3.1. Lift model for unbounded linear shear flow

First, we revisit the classical problem of the lift force acting on a translating sphere in a simple shear flow. We focus on the moderate Reynolds number regime ( $5 \leqslant Re \leqslant 20$ ) and $0.2\leqslant Sr\leqslant 2.5$ , and extend the existing model for a better prediction. The Saffman lift considering only the leading order can be expressed as

(3.1) \begin{equation} C_{\textit{L,s}}^{{\textit{Saffman}}} = \frac {{18}}{{{\pi ^2}}}{J_L}\varepsilon , \end{equation}

with the conditions $\varepsilon =\sqrt {\left | Sr \right |/Re} \gg 1,\ Re\ll 1$ and $Re\,|Sr|\ll 1$ . At $\left | Sr \right |\gg 1$ , a second-order term $-{{11}}\,Sr/8$ becomes comparable to the first term and cannot be neglected. McLaughlin (Reference McLaughlin1991) showed that the second term is also important at $\varepsilon \ll 1$ , and results in a significant decrease of the lift force. Theoretical extension of the $J_L$ function to arbitrary $\varepsilon$ was done by Asmolov (Reference Asmolov1990) and McLaughlin (Reference McLaughlin1991), and the explicit expression of $J_L (\varepsilon )$ in the limit of $\varepsilon \gg 1$ and $\varepsilon \ll 1$ is given by McLaughlin (Reference McLaughlin1991). Based on the numerical integration results provided by McLaughlin (Reference McLaughlin1991) for $0.1\leqslant \varepsilon \leqslant 20$ , Mei (Reference Mei1992) proposed the fitting function for $J_L$ as

(3.2) \begin{equation} J_L^{{\textit{Mei}}} = 0.6765\left \{ {1 + \tanh \left [ {2.5\left ( {\lg \varepsilon + 0.191} \right )} \right ]} \right \}\left \{ {0.667 + \tanh \left [ {6\left ( {\varepsilon - 0.32} \right )} \right ]} \right \}. \end{equation}

Note that ` $\lg$ ’ in the above equation denotes the base-10 logarithm, while ` $\ln$ ’ (used elsewhere) denotes the natural logarithm.  However, the above extensions of the Saffman lift predict a negative lift for a small value of $\varepsilon$ , which is opposite to the DNS results. Legendre & Magnaudet (Reference Legendre and Magnaudet1998) used their DNS data at $Sr=0.2$ and $0.5$ for $Re \lt 1$ , and the theoretical results of McLaughlin (Reference McLaughlin1991) for $\varepsilon \gt 0.8$ , to propose their empirical expression as

(3.3) \begin{equation} J_L^{\textit{L&M}} = 2.255{\left ( {1 + 0.2/{\varepsilon ^2}} \right )^{{ - 1}{.5}}}, \end{equation}

which avoids the negative value and thus indicates a positive leading-order term for (3.1). The DNS work of Cherukat et al. (Reference Cherukat, McLaughlin and Dandy1999) showed that for low Reynolds numbers, the inclusion of the second-order term can improve the agreement between the model prediction and their DNS data. For relatively high ${\textit{Re}}$ (before the lift force changes its sign at ${\textit{Re}} = 40{-}100$ ), a damping for the second term is added by Shi & Rzehak (Reference Shi and Rzehak2019) to avoid the negative lift, as follows:

(3.4) \begin{equation} C_{\textit{L,s}}^{\textit{Shi}} = \frac {{18}}{{{\pi ^2}}}{J_L}\varepsilon - \frac {{11}}{8}Sr\exp \left ( { - 0.5\,Re} \right ). \end{equation}

Although numerous DNS have been carried out to calculate the lift on a spherical particle in unbounded linear shear flow, the results deviate from each other significantly, especially when the magnitude of the lift coefficient is small, and few works have been devoted to $Sr\gt 1$ . As discussed in the literature (Kim Reference Kim2006; Homann, Bec & Grauer Reference Homann, Bec and Grauer2013), the deviation between different DNS results is probably due to the confinement effect of the computational domain. Our numerical data for the unbounded lift coefficients together with some of the above-mentioned lift models are plotted as a function of ${\textit{Sr}}$ for different ${\textit{Re}}$ in figure 3. The model by Legendre & Magnaudet (Reference Legendre and Magnaudet1998) is shown as a blue line by substituting the $J_L$ function (3.3) to Saffman’s first-order term (3.1). It predicts the lift at low Reynolds number and for low ${\textit{Sr}}$ at moderate Reynolds number well, while underestimating the lift for higher ${\textit{Sr}}$ . The finite ${\textit{Re}}$ model of Shi & Rzehak (Reference Shi and Rzehak2019) (3.4) using the $J_L$ function (3.3) also underestimates the lift force, since the additional second term is negative. Thus a correction is proposed for $J_L$ as

(3.5) \begin{equation} J_L^{\textit{Present}} = 2.255{\left ( {1 + 0.2/{\varepsilon ^2}} \right )^{ - 1.5\left \{ {1 - \tanh \left [ {\ln ( Re \, | {Sr} | )/2 - 1} \right ]} \right \}/2}}, \end{equation}

which recovers the original form of $J_L^{{L\&\!M}}$  (3.3) at a small shear Reynolds number $Re \, | Sr |$ , and approaches $J_L^{\textit{Saffman}}$  (3.4) with the increase of $Re\,| Sr|$ . Our model agrees well with the data from Ben Salem & Oesterle (Reference Ben Salem and Oesterle1998) and Cherukat et al. (Reference Cherukat, McLaughlin and Dandy1999) at low ${\textit{Re}}$ for $Sr \lt 1$ . The present model agrees well with the results of Kim (Reference Kim2006) for finite ${\textit{Re}}$ at low ${\textit{Sr}}$ . By applying the present $J_L$ to model (3.4), as shown by the black dashed lines in figure 3, the new model is in better agreement with the present data and those given by Bagchi & Balachandar (Reference Bagchi and Balachandar2002) at ${\textit{Re}}=5$ , while no distinctive difference occurs for other Reynolds numbers. Therefore, in the following, $C_{\textit{L,s}}$ takes the form

(3.6) \begin{equation} {C_{\textit{L,s2}}} = C_{\textit{L,s}}^{{\textit{Saffman}}}, \end{equation}

with $C_{\textit{L,s}}^{\textit{Saffman}}$ and $J_L$ given by (3.1) and (3.5), respectively. It is interesting that the computed lift approaches the Saffman lift as ${\textit{Sr}}$ increases to a value that corresponds to $\varepsilon$ being well below 1 for finite Reynolds numbers. Though the present model fits the data well, it should be used with caution, since the idea of modelling the lift at finite $\varepsilon$ by employing the $J_L$ function inside the leading-order term of the Saffman force is strictly valid only for low Reynolds numbers and $\varepsilon \gt 1$ , as pointed out by Shi & Rzehak (Reference Shi and Rzehak2019).

