symmetry of the sphere surface mesh and its surrounding cells in the bounding box are enforced by employing a ‘Lego’-type construction (figure 9 in the appendix) whereby groups of mesh ‘blocks’ are copied, translated, rotated and joined to construct the full computational domain. For example, a single block mesh ($B1$) is copied and then rotated about the $x=0$ and $y=0$ axes to create identically discretised blocks that surround the sphere ($B1_{x},B1_{y}$ and $B1_{xy}$). Similarly, the upper and lower mesh domain blocks are built using two separate block mesh entities, $B2$ and $B3$ as illustrated in figure 9, which are later rotated about $x=0$ axis. This mesh configuration ensures that we compute the zero lift force on a slip free particle under Stokes conditions down to $F_{L}^{\ast } \sim 10^{-15}$. The shear and slip Reynolds numbers considered in this study are sufficiently small to assume flow symmetry about the $z = 0$ plane. Hence a final domain size of [$-L, L$], [$L, -l$] and [$L, 0$] is used for simulations.

2.2.1. Domain size dependency

The domain size is first tested to determine a suitable choice of ${L^{\ast }}$ such that the lift and drag forces are negligibly affected by this parameter. We perform a series of simulations where ${L^{\ast }}$ is increased from $20$ to $100$. Simulations are performed for seven selected shear Reynolds numbers; $Re_{\gamma } = (1,2,4,6,8) \times 10^{-3}, 10^{-2}$ and $10^{-1}$; and five selected wall distances; ${l^{\ast }} = 1.2, 2, 4, 6$ and $9.5$; at $Re_{{slip}}=0$. The number of mesh points in the domain ($N_{d}$) is systematically increased with ${L^{\ast }}$ while maintaining a constant progression ratio ($\beta$) and a constant minimum cell thickness in the outer domain.

Results for the lift and drag coefficients, $C_{L,1}$ and $C_{D,1}$ (formerly defined in § 3.1) respectively, for the minimum and maximum separation distances (${l^{\ast }} = 1.2$ and $9.5$, respectively) and minimum and maximum shear Reynolds numbers ($Re_{\gamma } = 10^{-3}$ and $10^{-1}$) are shown in table 2. Also shown are the percentage differences ($\delta$) for these coefficients relative to the corresponding values obtained using the maximum domain size (${L^{\ast }}$). We use $\delta$ as an indicator of the coefficient accuracy, noting the limitations of this measure as we approach the maximum domain size.

Table 2. Effect of domain size on drag and lift coefficients for maximum and minimum separation distances (${l^{\ast }} = 1.2$ and $9.5$) and shear Reynolds number ($Re_{\gamma } = 10^{-3}$ and $10^{-1}$) at $Re_{{slip}}=0$. $\delta$ is the percentage error in coefficient, relative to results calculated using the largest domain size (${L^{\ast }}=100$).

For $Re_{\gamma } \geq 10^{-2}$, a domain size of ${L^{\ast }} = 50$ is sufficient to capture all the inertial effects responsible for the lift forces, with $\delta$ less than $1\,\%$ for all separation distances ($1.2 \leq {l^{\ast }} \leq 9.5$) under both non-rotating and freely rotating conditions (additional supporting data not shown). In contrast, for low shear Reynolds numbers, for example the conditions of $Re_{\gamma } = 10^{-3}$, non-rotating (freely rotating) particles and a domain size of ${L^{\ast }} = 50$, an increase of $\delta$ from ${\sim}0.96\,\%$$(0.64\,\%)$ to ${\sim}12\,\%$$(9.15\,\%)$ is observed as the separation distance increases from ${l^{\ast }} = 1.2$ to $9.5$. For smaller separation distances (i.e. ${l^{\ast }} = 1.2$), even at low $Re_{\gamma }$, smaller values for $\delta$ are expected as wall effects dominate outer boundary effects in this near-to-wall regime (Ekanayake et al. Reference Ekanayake, Berry, Stickland, Muir, Dower and Harvie2018). However, when the distance to the wall is large and $Re_{\gamma }$ low, the lift force is quite sensitive to the location of the outer boundary. Noting that the boundary layer thickness around a translating sphere in an unbounded environment is inversely proportional to $\sqrt {Re_{slip}}$ (Dandy & Dwyer Reference Dandy and Dwyer1990), it follows that larger domain sizes are required to minimise outer boundary effects when $Re_{\gamma }$ is small (Dandy & Dwyer Reference Dandy and Dwyer1990). Therefore, for the simulations where $Re_{\gamma } < 10^{-2}$, a larger domain size of ${L^{\ast }} = 100$ is used in this study, compared to the domain size of ${L^{\ast }} = 50$ used for the higher $Re_{\gamma }$ situations. Using this strategy we believe the accuracy of all presented lift coefficients to be better than $1\,\%$, with the possible exception of values calculated using the combination of the lowest $Re_{\gamma }$ and highest ${l^{\ast }}$ values considered.

For drag, the accuracy of the drag coefficient calculation is largely a function of wall separation distance. With the selected domain size of ${L^{\ast }} = 50$ that is used for the $Re_{\gamma } \geq 10^{-2}$ results, $\delta < 1\,\%$ only for ${l^{\ast }} \leq 4$ (additional data not shown). Further away from the wall, a notable dependency on domain size is found, with a $\delta$ variation of ${\sim }5\,\%$$(2\,\%)$ to $24\,\%$$(12\,\%)$ for non-rotating (freely rotating) conditions as ${l^{\ast }}$ increases from $6$ to $9.5$ for $Re_{\gamma } = 10^{-1}$. Results for $Re_{\gamma } = 10^{-3}$ are similar. Also at these larger separation distances, specifically at ${l^{\ast }}= 8$ and $9.5$, small positive drag coefficients are observed for $-C_{D,1}$ when $Re_{\gamma }=10^{-1}$ (noting that results for ${l^{\ast }} = 4, 6, 8$ are not presented in the table). The small positive drag coefficients and larger $\delta$ values found at large separation distances are caused by small errors in $F_{D,1}^{\ast }$ which are amplified when expressed as a relative drag coefficient error, because as the separation distance increases, both the coefficients and the force approach zero. Note, however, that while the drag coefficient errors are higher at the largest separation distances, an $11\,\%$ change in domain size from ${L^{\ast }}=90$ to $100$ results in less than a $1\,\%$ change in drag coefficient (for $Re_{\gamma }=10^{-3}$), suggesting that the results are close to independent of domain size. In summary, while the errors in the calculated drag coefficients are larger than for lift, the largest errors occur under conditions in which the drag force is low. Under conditions in which the drag force is appreciable, the error in the drag force coefficient is similar to that reported for the lift force coefficient.

