3.1. The generalized reciprocal theorem

To derive the generalized reciprocal theorem identity, we contract (2.4a) with $\boldsymbol {u}^t$ and (3.1a) with $\boldsymbol {u}'$, and subtract the resulting expressions (Ho & Leal Reference Ho and Leal1974) to obtain

(3.3)\begin{equation} (\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{\sigma}')\boldsymbol{\cdot} \boldsymbol{u}^t-(\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{\sigma}^t) \boldsymbol{\cdot} \boldsymbol{u}'=Re_p\,\boldsymbol{u}^t\boldsymbol{\cdot}\boldsymbol{f}, \end{equation}

where $\boldsymbol {f}=({\partial \boldsymbol {u}'}/{\partial t}+\boldsymbol {u}'\boldsymbol {\cdot }\boldsymbol {\nabla } \boldsymbol {u}'+\boldsymbol {u}'\boldsymbol {\cdot } \boldsymbol {\nabla } \boldsymbol {u}^\infty +\boldsymbol {u}^\infty \boldsymbol {\cdot } \boldsymbol {\nabla } \boldsymbol {u}')$. Integrating (3.3) over the fluid volume ($V^F$) between the channel walls gives

(3.4)\begin{align} \int_{V^F} \boldsymbol{\nabla}\boldsymbol{\cdot}(\boldsymbol{\sigma}'\boldsymbol{\cdot} \boldsymbol{u}^t-\boldsymbol{\sigma}^t\boldsymbol{\cdot}\boldsymbol{u}')\,{\rm d}V + \int_{V^F} (\boldsymbol{\sigma}^t\boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{u}'-\boldsymbol{\sigma}' \boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u}^t)\,{\rm d}V = Re_p\int_{V^F}\boldsymbol{u}^t\boldsymbol{\cdot}\boldsymbol{f}\,{\rm d}V. \end{align}

Using (2.4b) and (3.1b), along with the definitions of $\boldsymbol {\sigma }'$ and $\boldsymbol {\sigma }^t$, the second integral on the left-hand side of (3.4) can be shown to be identically zero. Applying the divergence theorem to the first integral yields

(3.5)\begin{equation} -\int_{S_p+S_w+S_\infty} (\boldsymbol{\sigma}'\boldsymbol{\cdot}\boldsymbol{u}^t-\boldsymbol{\sigma}^t\boldsymbol{\cdot} \boldsymbol{u}')\boldsymbol{\cdot}\boldsymbol{n}\,{\rm d}S = Re_p\int_{V^F}\boldsymbol{u}^t\boldsymbol{\cdot}\boldsymbol{f} \,{\rm d}V, \end{equation}

where $\boldsymbol {n}$ is the unit normal to all bounding surfaces pointing into the fluid domain. The set of bounding surfaces include the particle surface ($S_p$), the channel walls ($S_w$) and the surface at infinity ($S_\infty$); the latter can be thought of as the curved surface of a cylinder of radius $R$, with its axis along the gradient direction, in the limit $R\to \infty$. For $R\gg H$, the disturbance fields in the actual and test problems are exponentially small (see (A25) and (3.32)), and therefore, the integral over $S_\infty$ in (3.5) is vanishingly small. The integral $\int _{S_w} (\boldsymbol {\sigma }'\boldsymbol {\cdot }\boldsymbol {u}^t-\boldsymbol {\sigma }^t\boldsymbol {\cdot }\boldsymbol {u}')\boldsymbol {\cdot } \boldsymbol {n}\,{\rm d}S$ can be shown to be identically zero on using the no-slip conditions (2.5c) and (3.2c) on the channel walls. Thus, (3.5) reduces to

(3.6)\begin{equation} -\int_{S_p} (\boldsymbol{\sigma}'\boldsymbol{\cdot}\boldsymbol{u}^t-\boldsymbol{\sigma}^t\boldsymbol{\cdot} \boldsymbol{u}')\boldsymbol{\cdot}\boldsymbol{n}\,{\rm d}S =Re_p\int_{V^F}\boldsymbol{u}^t \boldsymbol{\cdot}\boldsymbol{f}\,{\rm d}V. \end{equation}

Applying the no-slip conditions (2.5a) and (3.2a) to (3.6),

(3.7)\begin{align} &-\boldsymbol{U}_p^t\boldsymbol{\cdot}\int_{S_p} \boldsymbol{\sigma}'\boldsymbol{\cdot}\boldsymbol{n}\,{\rm d}S - \boldsymbol{\varOmega}_p^t\boldsymbol{\cdot}\int_{S_p}\boldsymbol{r}\wedge(\boldsymbol{\sigma}' \boldsymbol{\cdot}\boldsymbol{n})\,{\rm d}S + \boldsymbol{\varOmega}_p \boldsymbol{\cdot}\int_{S_p} \boldsymbol{r}\wedge(\boldsymbol{\sigma}^t\boldsymbol{\cdot} \boldsymbol{n})\,{\rm d}S\nonumber\\ &\quad +\boldsymbol{U}_p\boldsymbol{\cdot}\int_{S_p} \boldsymbol{\sigma}^t\boldsymbol{\cdot} \boldsymbol{n}\,{\rm d}S-\int_{S_p} (\boldsymbol{\sigma}^t\boldsymbol{\cdot}\boldsymbol{n})\boldsymbol{\cdot}(\alpha+\beta r_2+\gamma r_2^2)\boldsymbol{1}_1\,{\rm d}S=Re_p\int_{V^F}\boldsymbol{u}^t\boldsymbol{\cdot}\boldsymbol{f} \,{\rm d}V. \end{align}