Figure 3.

Lift coefficient for unbounded linear shear flow: (a) ${\textit{Re}}=0.1$ , (b) ${\textit{Re}}=5$ , (c) ${\textit{Re}}=10$ , (d) ${\textit{Re}}=20$ .

There are typically three mechanisms to describe the lift force on a sphere. At low Reynolds numbers, there is a vorticity mechanism where the lift varies with the square of the maximal vorticity on the particle surface (Legendre & Magnaudet Reference Legendre and Magnaudet1997). The vorticity mechanism is replaced by two other mechanisms for relatively high Reynolds numbers: the ‘ $L$ ’ mechanism and the ‘ $S$ ’ mechanism. The ‘ $L$ ’ mechanism is related to the vorticity of the undisturbed flow, which results from the inviscid effect and exerts a lift force in the same direction as the low- ${\textit{Re}}$ Saffman lift (Lighthill Reference Lighthill1956; Auton Reference Auton1987). The ‘ $S$ ’ mechanism is related to the surface vorticity, which is negligible for a clean spherical bubble with free-slip surface, but causes the decrease in the lift force with increasing internal viscosity and even a negative lift at high particle Reynolds numbers (Adoua, Legendre & Magnaudet Reference Adoua, Legendre and Magnaudet2009; Shi, Climent & Legendre Reference Shi, Climent and Legendre2024). The lift coefficient and the contribution from pressure for unbounded linear shear flow at $Sr=0.2$ and $Sr=2.5$ are plotted in figure 4(a). One can see that the lift force decreases with increasing Reynolds number for both $Sr=0.2$ and $Sr=2.5$ . Nevertheless, the lift coefficient for $Sr=2.5$ is much larger than that for $Sr=0.2$ . This can be explained in two ways. The first is related to the ‘ $L$ ’ mechanism: a larger ${\textit{Sr}}$ corresponds to a larger vorticity of the undisturbed flow and thereby a larger lift force resulted from the pressure contribution, as shown in figure 4(a). The second is related to the observation in figure 4(a) that the contribution of viscous force to the lift is negative for $Sr=0.2$ , whereas it becomes positive for $Sr=2.5$ . Shi et al. (Reference Shi, Climent and Legendre2024) observed that the ‘ $S$ ’ mechanism played a role for a droplet at $Sr=0.2$ when the Reynolds number exceeds 5. Therefore, for the finite Reynolds number regime studied here (i.e. 5–20), the ‘ $S$ ’ mechanism may be regarded as the reason for particularly low lift coefficient being close to zero for low ${\textit{Sr}}$ , whereas the ‘ $L$ ’ mechanism causes a high lift coefficient being $O(1)$ for high ${\textit{Sr}}$ . The variations of the maximal vorticity on the particle surface with ${\textit{Sr}}$ for ${\textit{Re}} = 5$ and 10 are compared in figure 4(b), and the trend of the variation is similar.

Figure 4.

(a) The lift coefficient and the contribution from pressure for unbounded linear shear flow at $Sr=0.2$ (black lines) and $Sr=2.5$ (blue lines). (b) The variation of the maximal vorticity on the particle surface with ${\textit{Sr}}$ .

3.2. Lift model for single-wall-bounded linear shear flow

For a non-rotating particle translating in a single-wall-bounded linear shear flow, the Oseen and Saffman lengths are introduced as ${l_u} = {\nu }/{\left | {{u_{\textit{slip}}}} \right |}$ and ${l_{g} } = \sqrt {{\nu }/{\left | {\dot \gamma } \right |}}$ , respectively. The particle–wall separation distance normalised by the Oseen and Saffman lengths is represented as

(3.7) \begin{equation} {L_U} = l/{l_u} = Re\,L,\quad {L_G} = l/{l_{g} } = \sqrt {Re \, | {Sr} |}\, L. \end{equation}

The wall lies in the inner region when $ \max \{L_U,L_G \}\ll 1$ , and in the outer region when $ \max \{L_U,L_G \}\gg 1$ . In addition to the shear-slip-induced lift, there are another two kinds of lift forces in the wall-bounded case at low Reynolds numbers. The wall-slip-induced lift is caused by the interaction of the Stokeslet with the wall (Vasseur & Cox Reference Vasseur and Cox1977; Takemura & Magnaudet Reference Takemura and Magnaudet2003). The shear-wall-induced lift arising from the interaction between the stresslet and the inner-region wall is proportional to $Sr^2$ (Cherukat & Mclaughlin Reference Cherukat and McLaughlin1994; Magnaudet, Takagi & Legendre Reference Magnaudet, Takagi and Legendre2003). Thus for $Sr\gt O(1)$ with the wall lying in the inner region, the shear-wall-induced lift becomes non-negligible compared to the shear-slip-induced lift, which is proportional to ${\textit{Sr}}$ .