2.2.2. Mesh dependency

The effect of mesh resolution within the boundary layers surrounding the sphere is examined in this section. The number of inflation layers around the sphere and the number of cells on the sphere surface were adjusted by varying $N_{{p}}$ while maintaining the same progression rate $\alpha$ in the bounding box. Concurrently, $N_{{d}}$ was also changed, maintaining $N_{{d}} = N_{{p}}$ for all cases.

Figure 3 shows the resulting lift coefficients and non-dimensionalised lift forces for four mesh refinement levels around the sphere and for the conditions of ${l^{\ast }}=1.2$, a domain size of ${L^{\ast }}=50$ and a range of shear Reynolds numbers. As $F^{\ast }_{L,1}$ reduces to zero, $C_{L,1}$ asymptotes to different finite values as $Re_{\gamma }$ approaches zero. A considerable variation of $C_{L,1}$ with mesh refinement is observed, particularly at low $Re_{\gamma }$, however, the relative change in $C_{L,1}$ decreases as $N_{p}$ and $N_{d}$ increase. Indeed, increasing $N_{p}$ and $N_{d}$ from $25$ to $30$ only changes $C_{L,1}$ by a small amount, but concurrently increases the total cell count $N_{t}$ from 158 976 to 310 500, significantly increasing computational memory requirements ($\sim 1.2\ \textrm {Tb}$). Noting that for low $Re_{\gamma }$ the force will be negligible anyway, and that across the entire $Re_{\gamma }$ range the difference between the $N_{p}, N_{d}=25$ and $N_{p}, N_{d}=30$ results is small anyway, we employed the mesh with $N_{p},N_{d} = 25$ for the remainder of the study.

Figure 3. Effect of mesh resolution around the sphere on $C_{L,1}$ for a non-rotating particle at ${l^{\ast }}=1.2$. $N_{p},N_{d}$ = 15 (); 20 (); 25 (); 30 ().

3.1. New lift and drag model definitions

We define the lift force on a particle experiencing both slip and shear within a linear shear flow as

(3.1a)\begin{equation} {F_{L}^{\ast}} = C_{L,1}Re_{\gamma}^2+\text{sgn}({\gamma}^{\ast})C_{L,2}Re_{\gamma}Re_{{slip}}+C_{L,3}Re_{{slip}}^2, \end{equation}

which may also be written in the alternative form

(3.1b)\begin{equation} {F_{L}^{\ast}} = {F_{L,1}^{\ast}}+{F_{L,2}^{\ast}}+{F_{L,3}^{\ast}}. \end{equation}

The form chosen for our model is the same as used in the discussed previous inner region studies (Cox & Hsu Reference Cox and Hsu1977; Cherukat & McLaughlin Reference Cherukat and McLaughlin1994), but we now apply this model over the entire inner, outer and unbounded regions. Further, we provide unambiguous definitions for the three coefficients that allow the (3.1) to be a valid representation of the lift force at any separation distance. Namely, the force term ${F_{L,1}^{\ast }}=C_{L,1}Re_{\gamma }^2$ is defined by the force in a linear shear flow in the absence of slip. The corresponding force coefficient $C_{L,1}$ (the main focus of this study) depends only on shear rate and wall distance. The last term ${F_{L,3}^{\ast }}=C_{L,3}Re_{slip}^2$ provides the lift in a quiescent flow in the absence of shear, and the force coefficient $C_{L,3}$ is only a function of slip velocity and wall distance. The remaining term ${F_{L,2}^{\ast }}=\text {sgn}({\gamma }^{\ast })C_{L,2}Re_{\gamma } Re_{slip}$ captures the remaining lift contributions in the presence of both slip and shear. Hence, the corresponding force coefficient, $C_{L,2}$ depends upon slip, shear and wall distance.

In the same spirit, the net drag acting on a particle with a finite slip in a linear shear flow, is defined as

(3.2a)\begin{equation} {F_{D}^{\ast}} = -\text{sgn}(\gamma)C_{D,1}Re_{\gamma} -\text{sgn}(u_{slip})C_{D,2}Re_{{slip}}, \end{equation}

which may also be written in the alternative form

(3.2b)\begin{equation} {F_{D}^{\ast}} = {F_{D,1}^{\ast}}+{F_{D,2}^{\ast}}. \end{equation}

Again, while the form of this equation comes from previous work (Magnaudet et al. Reference Magnaudet, Takagi and Legendre2003), our definition of the coefficients ensures that it provides an exact expression for the drag force across all separation distances. Specifically, the force term ${F_{D,1}^{\ast }} =-\text {sgn}(\gamma )C_{L,1}Re_{\gamma }$ in (3.2) is defined as the drag force in a linear shear flow in the absence of slip. The corresponding force coefficient $C_{D,1}$, valid for all separation distances, is only a function of shear and wall distance. The term ${F_{D,2}^{\ast }}= -\text {sgn}({u_{slip}})C_{D,2}Re_{slip}$ in (3.2) captures the remaining drag contributions in the presence of both the slip and shear. The force coefficient $C_{D,2}$, is a function of slip, shear and wall distance.