The lift model for high ${\textit{Sr}}$ has been considered by Ekanayake et al. (Reference Ekanayake, Berry and Harvie2021), covering the ranges $2\times 10^{-3} \leqslant Re \leqslant 2\times 10^{-1}$ and $4\times 10^{-3} \leqslant Re\,| Sr| \leqslant 4\times 10^{-1}$ , and $0.6 \leqslant L \leqslant 4.75$ :

(3.8) \begin{equation} C_{\textit{L,}w\textit{s}}^{\textit{Ekanayake}} = \frac {2}{\pi }\left ( {\frac {{S{r^2}}}{4}{C_{\textit{L,1}}} - \frac {{Sr}}{2}{C_{\textit{L,2}}} + {C_{\textit{L,3}}}} \right ), \end{equation}

with

(3.9) \begin{align} {C_{\textit{L,1}}} {}&= \left [ {0.93\exp \left ( { - 0.88\,Re\, | {Sr} |/4} \right ) - 0.03\exp \left ( { - 1000\,Re\, | {Sr} |/4} \right )} \right ] {C_{\textit{outer}}} \nonumber \\ &\quad +\, \left ( {1 + \sqrt {Re\, | {Sr} |} /2} \right ){C_{\textit{inner}}}, \end{align}

(3.10) \begin{align} {C_{\textit{outer}}} &= 2.231\exp \left ( { - 0.1054L_G^2 - 0.3859{L_G}} \right ), \end{align}

(3.11) \begin{align} {C_{\textit{inner}}} &= 1.0575/\left ( {2L} \right ) - 2.4007/{\left ( {2L} \right )^2} + 1.3174/{\left ( {2L} \right )^3}, \end{align}

(3.12) \begin{align} {C_{\textit{L,2}}} &= - {h_L}\frac {9}{\pi }{J_L}\varepsilon \frac {2}{{\left | {Sr} \right |}} - 2.6729 - 0.8373/\left ( {2L} \right ) + 0.4683/{\left ( {2L} \right )^2}, \end{align}

(3.13) \begin{align} {C_{\textit{L,3}}} &= \frac {{18\pi }}{{32 + 2{L_U} + 3.8L_U^2 + 0.049L_U^3}} + 0.4757/( {2L} ) - 1.268/{( {2L})^2} + 0.683/{( {2L} )^3}, \end{align}

where $h_L$ takes the original form of Takemura et al. (Reference Takemura, Magnaudet and Dimitrakopoulos2009) as

(3.14) \begin{equation} h_L^{{{Takemura}}} = 1 - \exp \left ( { - \frac {{11}}{{96}}{\pi ^2}{L_G}/{J_L}} \right ), \end{equation}

$J_L$ is given in (3.3), and $L_{U}$ and $L_G$ are given in (3.7). Their model well captures the lift when the particle–wall separation distance lies between the inner and outer regions.

For higher Reynolds numbers, Shi et al. (Reference Shi, Rzehak, Lucas and Magnaudet2021) proposed the lift model for $0.1 \leqslant Re \leqslant 100,\ {-0.5} \leqslant Sr \leqslant 0.5$ and $0.75 \leqslant L \leqslant 4$ as

(3.15) \begin{equation} C_{\textit{L,}w\textit{s}}^{\textit{Shi}} = g_L^{{\textit{Shi}}}{C_{\textit{L,w}}^{\textit{Shi}}} + h_L^{{\textit{Shi}}}C_{\textit{L,s}}^{{\textit{Shi}}}, \end{equation}

where $C_{\textit{L,s}}^{\textit{Shi}}$ is given by the first term of (3.4), and

(3.16) \begin{align} g_L^{{\textit{Shi}}} &= {{\textrm e}^{ - 0.22\varepsilon ^{0.8}{L_G^{2.5}}\exp \left ( { - 0.01\,Re^2} \right )}}, \end{align}

(3.17) \begin{align} h_L^{{\textit{Shi}}} &= 1 - {\exp\left({ - \frac {{11}}{{96}}{\pi ^2}\frac {{L_G }}{{{J_L}}}\left ( {1 + 0.1771/L + 0.1608/L^2 - 0.035/L^3} \right )}\right)}, \end{align}

(3.18) \begin{align} {C_{\textit{L,w}}^{\textit{Shi}}} &= {C_{\textit{L,w0}}^{\textrm{T&M}}}\left [ {1 + \left ( {0.37\,Re - 0.12\,Re^{0.5}} \right )\tanh \left ( {0.2\,Re\,{L^2}} \right )} \right ]{\left ( {\frac {2}{3}L} \right )^{ - 2\tanh \left ( {0.01\,Re} \right )}}, \end{align}

(3.19) \begin{align} C_{\textit{L,}w\textit{0}}^{\textrm{T&M}} & = \begin{cases} \frac {9}{8}\left [ 1 + 0.13L_{U}\left ( L_{U} + 0.53 \right ) \right ]^{ - 1}& \textrm{for}\quad L_{U} \leqslant 10,\\ 8.94L_{U}^{ - 2.09}&\textrm{for}\quad L_{U} \gt 10. \end{cases} \end{align}

Their model has accounted for the nonlinear coupling of the wall-slip and shear-slip-induced lift force due to the finite Reynolds number effect. To propose a lift model for a finite Reynolds number with higher shear rate, we take the form given by Shi et al. (Reference Shi, Rzehak, Lucas and Magnaudet2021), with a correction term from the outer-region part of the shear-wall-induced lift model by Ekanayake et al. (Reference Ekanayake, Berry, Stickland, Dunstan, Muir, Dower and Harvie2020), as follows:

(3.20) \begin{equation} C_{\textit{L,}w\textit{s}} = g_L{C_{\textit{L,w}}} + h_LC_{\textit{L,s}}+C_{\textit{L,Sr2}}, \end{equation}

where

(3.21) \begin{align} C_{\textit{L,w}} &=C_{\textit{L,w}}^{{\textit{Shi}}},\quad C_{\textit{L,s}}=C_{\textit{L,s2}},\quad g_L = \left (g_L^{{\textit{Shi}}}\right )^{1+\tanh(Sr)},\quad h_L = h_L^{{\textit{Shi}}}, \end{align}

(3.22) \begin{align} {C_{\textit{L,Sr2}}} &= \frac {{S{r^2}}}{{2\pi }}\left [ {0.93\exp \left ( { - 0.88\,Re\, | {Sr} |/4} \right ) - 0.03\exp \left ( { - 1000\,Re\, | {Sr} |/4} \right )} \right ] {C_{\textit{outer}}}, \end{align}

with $C_{\textit{L,w}}^{{\textit{Shi}}}$ , $C_{\textit{L,s2}}$ , $g_L^{{\textit{Shi}}}$ , $h_L^{{\textit{Shi}}}$ and $C_{\textit{outer}}$ given by (3.18), (3.6), (3.16), (3.17) and (3.10), respectively.