In the present study, our primary aim is to investigate forces on a particle under zero-slip conditions, and hence provide correlations for the lift and drag force coefficients, $C_{L,1}$ and $C_{D,1}$ respectively. However, in § 4 we do use the full form of (3.1) and (3.2) when applying our new correlations to analyse the movement of force-free particles translating near a wall.

3.2. Lift force

In figure 4, lift coefficients $C_{L,1}$ computed for both non-rotating and a freely rotating particles are plotted as a function of separation distance (${l^{\ast }}$). The numerical results are compared with the available inner region correlations listed in table 1 that are valid for $Re_{\gamma } \ll 1$ and $Re_{{slip}} = 0$. For $Re_{\gamma } < 10^{-2}$, the numerically computed lift forces in the region close to the wall (${l^{\ast }} < 2$) agree reasonably well with the asymptotic values predicted by the analytical solutions derived for low Reynolds numbers (Cherukat & McLaughlin Reference Cherukat and McLaughlin1994; Krishnan & Leighton Reference Krishnan and Leighton1995). Specifically, the lowest shear Reynolds number simulation conducted at the smallest distance to the wall (${l^{\ast }} = 1.2$) gives a $C_{{L,1}}$ of $1.993$ ($1.787$) for a non-rotating (freely rotating) particle, which is only ${\sim }1.3\,\%$ ($5.2\,\%$) higher than the asymptotic value of 1.9680 (1.6988) predicted for a non-rotating (freely rotating) particle at ${l^{\ast }} = 1$ (Krishnan & Leighton Reference Krishnan and Leighton1995). The computed lift coefficients also agree very well with the low Reynolds number theoretical values of 1.9834 (1.7854) for non-rotating (freely rotating) particles at ${l^{\ast }} = 1.2$, as predicted by Cherukat & McLaughlin (Reference Cherukat and McLaughlin1994). However, as illustrated in figure 4(b), for a freely rotating particle the Magnaudet et al. (Reference Magnaudet, Takagi and Legendre2003) lift correlation predicts a larger coefficient of 2.81 at ${l^{\ast }} = 1$ which is inconsistent with the numerical data and the available theories in this region. As explained in Shi & Rzehak (Reference Shi and Rzehak2020), this over-prediction of the lift coefficient near the wall is due to the neglect of the higher-order separation distance terms (${O}(1/{l^{\ast }}) > 2$) that are significant when representing lift in the vicinity of the wall. At large $l^{\ast }$ the Magnaudet et al. (Reference Magnaudet, Takagi and Legendre2003) lift correlation gives $C_{{L,1}}=1.8$, a result that is consistent with the large ${l^{\ast }}$ value from the Cox & Hsu (Reference Cox and Hsu1977) lift correlation. However, like all inner region-based models, this large separation limit only has practical relevance for very small Reynolds numbers.

Figure 4. Lift coefficient ($C_{{L,1}}$) for different shear Reynolds number as a function of non-dimensional separation distance (${l^{\ast }}$) for $(a)$ non-rotating and $(b)$ freely rotating spheres. Simulations: inner region (), outer region (). Analytical (1.3), (1.4) (green) from Cox & Hsu (Reference Cox and Hsu1977), (1.5), (1.6) (red) from Cherukat & McLaughlin (Reference Cherukat and McLaughlin1994), (1.7), (1.8) (blue) from Krishnan & Leighton (Reference Krishnan and Leighton1995), (1.9) (pink) from Magnaudet et al. (Reference Magnaudet, Takagi and Legendre2003). Present numerical fit: (3.3) (grey). Present correlation results for $Re_{\gamma }=0$ are coincident with the Cherukat & McLaughlin (Reference Cherukat and McLaughlin1994, Reference Cherukat and McLaughlin1995) results and not shown on this figure.

As $Re_{\gamma }$ increases and inertial effects become more significant, the computed lift coefficients deviate significantly from the available theoretical values. Even when the particle is far away from the wall but still within the inner region (e.g. results for $Re_{\gamma } < 10^{-2}$), the computed lift force coefficient decreases with shear rate, in contrast to the available inner region models which give coefficients that are independent of $Re_{\gamma }$. For example, the Cherukat & McLaughlin (Reference Cherukat and McLaughlin1994) model gives a $C_{{L,1}}$ of 2.0069 (1.8065) for a non-rotating (freely rotating) particle as ${l^{\ast }} \rightarrow \infty$. Eventually, with increasing shear rate and increasing separation distance, the walls move to the outer region ($l^{\ast }/{L_{G}}^{\ast } > 1$) and, unsurprisingly, the inner region-based models do not capture the lift coefficient variation well at all. Hence the available inner region-based theoretical lift models overestimate the lift coefficient for all but the lowest shear Reynolds numbers, for particles both near and (especially) further away from the wall.

The discrepancy between simulation and theory arises for the inner region-based models because they use a first-order expansion for the disturbance velocity that cannot satisfy the boundary conditions at infinity (Cherukat & McLaughlin Reference Cherukat and McLaughlin1994). For many applications relevant to cell sorting in microfluidics, or to understand mechanical phenomena like particle deposition and fouling at $Re_{\gamma } \lesssim {O}(1)$, this is problematic, and a single equation for the wall-induced lift force valid across both the inner and outer regions would be beneficial. Therefore, as well as providing a shear-based inertial correction for the analytical inner region model of Cherukat & McLaughlin (Reference Cherukat and McLaughlin1994), we here propose a model that simultaneously captures the outer region behaviour.

In this vein, we first plot the computed lift force coefficients against the separation distance, now normalised using Saffman's length scale ($L_{{G}}^{\ast }$). Recall that this length scale defines the boundary between the inner and outer regions in this problem. These results, shown in figure 5, indicate that the outer region data $(l^{\ast }/L_{{G}}^{\ast }> 1)$ collapse to a single curve for the selected range of shear Reynolds numbers, similar to the results given for a freely rotating particle at $Re_{\gamma } \ll 1$ presented by Asmolov (Reference Asmolov1999). The difference between a non-rotating and freely rotating lift coefficient value is less significant in the outer region, particularly at $l^{\ast }/L_{{G}}^{\ast } \gg 1$. Utilising this outer region behaviour, together with the existing low $Re_{\gamma }$ inner region Cherukat & McLaughlin (Reference Cherukat and McLaughlin1994) results, the following lift correlation is proposed for particles moving near a wall under zero-slip conditions:

(3.3)\begin{equation} {C_{L,1}} = f_{1}(Re_{\gamma}){C_{outer}}(l^{\ast}/{L_{G}}^{\ast}) + f_{2}(Re_{\gamma}){C_{inner}}({1/l^{\ast}}), \end{equation}

where

(3.4)\begin{gather} f_{1}(Re_{\gamma}) = {\lambda_{1}}\exp{({\lambda_{2}} Re_{\gamma})}+{\lambda_3}\exp{({\lambda_4} Re_{\gamma})}; \end{gather}

(3.5)\begin{gather} {C_{outer}}(l^{\ast}/L_{G}^{\ast}) = {\lambda_5}\exp\left[{{\lambda_6}\left({l^{\ast}}/{L_{G}^{\ast}}\right)^2+{\lambda_7}\left({l^{\ast}}/{L_{G}^{\ast}}\right)}\right]; \end{gather}

(3.6)\begin{gather} f_{2}(Re_{\gamma}) =1+\sqrt{Re_{\gamma}}; \end{gather}

(3.7)\begin{gather} {C_{inner}}({1/l^{\ast}})={\lambda_8}(1/{l^{\ast}})+{\lambda_9}(1/{l^{\ast}})^{2}+{\lambda_{10}}(1/{l^{\ast}})^{3}. \end{gather}

Coefficients for the above functions are listed in table 3 for both non-rotating and freely rotating particles.

Figure 5. Lift coefficient ($C_{{L,1}}$) for shear Reynolds number values of 1, 2, 4, 6, 8, 10, 20, 40, 60, 80 and 100; $\times 10^{-3}$ (the arrow indicates the direction of increasing $Re_{\gamma }$) as a function of separation distance non-dimensionalised by the Saffman length scale ($l^{\ast }/{L_{{G}}}^{\ast }$) for a $(a)$ non-rotating and $(b)$ freely rotating particle. Simulations: inner region (), outer region (). Present numerical fit: (3.5) (blue). Asmolov (Reference Asmolov1999) results for outer region (pink).

Table 3. Coefficients for (3.3) for a non-rotating and freely rotating particle.

$^{a}$Coefficients from Cherukat & McLaughlin (Reference Cherukat and McLaughlin1994) and Cherukat & McLaughlin (Reference Cherukat and McLaughlin1995).

The functions used in (3.3) represent different limits. Within the inner region, the first term of (3.3) rapidly reduces to zero as both $Re_{\gamma } \rightarrow 0$ and $l^{\ast } \rightarrow 1$, leaving the second part of the equation to be dominant. The function ${C_{inner}}$ consists of the non-zero degree polynomial terms of the Cherukat & McLaughlin (Reference Cherukat and McLaughlin1994) model. Here, $f_{2}(Re_{\gamma })$, which is independent of wall distance, captures inertial effects when the particle is close to the wall. At ${l^{\ast }} = 1$ and $Re_{\gamma } = 0$, the product of $f_{1}(Re_{\gamma }){C_{outer}}$ reduces to the constant value of $2.0069$ ($1.8065$), consistent with the Cherukat & McLaughlin (Reference Cherukat and McLaughlin1994) result for a non-rotating (freely rotating) particle. With increasing shear rate and wall distance, the first term of the (3.3) becomes dominant and captures the outer region variation. This term gives zero $C_{{L,1}}$ as ${l^{\ast }} \rightarrow \infty$.

Figure 4 shows that the proposed model captures the simulated lift coefficients accurately over most of the $Re_{\gamma }$ range and separation distances considered in this study. For low shear rates ($Re_{\gamma } \sim 10^{-3}$), the model performs well, slightly underestimating the simulated results with a maximum deviation of $1.6\,\%$$(3.2\,\%)$ for a non-rotating (freely rotating) particle at distances far from the wall. For high shear rates ($Re_{\gamma } \sim 10^{-1}$), the model generally overestimates the simulated results, with a maximum deviation of $41\,\%$$(21\,\%)$ for a non-rotating particle (freely rotating particle) occurring when the particle is furthest from the wall, that is at $l^{\ast } = 9.5$. This deviation rapidly reduces down to 4.5 % ($5.6\,\%$) for a non-rotating (freely rotating) particle when the separation distance decreases to $l^{\ast } = 6$, showing that even for the relatively high $Re_{\gamma } = 10^{-1}$, the proposed model still accurately captures the lift force variation provided that the particle is within a few diameters of the wall. Note that as discussed in § 2.2.1, we expect the numerical data to have the highest errors under the same high $Re_{\gamma }$ and $l^{\ast }$ conditions that generate the largest deviations between correlation and simulation – conditions that also result in low lift forces. Hence, the proposed model captures the variation of lift coefficient with $Re_{\gamma }$ and separation distance reasonably well across all conditions considered, but is most accurate when the resulting lift force is most significant.

3.3. Drag force

The drag force on a spherical particle moving parallel to a wall and in the absence of slip is considered in this section. Under these conditions the presence of the wall (combined with the positive shear rate) produces a force on the particle that is in the opposite direction to the flow, and which rapidly decays as the separation distance between the sphere and wall increases. Note that this wall-shear drag causes force-free particles to lag the flow when moving in close proximity to a wall.

Figure 6 shows the variation in drag coefficient $C_{D,1}$ for shear Reynolds numbers in the range $Re_{\gamma } = 10^{-3}{-}10^{-1}$ as a function of wall separation distance. Results for both non-rotating and freely rotating conditions are shown.