The comparison between the data from Ekanayake et al. (Reference Ekanayake, Berry and Harvie2021) and the above models is shown in figure 5. The model of Ekanayake et al. (Reference Ekanayake, Berry and Harvie2021) fits the data better at small particle–wall separation distance at relative small ${\textit{Sr}}$ . Our model fits well with most of the data and gives better prediction for those near the outer region. At small ${\textit{Sr}}$ , our model approaches the model of Shi et al. (Reference Shi, Rzehak, Lucas and Magnaudet2021), which could consequently reduce to the outer-region shear-slip (wall-slip) lift model at low Reynolds numbers with the vanishing wall (shear) effect. Our model is promising for higher ${\textit{Sr}}$ since the $Sr^2$ term is included in our model.

Figure 5.

Lift coefficient for single-wall-bounded linear shear flow at ${\textit{Re}}=0.2$ and (a) $Sr=0.2$ , (b) $Sr=2$ , (c) $Sr=-0.2$ , (d) $Sr=-2$ .

Figure 6 shows the lift coefficients as functions of Reynolds number. Our numerical results are compared to the above-mentioned single-wall-bounded lift model for low ${\textit{Sr}}$  (3.15), and the unbounded lift models (3.1) and (3.6). As shown by the black lines in figure 6, the $Sr^2$ term is important for small $L$ , especially for negative ${\textit{Sr}}$ when the shear-slip-induced lift and wall-slip-induced lift are in the opposite direction. Our model reduces to the model of Shi et al. (Reference Shi, Rzehak, Lucas and Magnaudet2021) as ${\textit{Sr}}$ approaches zero, which is in excellent agreement with our DNS results for $Sr \leqslant 0.5$ . The present model for the unbounded case (3.6) gives a better prediction for relatively high Reynolds numbers ( $10 \leqslant Re \leqslant 20$ ). Our wall-bounded model converges to the unbounded shear flow model with increasing $L$ or ${\textit{Re}}$ , which demonstrates that the wall effect becomes negligible.

Figure 6.

Lift coefficient for single-wall-bounded linear shear flow: (a,c,e,g) $L=1$ , (b,d,f,h) $L=6$ . Subplots from the top to the bottom correspond to $Sr = 2.5, 1.5, -1.5, -2.5$ , respectively. The present model is given in (3.20).

3.3. Lift model for plane Poiseuille flow

For the case of Poiseuille background flow, the ratio of channel half-width to particle diameter $H$ should be considered as the fourth dimensionless parameter. Here, we employ two ways to formulate the lift coefficient for the Poiseuille flow.

For the first approach, we utilise the model developed for Couette flow (Wang et al. Reference Wang, Lin, Guo and Yu2025). The lift force for Poiseuille flow is obtained by correcting the lift for Couette flow as follows:

(3.23) \begin{equation} {C_{\textit{L,Poiseuille}}} = {C_{\textit{L,Couette}}} + {C_{\textit{L,ps1}}}, \end{equation}

where $C_{\textit{L,Couette}}$ is evaluated with the model proposed in our previous study (Wang et al. Reference Wang, Lin, Guo and Yu2025).

For the second approach, inspired by the form of the single-wall-bounded lift model, the lift coefficient for Poiseuille flow is decomposed into

(3.24) \begin{equation} {C_{\textit{L,Poiseuille}}} = {g_{L}}C_{\textit{L,w}} + {h_{L}}{C_{\textit{L,ps2}}}+C_{\textit{L,Sr2}} + \Delta {C_{\textit{L,w}}}, \end{equation}

where $g_{L}$ , $h_{L}$ , $C_{\textit{L,w}}$ and $C_{\textit{L,Sr2}}$ are given in (3.21) and (3.22), respectively. The shear-slip-induced lift term $C_{\textit{L,s}}$ is replaced with $C_{\textit{L,ps2}}$ to account for the effect of the parabolic shear. Here, $\Delta C_{\textit{L,w}}$ is the correction term due to the far side wall, and its model is adopted from our previous study (Wang et al. Reference Wang, Lin, Guo and Yu2025) as

(3.25) \begin{align} \Delta {C_{\textit{L,w}}} &= - \exp \left ( {{\alpha _w}\,Re} \right )(0.0373H + 1.1436){\left ( {\frac {L}{H}} \right )^{0.2098H + 1.2434}}, \end{align}

(3.26) \begin{align} {\alpha _w} &= - \frac {{0.4245{H^{0.2248H - 0.67}}}}{{{L^{0.2248H + 2.189}}}} - 0.4116H + 0.5725. \end{align}

For both methods of lift decomposition, we ensure that $C_{\textit{L,Poiseuille}}$ converges to the single-wall-bounded flow case with the increase of channel width.

3.3.1. Modelling of $C_{\textit{L,ps1}}$

To facilitate the modelling, we model $C_{\textit{L,ps1}}$ at positive and negative ${\textit{Sr}}$ separately. The parameter ranges considered are the same as for the Couette flow case (Wang et al. Reference Wang, Lin, Guo and Yu2025), i.e. $0.1 \leqslant Re \leqslant 100,\ {-0.5} \leqslant Sr \leqslant 0.5,\ 1 \leqslant L \leqslant 6,\ 4 \leqslant H \leqslant 12$ . For positive ${\textit{Sr}}$ , as shown in figures 7(a–f), the correction of the lift force for the Poiseuille flow is negative for relatively low ${\textit{Re}}$ , and becomes positive when ${\textit{Re}}$ exceeds a critical value. From the present DNS data, the critical Reynolds number at which $C_{\textit{L,ps1}}$ changes sign primarily depends on $H$ , with the critical Reynolds number decreasing as $H$ increases. The negative value at low Reynolds numbers indicates that when the fluid inertial effect is small, the curvature effect of the parabolic shear flow produces an addition lift opposite to the lift produced by the linear shear flow, which is more pronounced for larger $L$ . It is reasonable that this part of the lift may overtake the positive part when the particle becomes neutrally buoyant, causing the two-way particle migration (Segré & Silberberg Reference Segré and Silberberg1961).

Figure 7.

Lift coefficient correction term $C_{\textit{L,ps1}}$ for the Poiseuille flow: (a,d,g,j) $H=4$ , (b,e,h,k) $H=8$ , (c,f,i,l) $H=12$ . Subplots from the top to the bottom correspond to $Sr=0.2,0.5,-0.2,-0.5$ , respectively. The lines represent our model (3.27). Symbols and lines with the same colour correspond to the same $L$ .