Figure 6. Magnitude of drag force coefficient in the absence of slip ($C_{{D,1}}$) $(a)$ for a non-rotating $(b)$ freely rotating particle. The arrow indicates the direction of increasing $Re_{\gamma }$. Simulations: inner region (), outer region (). Analytical prediction of Magnaudet et al. (Reference Magnaudet, Takagi and Legendre2003) (1.11) (pink). Present numerical fits for $Re_{\gamma } \ll 1$: (3.8) (blue) and finite $Re_{\gamma }$ (3.9) (grey).

Figure 6(a) is specific to non-rotating particles. The highest drag coefficient magnitudes are observed close to the wall, with a rapid decrease occurring with increasing wall distance or increasing $Re_{\gamma }$. For the largest separation considered (${l^{\ast }} = 9.5$), small positive $-C_{D,1}$ values of ${O}(10^{-3}){-}{O}(10^{-2})$ are reported for $Re_{\gamma }=10^{-1}, 8 \times 10^{-2}$ as illustrated in the inset of figure 6(a). As discussed in § 2.2.1, these small positive drag coefficients are more likely to be due to the limited domain size employed in the simulations (numerical accuracy), rather than being physically meaningful.

Figure 6(b) shows the drag coefficient for a freely rotating particle as a function of wall distance. The highest negative values for $-C_{D,1}$ are obtained when the particle is close to the wall, reducing to zero with increasing wall distance, in a manner similar to that observed for a non-rotating particle. The highest reported $-C_{D,1}$ values are ${\sim }6\,\%$ less than the values reported for a non-rotating particle. The simulated drag coefficient results are also compared with the inner region analytical drag correlation of Magnaudet et al. (Reference Magnaudet, Takagi and Legendre2003) for a freely rotating particle at low $Re_{\gamma } \ll 1$. The computed drag coefficients are in good agreement with the correlation values for Reynolds numbers $Re_{\gamma } < 10^{-2}$ and for wall distances ${l^{\ast }>1.6}$. However, as the separation distance decreases (${l^{\ast }} \rightarrow 1$), the theoretical predictions of (1.11) underestimate the drag, resulting in a difference of ${\sim }32\,\%$ at ${l^{\ast }} = 1.2$. This could again be due to the neglect of higher-order terms in the analytical solution, (1.11). In support of this, in the same study by Magnaudet et al. (Reference Magnaudet, Takagi and Legendre2003), but for zero-shear flows ($Re_{\gamma } = 0$), the effect of higher-order ${l^{\ast }}$ terms on $C_{D,2}$ was examined in the context of small separation distances (${l^{\ast }} < 3$), and it was found that a $\sim 60\,\%$ difference in this drag coefficient occurred when using ${O}(({1/l^{\ast }})^3)$ over ${O}(({1/l^{\ast }})^5)$ terms in their analysis, for ${l^{\ast }} \rightarrow 1$. Similar higher-order terms for the $C_{D,1}$ drag coefficient examined in our study are not available in the literature, however.

The freely rotating drag coefficients shown in figure 6(b) also display a dependence on shear rate, as well as separation distance. Generally studies of force-free particles and fixed particles give inertial corrections for $C_{D,1}$ and $C_{D,2}$ in terms of the slip velocity ($Re_{slip}$) (Kurose & Komori Reference Kurose and Komori1999; Magnaudet et al. Reference Magnaudet, Takagi and Legendre2003). However, in the present zero-slip context these inertial corrections are not relevant, and instead an inertial correction based on $Re_{\gamma }$ is required.

Hence, to accurately predict the variation of $C_{D,1}$ in (3.2) over the shear rates and separation distances considered in this study, we propose a new correlation that is partially based on existing inner region results (Magnaudet et al. Reference Magnaudet, Takagi and Legendre2003), but includes two modifications.

Firstly, to capture the drag force variation close to the wall at $Re_{\gamma } \ll 1$, (1.11) is modified as

(3.8)\begin{align} C_{D,1}^{\prime}=\dfrac{15{\rm \pi}}{8}\left(\dfrac{1}{l^{\ast}}\right)^2\left[1+\dfrac{9}{16}\left(\dfrac{1}{l^{\ast}}\right)+0.5801\left(\dfrac{1}{l^{\ast}}\right)^2 -3.34\left(\dfrac{1}{l^{\ast}}\right)^3+4.15\left(\dfrac{1}{l^{\ast}}\right)^4\right], \end{align}

where $C_{D,1}^{\prime}$ consists of the terms given by Magnaudet et al. (Reference Magnaudet, Takagi and Legendre2003) in (1.11), combined with three new higher-order terms in separation distance derived through a numerical fitting to our simulation results for $Re_{\gamma } \ll 1$. The proposed correlation accurately captures the computed drag coefficient down to a minimum separation distance of ${l^{\ast }=1.2}$. When the particle is in contact with the wall ($l^{\ast }=1$), the current model predicts a finite drag coefficient $-C_{D,1}$ of $-17.392$ (compared to a value of $-9.204$ from (1.11)). Whilst the proposed model performs better than the Magnaudet et al. (Reference Magnaudet, Takagi and Legendre2003) correlation at $l^{\ast }=1.2$, for the limiting case of $l^{\ast }=1$ there is no asymptotic value available for $C_{D,1}$ upon which to validate the new expression.

Secondly, we present an inertial correction to the proposed low $Re_{\gamma }$ equation (3.2), based on fits to our data, that predicts the variation of $C_{D,1}$ with finite $Re_{\gamma }$. The final wall-shear drag correlation can be written

(3.9)\begin{equation} C_{D,1}=C_{D,1}^{\prime}+(3.001Re_{\gamma}^2 -1.025Re_{\gamma}), \end{equation}

where $C_{D,1}^{\prime}$ is given via (3.8). Although, the new inertial correction accurately captures the shear dependence of $C_{D,1}$ across all considered separations at low $Re_{\gamma }$, far away from the wall ($l^{\ast } > 8$) the coefficient is underestimated for $Re_{\gamma } > 6 \times 10^{-2}$. Note, however, that these are the same conditions that result in the poorest accuracy of the computed $C_{D,1}$, as previously discussed.