At moderate Reynolds numbers, the positive $C_{\textit{L,ps1}}$ has not been mentioned in any previous studies, to our knowledge, which induces a significant enhancement in the lift, especially when $C_{L,Couette}$ approaches 0 at $Re \sim O (10 )$ and $L \geqslant 3$ . Here, $C_{\textit{L,ps1}}$ monotonically approaches 0 as ${\textit{Re}}$ further increases, and becomes negligibly small for $Re \geqslant 50$ , except for the case $L=3$ and $H=4$ in figure 7(d), where the quadratic effect remains pronounced. We propose a correlation of $C_{\textit{L,ps1}}$ for $Sr\gt 0$ as

(3.27) \begin{align} {C_{\textit{L,ps1}}} & = S{r^{1 + \left ( {0.5 - Sr} \right )Re/H}}{\left ( {\frac {L}{H}} \right )^{2 + \tanh \left ( {0.067\,Sr\,{{10}^{H - 4}}} \right )\lg \left ( Re \right )}}\notag \\ & \times\left \{ {\frac {{ - 128{H^{ - 1.53}}}}{{Re + 37.3{H^{ - 1.78}}}}\left [ {1 + \left ( {1.2 - Sr} \right )\exp \left ( {4 - H} \right )\left ( {1 - \tanh \left ( {4L - 7.6} \right )} \right )/2} \right ] + 3} \right \}\notag \\ & \times\left \{ {1 - \tanh \left [ {0.8\left ( {\ln (Re) + 0.253{{(\ln H)}^3} - 2.08{{(\ln H)}^2} + 5.72\ln H - 7.02} \right )} \right ]} \right \}. \end{align}

For $Sr\lt 0$ , as shown in figures 7(g–l), the way $C_{\textit{L,ps1}}$ varies with ${\textit{Re}}$ is similar to the case $Sr\gt 0$ to some extent. Here, $C_{\textit{L,ps1}}$ is always opposite to $C_{\textit{L,s}}$ , indicating that the effect from the parabolic shear tends to suppress the linear-shear-induced lift at low Reynolds numbers, irrespective of the sign of ${\textit{Sr}}$ . For moderate Reynolds numbers, in contrast to $Sr\gt 0$ , there scarcely exists a change of sign for most cases, and the magnitude of $C_{\textit{L,ps1}}$ monotonically decreases to 0 as ${\textit{Re}}$ increases. Here, we propose the model of $C_{\textit{L,ps1}}$ for $Sr \lt 0$ as

(3.28) \begin{align} {C_{\textit{L,ps1}}} &= {{Sr\left [ {{a_a}{{\left ( {\frac {L}{H}} \right )}^{{b_a}}} + {c_a}} \right ]} \mathord {\left / {\vphantom {{Sr\left [ {{a_a}{{\left ( {\frac {L}{H}} \right )}^{{b_a}}} + {c_a}} \right ]} {\left \{ {\exp \left [ {{a_b}{{\left ( {\frac {L}{H}} \right )}^{{b_b}}} + {c_b}} \right ]{{{\mathop {\textrm{Re}}\nolimits } }^{1.5}} + 1} \right \}}}} \right . } {\left \{ {\exp \left [ {{a_b}{{\left ( {\frac {L}{H}} \right )}^{{b_b}}} + {c_b}} \right ]{{{\mathop {{Re}}\nolimits } }^{1.5}} + 1} \right \}}}, \end{align}

(3.29) \begin{align} {a_a} &= 44.3{H^{ - 0.5471}} - 18.97,\quad {b_a} = 2.526{H^{0.3053}} - 4.215, \end{align}

(3.30) \begin{align} {c_a} &= - 2149/{H^3} + 621.4/{H^2} - 36.24/H, \end{align}

(3.31) \begin{align} {a_b} &= - 7.907{H^{ - 2.119}},\quad {b_b} = - 0.06425H - 0.7797, \end{align}

(3.32) \begin{align} {c_b} &= 0.01678{H^2} - 0.2107H + 0.6111. \end{align}

The lift force for the Poiseuille flow can now be calculated from (3.23). As shown in figure 8, for small $H$ , $C_{\textit{L,Poiseuille}}$ varies slowly at small ${\textit{Re}}$ . The present model is in good agreement with the DNS data. At low ${\textit{Re}}$ , the maximum relative difference between the lifts for the Poiseuille and Couette (dashed lines) flows is approximately 50 % for positive ${\textit{Sr}}$ , and even larger for negative ${\textit{Sr}}$ , in which case the direction of the lift may be different for two flows, as shown in figures 8(g,j).

Figure 8.

Lift coefficients for the Poiseuille flow, obtained from the correction of the lift coefficient for the Couette flow: (a,d,g,j) $H=4$ , (b,e,h,k) $H=8$ , (c,f,i,l) $H=12$ . Subplots from the top to the bottom correspond to $Sr=0.2,0.5,-0.2,-0.5$ , respectively. Solid and dashed lines correspond to the present model (3.23) and the model for the Couette flow (Wang et al. Reference Wang, Lin, Guo and Yu2025). Symbols and lines with the same colour correspond to the same $L$ .

3.3.2. Modelling of $C_{\textit{L,ps2}}$

Here, we first propose the lift model for the unbounded Poiseuille flow based on the low Reynolds number theoretical results. Asmolov (Reference Asmolov1999) theoretically solved the problem for a large channel Reynolds number by the method of matched asymptotic expansion. The problem is considered as ’unbounded’ when the wall effect is negligible and the lift was obtained by numerical integration. The lift coefficient for unbounded parabolic flow could take the form

(3.33) \begin{equation} {C_{\textit{L,ps2}}} = \frac {{{18}}}{{{\pi ^{2}}}}{J_{\textit{L,ps}}}\varepsilon , \end{equation}

where $J_{\textit{L,ps}}$ is fitted based on the results by Asmolov (Reference Asmolov1999) as

(3.34) \begin{equation} {J_{\textit{L,ps}}} = 2.255\left ( {1 + {\sigma ^{0.9}}/0.8} \right )^{-1}{\left [ {1 + 0.2{{\left ( {{\textrm{sgn}(u_{\textit{slip}})\,\varepsilon ^{ - 1}} - 1.7{\sigma ^{0.6}}} \right )}^2}} \right ]^{ - 2/(1+2\sigma )}}. \end{equation}

In our simulations, the particle is always located in the lower half-channel, thus $\textrm{sgn}(u_{\textit{slip}})=\textrm{sgn}(Sr)$ . Here, $\sigma$ is given by

(3.35) \begin{equation} \sigma = {\left ( {2\left ( {H - L} \right )\sqrt {Re\left | {Sr} \right |} } \right )^{{ - 1}}}, \end{equation}

and $J_{\textit{L,ps}}$ reduces to the form for linear shear when $(H-L)$ or $Re\,|Sr|$ is large, corresponding to small curvature or the large Reynolds number limit.

Then we consider the second way of modelling, as given by (3.24). Compared to the first way, here we focus on relatively lower slip velocities and a wider range of shear rates as $0.1 \leqslant Re \leqslant 20$ and $-2.5 \leqslant Sr \leqslant 2.5$ . Since small particles are more widely encountered in engineering applications, the minimum width of the channel $H=8$ is considered here.

Assuming that the shear effect and the wall effect are coupled in a similar way as in the linear shear case, we can obtain the the lift model for the Poiseuille flow by substituting (3.33) into (3.24). The present model is plotted as solid lines in figure 9, compared with the present data and the single-wall-bounded linear shear model (3.20). Here, $C_{\textit{L,}w\textit{s}}$ provides good predictions for moderate Reynolds numbers, while for low Reynolds numbers, $C_{\textit{L,}w\textit{s}}$ fails to reproduce the lift reduction as $L$ increases to 6 for $H=8$ and $Sr\gt 0$ . This discrepancy arises not only from the confinement effect of the second wall but primarily from the non-negligible negative quadratic-shear-induced lift. Though the second model here does not give predictions of lift as precise as in the first model within the parameter space that we explored, it captures most of the lift coefficients well. The second model takes advantage of its simplicity and promising capability to capture the lift over a wider range of $L$ and ${\textit{Sg}}$ , which benefits from its semi-empirical essence. With the increase of $H, 1/L,Re, Sr$ , $C_{\textit{L,Poiseuille}}$ converges to $C_{\textit{L,}w\textit{s}}$ . Further adjustments could be made by considering the interaction of the shear effect and wall effect more precisely.

Figure 9.

Lift coefficients for the Poiseuille flow: (a–d) $Sr=0.2,0.5,1.5,2.5$ with $H=8$ ; (e–h) $Sr=0.2, 0.5,1.5,2.5$ with $H=12$ ; (i–l) $Sr=-0.2,-0.5,-1.5,-2.5$ with $H=8$ ; (m–p) $Sr=-0.2,-0.5, -1.5,-2.5$ with $H=12$ , respectively. Solid and dashed lines correspond to the present model for $C_{\textit{L,Poiseuille}}$  (3.24) and $C_{\textit{L,}w\textit{s}}$  (3.20). Symbols and lines of the same colour correspond to the same $L$ .

3.4. Lift and torque from rotation

The rotation-induced lift is calculated by subtracting the lift on a non-rotating particle from the lift when the particle is forced to rotate. For linear shear flow, it has been recognised that $C_{L,\varOmega }$ is proportional to the particle rotation rate, while being independent of the fluid shear rate in a wide range of ${\textit{Re}}$ and $Rr$ (Kurose & Komori Reference Kurose and Komori1999; Bagchi & Balachandar Reference Bagchi and Balachandar2002; Wang et al. Reference Wang, Lin, Guo and Yu2025). For the Couette flow, the far side wall was found to have negligible effects on $C_{L,\varOmega }$ , while the near side wall effect was taken into consideration by our previous model (Wang et al. Reference Wang, Lin, Guo and Yu2025), in the form

(3.36) \begin{equation} C_{L,\varOmega }^{ws} = \left [ {\left ( {\frac {{ - 1}}{{0.29 + 2.9L}} + 0.55} \right ){{\textrm e}^{\left ( {0.1{L^{ - 2.1}} - 0.16} \right )Re}} + 0.45\,{{\textrm e}^{0.0032\,Re}}} \right ]Rr. \end{equation}

For the Poiseuille flow, figure 10 shows the variation of the rotation-induced lift coefficient with ${\textit{Sg}}$ , which is the non-dimensional curvature of the velocity profile (i.e. the gradient of the fluid shear rate in the wall-normal direction) and is represented as

(3.37) \begin{equation} Sg = \frac {{{\partial ^2}{u_f}}}{{\partial {y^2}}}\frac {{{d^2}}}{{{u_{\textit{slip}}}}} = \begin{cases} - \dfrac {{Sr}}{{H - L}}\quad(L \lt H)& \textrm{Poiseuille},\\ 0&{\textrm{Couette}}. \end{cases} \end{equation}

The rotation-induced lift for the Poiseuille flow ( $C_{L,\varOmega }^{ps}$ ) approaches the value for the Couette flow ( $C_{L,\varOmega }^{ws}$ ) as $|Sg|$ decreases. Note that the zero value of ${\textit{Sg}}$ is only strictly reached when the sphere is in the Couette flow. In figure 10, the straight lines can approximately fit the data points at different ${\textit{Sg}}$ , especially for low Reynolds numbers. The slope decreases with increasing ${\textit{Re}}$ , while it increases with increasing $L$ , indicating that the rotation-induced lift for the Poiseuille flow $C_{L,\varOmega }^{ps}$ becomes more sensitive to the curvature when the particle is located closer to the channel centre. Based on these observations and the results in figure 10, we propose a correlation for the rotation-induced lift coefficient for the Poiseuille flow as

(3.38) \begin{equation} C_{L,\varOmega }^{ps} = \left [ {0.143\,{Re^{ - 0.282}}L - 0.045\,{Re^{ - 0.78}} + 0.2/\left ( {1 + {Re^2}} \right )} \right ]Sg\,Rr + C_{L,\varOmega }^{ws}. \end{equation}

Figure 10.

The rotation-induced lift coefficient normalised with $Rr$ ( $-0.1\text{ to }0.2$ ) for $H=4{-}25$ at (a) ${\textit{Re}}=0.1$ , (b) ${\textit{Re}}=0.5$ , (c) ${\textit{Re}}=2$ , (d) ${\textit{Re}}=10$ , (e) ${\textit{Re}}=50$ , (f) ${\textit{Re}}=100$ . The lines represent the model (3.38). Symbols and lines of the same colour correspond to the same $L$ .