4.1. Slip velocity

The slip velocity of a force-free particle is first calculated by setting the net drag force (${F_{D}}^{\ast }$) in (3.2) to zero. The resulting non-dimensional velocity $u^{\ast }_{slip} (=Re_{slip}/Re_{\gamma }$) is

(4.1)\begin{equation} u^{\ast}_{slip}=-\dfrac{C_{D,1}}{C_{D,2}}. \end{equation}

The present numerical drag correlation for $C_{D,1}$ provided by (3.9) is used to evaluate the slip velocity for three finite shear Reynolds numbers ($Re_{\gamma } = 10^{-1}$, $10^{-2}$, $10^{-3}$) as well as $Re_{\gamma } = 0$. In these calculations the Faxen drag correlation given by Happel & Brenner (Reference Happel and Brenner1981) (with higher-order terms up to ${O}(({1/l^{\ast }})^5)$) is used to evaluate $C_{D,2}$. Note that this correlation has been derived for the inner region of quiescent flows ($Re_{\gamma } = 0$) at low slip Reynolds numbers ($Re_{slip} \ll 1$), and yet we are applying it in this section for finite slip Reynolds number in the presence of shear, and across all three regions (inner, outer and unbounded regions). However, although this Faxen drag model is derived for $Re_{slip} \ll 1$, Ambari, Gauthier & Guyon (Reference Ambari, Gauthier and Guyon1984) and Takemura (Reference Takemura2004) experimentally showed that this correlation is valid for $Re_{slip} \leq 0.1$ in quiescent flows, partly justifying the use of this correlation for the slip Reynolds numbers considered here. Further, in unbounded linear shear flows, Kurose & Komori (Reference Kurose and Komori1999) showed that $C_{D,2}$ is nearly independent of shear rate for $Re_{slip} < 5$ and ${O}(Re_{\gamma }/Re_{slip}) < 1$. It is also worth mentioning that in quiescent flows, $L_{S}$ has been used to define the boundary of the inner and outer regions. However, for non-zero-shear flows (as in this case), the inner and outer region boundary is defined based on the $\min (L_{S}, L_{G})$ condition. For this specific case, $L_{S}$ decays rapidly with separation distance, resulting $L_{S} \gg L_{G}$ for all the separation distances and all $Re_{\gamma }$ values considered in this study. Hence, $L_{G}$ is chosen to define the inner and outer regions. However, with respect to $L_{S}$ (as defined for quiescent flows), the walls are well within the inner region, which may again partly justify the use of $C_{D,2}$ when $L_{G}<l<L_{S}$. Hence, while we are applying this $C_{D,2}$ correlation outside its range of validity (i.e. across all three regions and for $Re_{\gamma } \neq 0$ and $Re_{slip}\neq 0$), evidence suggests that the associated error should not be large.

The calculated $u^{\ast }_{slip}$ values are plotted as a function of separation distance in figure 7(a). The velocities are compared with: (i) our numerical predictions; (ii) numerical predictions by Fischer & Rosenberger (Reference Fischer and Rosenberger1987) based on a boundary element method that included small inertial effects ($Re_{\gamma } \ll 1$); (iii) theoretical/numerical predictions by Goldman et al. (Reference Goldman, Cox and Brenner1967b) applicable to Stokes flow; (iv) slip velocities again calculated using (4.1), but with $C_{D,1}$ evaluated using the Magnaudet et al. (Reference Magnaudet, Takagi and Legendre2003) correlation of (1.11).

Figure 7. Analysis of a force-free particle translating in a linear shear flow near a wall. $(a)$ Non-dimensional slip velocity when $C_{D,1}$ is evaluated using the present numerical correlation, (3.9) for $Re_{\gamma } =0$ (black), $Re_{\gamma }=10^{-3}$ (pink), $Re_{\gamma }=10^{-2}$ (green), $Re_{\gamma }=10^{-1}$ (red). Dashed lines indicate results evaluated outside the strict range in which the new correlation has been validated. Slip velocity when $C_{D,1}$ is evaluated using the Magnaudet et al. (Reference Magnaudet, Takagi and Legendre2003) correlation, (1.11) for $Re_{\gamma } = 0$ (blue); Fischer & Rosenberger (Reference Fischer and Rosenberger1987) numerical results for $Re_{\gamma } \ll 1$ ($\circ$); Goldman et al. (Reference Goldman, Cox and Brenner1967b) numerical results for Stokes flow ($\bullet$); present force-free numerical results (coloured hollow circles). (b–d) Different contributions to the net lift force acting on the particle, ${F_{L,1}^{\prime }}, {F_{L,2}^{\prime }}$ and ${F_{L,3}^{\prime }}$ representing shear, slip-shear and slip effects, respectively. (e) Net lift force non-dimensionalised by shear Reynolds number when ${F_{L,tot}^{\prime }}$ is evaluated using (4.3) for $Re_{\gamma } =0$ (black), $Re_{\gamma }=10^{-3}$ (pink), $Re_{\gamma }=10^{-2}$ (green), $Re_{\gamma }=10^{-1}$ (red); Fischer & Rosenberger (Reference Fischer and Rosenberger1987) numerical results for $Re_{\gamma } \ll 1$ ($\ast$); present force-free numerical results (coloured hollow circles).