It should be noted that in our simulations the particle is always located in the lower half-channel where the fluid shear rate is positive, and positive ${\textit{Sr}}$ means lagging particle defined as positive slip velocity, and negative ${\textit{Sr}}$ means leading particle defined as negative slip velocity. Thus both Saffman-type (shear-slip-induced) lift force and the Magnus force on the clockwise-rotating (same rotation direction as the local fluid) particle point to the channel centre (i.e. positive lift) for positive ${\textit{Sr}}$ , and point to the near wall (i.e. negative lift) for negative ${\textit{Sr}}$ . In figure 10, ${\textit{Sg}}$ is varied by changing ${\textit{Sr}}$ , being positive for negative ${\textit{Sr}}$ , and negative for positive ${\textit{Sr}}$ from the definition (3.37). The rotation of the particle is always clockwise, then the dimensionless particle rotational velocity $Rr$ is negative for positive ${\textit{Sg}}$ , which corresponds to negative ${\textit{Sr}}$ and negative slip velocity, and is positive for negative ${\textit{Sg}}$ , which corresponds to positive ${\textit{Sr}}$ and positive slip velocity. For $Sg\gt 0$ , both the lift and $Rr$ are negative, thus their ratio is positive. From figure 10, the effect of a positive (negative) ${\textit{Sg}}$ is to exert an additional force pointing to the same (opposite) direction as predicted by the Magnus effect. Consequently, the predicted rotation-induced lift force changes sign when ${\textit{Sg}}$ is below a certain negative value (only in the case $Sg\lt 0$ for a lagging particle). Thus unlike the inverse Magnus effect, which is associated with the turbulence transition of the boundary layer on the advancing side (Kray, Franke & Frank Reference Kray, Franke and Frank2005; Muto, Tsubokura & Oshima Reference Muto, Tsubokura and Oshima2012; Kim et al. Reference Kim, Choi, Park and Yoo2014), we observe that an inverse Magnus effect occurs at a low Reynolds number, as shown in figure 10(a) for ${\textit{Re}}=0.1$ and $L=6$ .

Then we attempt to investigate the reason for the direction change of the rotation-induced lift. The rotation-induced local viscous stress and pressure coefficients are defined as the non-dimensional viscous stress and pressure difference between rotating and non-rotating sphere in the $y$ -direction:

(3.39) \begin{equation} C_{f,\varOmega }=\left .\frac {\boldsymbol n\boldsymbol{\cdot }\boldsymbol \tau \boldsymbol{\cdot }\boldsymbol{e_y}}{\rho u_{\textit{slip}}^2} \right |_{{\varOmega _z\neq 0}}-\left .\frac {\boldsymbol n\boldsymbol{\cdot }\boldsymbol \tau \boldsymbol{\cdot }\boldsymbol{e_y}}{\rho u_{\textit{slip}}^2} \right |_{\varOmega _z = 0},\quad\!\! C_{p,\varOmega }=\left .\frac { -p \boldsymbol n \boldsymbol{\cdot }\boldsymbol{e_y}}{\rho u_{\textit{slip}}^2} \right |_{\varOmega _z\neq 0}-\left .\frac { -p \boldsymbol n \boldsymbol{\cdot }\boldsymbol{e_y}}{\rho u_{\textit{slip}}^2} \right |_{\varOmega _z = 0}. \end{equation}

To examine the effect of the parabolic shear on the lift force, we calculate the difference in the local rotation-induced stress coefficients between the Poiseuille and Couette flows. The surface contours of viscous stress difference ( $\Delta C_{f, \varOmega }$ ) and pressure difference ( $\Delta C_{p, \varOmega }$ ) seen from the positive $z$ -axis (i.e. spanwise) direction are shown in figure 11. For ${\textit{Re}}=0.1$ and $Sr=0.5$ , the Magnus force points to the channel centre. From figures 11(a,c) for this case, it is the viscous stress that produces the negative lift (pointing to the near wall) and leads to an inverse Magnus force at a relatively large magnitude of ${\textit{Sg}}$ . For ${\textit{Re}}=100$ and $Sr=-0.5$ , the Magnus force points to the channel wall. From figures 11(b,d) for this case, it is the pressure that produces the negative lift (pointing to the near wall) to enhance the Magnus lift.

Figure 11.

Surface contours showing the differences in the $y$ -component of the rotation-induced viscous stress and pressure coefficients between Couette and Poiseuille flow configurations at $L=3,\ H=4$ , seen from the positive $z$ -axis (i.e. spanwise) direction. Viscous stress coefficient difference $\Delta C_{f, \varOmega }$ at (a) ${\textit{Re}}=0.1,\ Sr=0.5,\ Rr=0.1$ , and (b) ${\textit{Re}}=100,\ Sr=-0.5,\ Rr=-0.1$ ; pressure coefficient difference $\Delta C_{p, \varOmega }$ at (c) ${\textit{Re}}=0.1,\ Sr=0.5,\ Rr=0.1$ , and (d) ${\textit{Re}}=100,\ Sr=-0.5,\ Rr=-0.1$ .

We have shown that the rotation-induced lift can be opposite to the Magnus force for a lagging particle in Poiseuille flow. However, it is not clear whether the total lift force can be opposite to the Magnus force. Figure 12 shows the linear variation of lift and torque coefficients with $Rr$ for both Couette and Poiseuille flows in a wide range of $Rr$ at ${\textit{Re}}=0.1$ . The direction of the rotation-induced lift force is altered by the curvature effect of the velocity profile. At $Rr=10$ , the rotation-induced lift exceeds the shear-slip-induced lift on a non-rotating particle, resulting in a total lift force with direction opposite to that of Magnus force. The torques for two flows have no discernible difference at low ${\textit{Re}}$ , and the equilibrium state of torque-free rotation is achieved approximately at $Rr=1.25$ .

Figure 12.

The total lift and torque coefficients versus the non-dimensional rotation rate at ${\textit{Re}}=0.1,\ Sr=2.5,\ L=6$ and $H=8$ .