Consistent with earlier studies (Goldman et al. Reference Goldman, Cox and Brenner1967b; Takemura & Magnaudet Reference Takemura and Magnaudet2009), the negative slip velocities obtained confirm that a force-free particle translating in a linear shear flow lags the fluid when in close proximity to a wall. The present correlation, (3.9), used for evaluating $C_{D,1}$ predicts a slip velocity that is consistent with the all available numerical results down to reasonably small separation distances ($l^{\ast } \sim 1.1$). In comparison the results obtained using the Magnaudet et al. (Reference Magnaudet, Takagi and Legendre2003) correlation ((1.11)) diverge from the numerical results at $l^{\ast } \sim 1.5$. At small separation distances ($l^{\ast } \lesssim 1.1$) the two numerical results diverge. The predicted Stokes flow velocities from Goldman et al. (Reference Goldman, Cox and Brenner1967b) exhibit large negative values when the particle is almost in contact with the wall ($l^{\ast } \sim 1$), whereas the few results of Fischer & Rosenberger (Reference Fischer and Rosenberger1987) in this region, that are valid for small but finite inertial effects, show smaller slip velocities. The difference between Fischer & Rosenberger (Reference Fischer and Rosenberger1987) and Goldman et al. (Reference Goldman, Cox and Brenner1967b) may be due to insufficient numerical accuracy of a Gauss–Legendre product formula used by Fischer & Rosenberger (Reference Fischer and Rosenberger1987), as suggested by Shi & Rzehak (Reference Shi and Rzehak2020). We make no further comment on this difference, but note that the extrapolation of our correlation (indicted by the dashed lines in figure 7a) appears to be more consistent with the Fischer & Rosenberger (Reference Fischer and Rosenberger1987) results at $l^{\ast } = 1.05$ than those of Goldman et al. (Reference Goldman, Cox and Brenner1967b).

Slip velocity curves for the selected range of Reynolds numbers suggest that the shear rate has only a small effect ($\sim {O}(10^{-3})$) on $u^{\ast }_{slip}$ for all separation distances. As expected, predicted slip velocities reduce to small values with increasing separation distance for the low $Re_{\gamma } = 10^{-3}, 10^{-2}$ cases. However, for $Re_{\gamma } = 10^{-1}$ the effect of inertia is relatively more important and the slip velocity further reduces, becoming approximately zero. In fact, small positive slip velocities are predicted under the conditions $Re_{\gamma } = 10^{-1}$ and $l^{\ast } > 9.2$. However, as mentioned in § 3.3, numerical inaccuracy of the fitted correlation, (3.9), at large separation distances for high $Re_{\gamma }$, is more likely to be the main reason that such positive slip velocities are obtained, rather than being physically meaningful. Significantly, the slip velocities are very small in this high separation distance region for all $Re_{\gamma }$ considered.

4.2. Lift force and migration velocity

By using the slip velocity calculated via (4.1) and the lift formulation given in (3.1), the lift force on a force-free particle experiencing both slip and shear within a linear shear flow can be written

(4.2)\begin{equation} \frac{F_{L}^{\ast}}{Re_{\gamma}^2} = C_{L,1}+C_{L,2} u^{\ast}_{slip}+C_{L,3} {u^{\ast}_{slip}}^2, \end{equation}

which may also be written in the alternative form

(4.3)\begin{equation} {F_{L,tot}^{\prime}} = {F_{L,1}^{\prime}}+{F_{L,2}^{\prime}}+{F_{L,3}^{\prime}}. \end{equation}

Again, note that the form of (4.3) is valid for all three regions (inner, outer, unbounded), provided that the lift coefficients used are valid in all three regions. Here, however, lift force coefficients for $C_{L,2}$ and $C_{L,3}$ are evaluated using the Cherukat & McLaughlin (Reference Cherukat and McLaughlin1995) inner region-based correlations as there are no correlations available for these coefficients which span all three regions. $C_{L,1}$ is evaluated using our freely rotating particle correlation, as given by (3.3), which is valid across all three regions. Here ${F_{L,tot}^{\prime }} (={F_{L}^{\ast }}/{Re_{\gamma }^2}$) is the net lift force non-dimensionalised by shear Reynolds number.

The lift contributions, as well as the net lift force, are plotted in figure 7 for $Re_{\gamma } \ll 1$ and $Re_{\gamma } = 10^{-3}, 10^{-2}, 10^{-1}$. The numerical lift results obtained from our force-free and torque-free particles simulations are also given in figure 7(e). For separation distances $l^{\ast } \gtrsim 2$, the force contributions illustrate that ${F_{L,1}^{\prime }} > {F_{L,2}^{\prime }} \gg {F_{L,3}^{\prime }}$ for a force-free particle. This indicates that the wall-shear lift contribution, ${C_{L,1}}$, either dominates or is similar to the other forces in this high force region, with the consequence that an accurate inertial correction for ${C_{L,1}}$ is required to accurately predict the net lift force acting on a force-free particle. Figure 7(e) also shows that the estimated ${F_{L,tot}^{\prime }}$ using our correlations agrees well with the low $Re_{\gamma }$ results of Fischer & Rosenberger (Reference Fischer and Rosenberger1987) (using $Re_{\gamma } = 0$ in our correlation), but deviates significantly from these results as the shear Reynolds number and separation distance increase. Most of our numerical results agree reasonably well with (4.3) predictions for the selected separation distances and shear Reynolds number ranges. The present numerical results clearly exhibit the inertial dependence predicted by new lift model, and thus reinforces the importance of the Reynolds number correction to ${C_{L,1}}$.

Interestingly, the inset of figure 7(e) illustrates that the net lift force, ${F_{L}^{\ast }}$, while increasing with shear rate, is relatively independent of separation distance, except at the highest $Re_{\gamma }$ considered. When $Re_{\gamma }$ is low, the net lift (and the lift coefficient) does not reduce to zero until well into the outer region, which for low $Re_{\gamma }$ is a large distance from the wall: for $Re_{\gamma } = 10^{-2}$ and $10^{-3}$, the transition from the inner to the outer region ($L_{G}^{\ast }$) occurs at $l^{\ast } = 10$ and $31.62$, respectively. Therefore, a clear lift dependence on the separation distance is not visible in figure 7(e) as the maximum $l^{\ast }$ indicated is 10. Nevertheless, a strong lift dependence on the separation distance is visible for the largest particle shear Reynolds number ($Re_{\gamma } = 10^{-1}$) as the transition from the inner to outer region occurs at $l^{\ast } = 3.16$, well within the region of analysis.