The lift is dependent on the particle rotational velocity, which depends on the torque on the particle. Therefore, the model for the torque is needed for the application of the lift model to a rotating particle. The torque coefficient for a rotating particle can be modelled as

(3.40) \begin{equation} {C_T} = \left ( {Rr_{\textit{eq}} - Rr} \right ){f_T}, \end{equation}

where $Rr_{\textit{eq}}$ is the non-dimensional particle rotational velocity at the torque-free equilibrium state, and $f_T$ is a torque factor. For low ${\textit{Sr}}$ ( $\leqslant 0.5$ ), Shi et al. (Reference Shi, Rzehak, Lucas and Magnaudet2021) modified the model of Bagchi & Balachandar (Reference Bagchi and Balachandar2002) for the unbounded linear shear flow to account for the influence of the nearby wall, while the wall effect is insignificant when the particle–wall separation distance is greater than one particle diameter. We propose a new model for the wall-bounded linear shear flow based on their models and our DNS data, as

(3.41) \begin{equation} R{r_{\textit{eq,}w\textit{s}}} = Rr_{\textit{eq,}w\textit{s}}^{ls}\left ( {1 + 0.015\tanh \left ( {0.2\,Re} \right ) {{\left | {Sr} \right |}^{2.2}}} \right ), \end{equation}

with

(3.42) \begin{equation} Rr_{\textit{eq,}w\textit{s}}^{ls} = \begin{cases} Rr_{\textit{eq,}w\textit{s}}^{\textit{Shi}},&L\leqslant 1,\\[5pt] Rr_{\textit{eq,}w\textit{s}}^{{B\&B}},&L \gt 1, \end{cases} \end{equation}

where

(3.43) \begin{align} Rr_{\textit{eq,}w\textit{s}}^{{\textit{Shi}}} = {}& - \frac {{{C_{\textit{L,w0}}}}}{{96{L^4}}}\left ( {1 - \frac {3}{{16L}} + \frac {9}{{80{L^3}}}} \right ) \notag \\ &{}+ \frac {{Sr}}{2}\left \{ {{f_\varOmega } + \frac {1}{{32{L^4}}}\tanh \left ( {\frac {{Re}}{2}} \right )\left [ {1 + {\mathop {\textrm{sgn}}} \left ( {Sr} \right )} \right ]} \right \}\left \{ {1 - \frac {5}{{128{L^3}}}\,{{\textrm e}^{ - Re/2}}} \right \} \end{align}

and

(3.44) \begin{equation} Rr_{\textit{eq,}w\textit{s}}^{{\textit{B&B}}} = \frac {{Sr}}{2}{f_\varOmega }, \end{equation}

with

(3.45) \begin{equation} {f_\varOmega } = {\begin{cases} 1 - 0.0364\,Re^{0.95},&0.5 \leqslant Re \leqslant 5,\\ 1 - 0.0755\,Re^{0.455},&5 \lt Re. \end{cases}} \end{equation}

From figure 13, the model of Bagchi & Balachandar (Reference Bagchi and Balachandar2002) is in good agreement with our DNS data. However, it treats $Rr_{\textit{eq, ws}}$ as a linear function of ${\textit{Sr}}$ , and appears to underestimate the free rotation rate at relatively high ${\textit{Sr}}$ and ${\textit{Re}}$ , as shown in figure 13(b). Thus a correction is made to the models of Bagchi & Balachandar (Reference Bagchi and Balachandar2002) and Shi et al. (Reference Shi, Rzehak, Lucas and Magnaudet2021) to obtain our model (3.41), which captures the nonlinear increase of $Rr_{\textit{eq, ws}}$ with ${\textit{Sr}}$ , as shown in figure 13.

Figure 13.

The non-dimensional free rotation rate for wall-bounded linear shear flow at (a) ${\textit{Re}}=0.1$ , (b) ${\textit{Re}}=10$ , (c) ${\textit{Re}}=100$ . The present model is (3.41). Symbols and lines of the same colour correspond to the same $L$ .

Then, based on our DNS data, we obtain a new model for the free rotation rate for a particle in the Poiseuille flow by modifying the model for the wall-bounded linear shear flow:

(3.46) \begin{equation} R{r_{\textit{eq,Poiseuille}}} = R{r_{\textit{eq,}w\textit{s}}} \left ( 1.3 - 0.3\exp\left( - Sg\left [ \tanh \left ( \ln \left ( {Re} \right ) - 3 \right ) + 1 \right ] \left [ 1 - \tanh \left ( {4\left | {Sr} \right | - 3} \right ) \right ]/2\right) \right ). \end{equation}

As shown in figure 14, the free rotation rate model proposed by Bagchi & Balachandar (Reference Bagchi and Balachandar2002) for linear shear flow well captures the free rotation rate for the Poiseuille flow at low Reynolds numbers, whereas our model is able to predict the nonlinear variation of the free rotation rate at relatively high ${\textit{Re}}$ and ${\textit{Sr}}$ .

Figure 14.

The non-dimensional rotation rate for Poiseuille flow at (a) ${\textit{Re}}=0.1$ , (b) ${\textit{Re}}=2$ , (c) ${\textit{Re}}=10$ , (d) ${\textit{Re}}=20$ , (e) ${\textit{Re}}=50$ , (f) ${\textit{Re}}=100$ . Symbols and lines of the same colour correspond to the same $L$ as in figure 13.

Regarding the lift factor $f_T$ , we employ the same model as for the Couette flow proposed in our previous study (Wang et al. Reference Wang, Lin, Guo and Yu2025), which is a function of ${\textit{Re}}$ and $L$ as

(3.47) \begin{equation} {{f_{T}} = \frac {{16}}{{Re}}\left ( {1 + \frac {5}{{128{L^3}}} + {{\textrm e}^{ - 5.4\,Re^{ - 0.3}}}} \right ).} \end{equation}

As shown in figure 15, the torque coefficients at different $L$ almost collapse into a single line, and are well predicted by the present model (3.40), (3.46) and (3.47).

Figure 15.

The torque coefficients for (a) ${\textit{Re}} = 0.1$ , (b) ${\textit{Re}} = 0.5$ , (c) ${\textit{Re}} = 2$ , (d) ${\textit{Re}} =10$ , (e) ${\textit{Re}} = 50$ , (f) ${\textit{Re}} = 100$ . Symbols represent DNS data, and the lines represent our model (3.40), (3.46) and (3.47). Symbols and lines of the same colour correspond to the same $L$ .