The minor deviations between numerical and correlation results observed in the predicted lift force for the highest $Re_{\gamma }$ could be a result of two factors. Firstly, as discussed in § 3.2, at high $Re_{\gamma }$ our freely rotating $C_{L,1}$ correlation slightly overestimates the lift force, particularly far from the wall. Secondly, there are errors in using the inner region-based correlations for $C_{L,2}$ and $C_{L,3}$ in our analysis. The main draw back is the overestimation of the slip-based lift forces when the particle is far from the wall. This happens as a result of $C_{L,2}$ and $C_{L,3}$ reducing to the outer boundary values of the inner region, instead of reducing to the unbounded values as $l^{\ast } \rightarrow \infty$. Additionally, the inertial dependence of ${F_{L,2}^{\prime }}$ and ${F_{L,3}^{\prime }}$ is weakly captured with the current coupling, as the inner region models for $C_{L,2}$ and $C_{L,3}$ are derived for $Re_{\gamma }, Re_{slip} \ll 1$ and $Re_{slip} \ll 1$ conditions, respectively. However, for the present case of a force-free particle, these two limitations are not that significant because $u_{slip}^{\ast }$ rapidly reduces with increasing $l^{\ast }$ (figure 7a). Therefore the slip-based lift forces, ${F_{L,2}^{\prime }}$ and ${F_{L,3}^{\prime }}$ contribute only a small amount to the overall lift force for $l^{\ast } > 2$. Whenever slip becomes significant for the case of a force-free particle, the walls are well within the inner region ($l^{\ast } < 2$), and therefore the net lift correlation reduces to the original Cherukat & McLaughlin (Reference Cherukat and McLaughlin1995) inner region correlation. However, for relatively large slip velocities (e.g. buoyant particles near walls), where ${F_{L,2}^{\prime }}$ and ${F_{L,3}^{\prime }}$ are more significant, predictions of (4.3) with inner region-based slip lift coefficients will be not necessarily be accurate. This is a limitation which needs to be addressed in future studies in order to completely generalise the net lift model given by (3.1).

Dimensionless migration velocities $u_{mig}^{\ast } (=u_{mig}/\gamma a)$ obtained by balancing the lift force and the wall-normal drag force,

(4.4)\begin{equation} u_{mig}^{\ast}=\frac{F_{L}^{\ast}}{6{\rm \pi} Re_{\gamma} (1+C_{{D}\perp})} \end{equation}

are presented in figure 8(a) for the same set of shear Reynolds numbers. Here, $C_{{D}\perp }$, the wall-bounded drag coefficient of a particle translating normal to a wall, is evaluated via the analytical inner region correlation of Faxen (Reference Faxen1922) and Happel & Brenner (Reference Happel and Brenner1981). The migration velocities obtained by balancing the lift with Stokes unbounded drag ($C_{{D}\perp } = 0$ in (4.4)) are also shown in the same figure using dashed lines. It is evident that the wall-normal correction to the Stokes drag greatly decreases the migration velocity close to the wall. Noting that $C_{{D}\perp }$ is an inner region model which rapidly decays to zero we assume that $C_{{D}\perp }$ can be used for all the separation distances to predict the migration velocity of a particle. Based on the migration velocity normal to the wall and the particle velocity parallel to the wall, we can calculate the lateral displacement in the wall-normal direction ($h$) and longitudinal displacement in the flow direction ($d$) as illustrated in figure 8(a). The calculated lateral displacement per unit longitudinal displacement ${\hat {h}} (=h/d)$ of a particle located at $l^{\ast }$ is also given in figure 8(b). For all shear Reynolds numbers the lateral displacement of a force-free particle closer to the wall is much higher than that of a particle travelling further away from the wall. The maximum lateral displacement $\hat {h}\sim 0.035$ that occurs when the particle is closer to the wall ($l^{\ast }\sim 1.5$) would quite quickly push a free particle away from the wall. The lateral displacement also displays a weak dependence on the shear rate close to the wall ($l^{\ast } \sim 1$), however, when a particle is away from the wall $\hat {h}$ decreases significantly as the shear Reynolds number increases.

Figure 8. $(a)$ Dimensionless migration velocity and $(b)$ Lateral displacement per unit movement in the flow direction of a force-free particle for $Re_{\gamma } \ll 1$, $Re_{\gamma }=10^{-3}$, $Re_{\gamma }=10^{-2}$ and $Re_{\gamma }=10^{-1}$. Dashed lines show the values obtained when $C_{{D}\perp } = 0$ in (4.4). Here, $\hat {h} = h/d$, $l^{\ast } = l/a$ and $u^{\ast }_{mig} = u_{mig}/\gamma a$.

Figure 9. A detailed two-dimensional frame at $z=0$ of a ‘Lego’ mesh showing how individual blocks are duplicated to form the complete mesh. $N_{d}, N_{p}$ and $N_{w}$ indicate number of points, while $L^{\ast }, l^{\ast }$ are dimensionless domain sizes.

Although the present study focuses on linear shear flows, it is interesting to briefly discuss the behaviour of lift coefficients of force-free particles in Poiseuille flows at $Re_{\gamma } < 1$. Direct numerical studies on planar Poiseuille flows have shown that the net lift coefficient (${F_{L,tot}^{\prime }}$) of force-free particles is relatively independent of $Re_{\gamma }$ even when $Re_{\gamma } \sim {O}(10^{-1})$ (Hood, Lee & Roper Reference Hood, Lee and Roper2015; Asmolov et al. Reference Asmolov, Dubov, Nizkaya, Harting and Vinogradova2018). This discrepancy is possibly due to the weak inertial dependency of the double wall-bounded shear gradient lift force (Schonberg & Hinch Reference Schonberg and Hinch1989; Asmolov Reference Asmolov1999) which is not relevant for linear flows. However, as shown in the present linear shear flow study which has a uniform local $Re_{\gamma }$ throughout the separation distance, the net lift is clearly dependent on $Re_{\gamma }$, particularly away from the wall.