The hydrodynamic force and torque experienced by the particle may be written as $\int _{S_p} \boldsymbol {\sigma }'\boldsymbol {\cdot }\boldsymbol {n}\,{\rm d}S=Re_p\,{\rm d}\boldsymbol {U}_p/{\rm d}t$ and $\int _{S_p} \boldsymbol {r}\wedge (\boldsymbol {\sigma }'\boldsymbol {\cdot }\boldsymbol {n})\,{\rm d}S=Re_p\, {\rm d}(\boldsymbol {I}_p\boldsymbol {\cdot }\boldsymbol {\varOmega }_p)/{\rm d}t$, $\boldsymbol {I}_p$ being the spheroid moment of inertia tensor. Next, to $O(Re_p)$, one may replace $\boldsymbol {u}'$ by $\boldsymbol {u}_s$ in the volume integral in (3.7), $\boldsymbol {u}_s$ being the corresponding Stokesian approximation. The validity of such a replacement requires the resulting volume integral to be convergent, this being related to inertia acting as a regular perturbation. Detailed arguments in this regard are given in the next subsection. Furthermore, noting that $\int _{S_p} \boldsymbol {r}\wedge (\boldsymbol {\sigma }^t\boldsymbol {\cdot }\boldsymbol {n})\,{\rm d}S = 0$ owing to the spheroid being torque free in the test problem, one obtains

(3.8)\begin{align} &-Re_p\,\boldsymbol{U}_p^t\boldsymbol{\cdot}\frac{{\rm d}\boldsymbol{U}_p}{{\rm d}t} -Re_p\, \boldsymbol{\varOmega}_p^t\boldsymbol{\cdot}\frac{{\rm d}(\boldsymbol{I}_p\boldsymbol{\cdot} \boldsymbol{\varOmega}_p)}{{\rm d}t}+\boldsymbol{U}_p\boldsymbol{\cdot}\int_{S_p} \boldsymbol{\sigma}^t\boldsymbol{\cdot}\boldsymbol{n}\,{\rm d}S\nonumber\\ &\quad-\int_{S_p} (\boldsymbol{\sigma}^t\boldsymbol{\cdot}\boldsymbol{n})\boldsymbol{\cdot}(\alpha+\beta r_2+\gamma r_2^2)\boldsymbol{1}_1\,{\rm d}S=Re_p\int_{V^F}\boldsymbol{u}^t \boldsymbol{\cdot}\boldsymbol{f}_s\,{\rm d}V, \end{align}

where $\boldsymbol {f}_s$ denotes the approximation of $\boldsymbol {f}$ based on replacing $\boldsymbol {u}'$ by $\boldsymbol {u}_s$, being given by

(3.9)\begin{equation} \boldsymbol{f}_s=\frac{\partial\boldsymbol{u}_s}{\partial t}+\boldsymbol{u}_s\boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{u}_s+\boldsymbol{u}_s\boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{u}^\infty+\boldsymbol{u}^\infty\boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{u}_s. \end{equation}

While $\boldsymbol {u}_s$ above is the Stokesian disturbance field induced by an arbitrarily oriented neutrally buoyant spheroid in a bounded plane Poiseuille flow, as will be seen in § 3.2, at leading order in $\lambda$, the spheroid may be replaced by the corresponding stresslet singularity, with the quadrupolar component of the disturbance field (that arises in response to the quadratic part of the ambient flow) being neglected.

One may now use the small-$Re_p$ expansions, $\boldsymbol {U}_p=\boldsymbol {U}_{p0}+Re_p \boldsymbol {U}_{p1}+O(Re_p^2)$ and $\boldsymbol {\varOmega }_p=\boldsymbol {\varOmega }_{p0}+Re_p \boldsymbol {\varOmega }_{p1}+O(Re_p^2)$, for the spheroid translational and angular velocities, in (3.8), and obtain the following relations at successive orders in $Re_p$.

1. (i) $O(1)$:

   (3.10)\begin{equation} \boldsymbol{U}_{p0}\boldsymbol{\cdot}\int_{S_p} \boldsymbol{\sigma}^t\boldsymbol{\cdot} \boldsymbol{n}\,{\rm d}S=\int_{S_p} (\boldsymbol{\sigma}^t\boldsymbol{\cdot} \boldsymbol{n})\boldsymbol{\cdot}(\alpha+\beta r_2+\gamma r_2^2) \boldsymbol{1}_1\,{\rm d}S; \end{equation}

2. (ii) $O(Re_p)$:

   (3.11)\begin{equation} -\boldsymbol{U}_p^t\boldsymbol{\cdot}\frac{{\rm d}\boldsymbol{U}_{p0}}{{\rm d}t}- \boldsymbol{\varOmega}_p^t\boldsymbol{\cdot}\frac{{\rm d}(\boldsymbol{I}_p\boldsymbol{\cdot} \boldsymbol{\varOmega}_{p0})}{{\rm d}t}+\boldsymbol{U}_{p1}\boldsymbol{\cdot} \int_{S_p} \boldsymbol{\sigma}^t\boldsymbol{\cdot}\boldsymbol{n}\,{\rm d}S= \int_{V^F}\boldsymbol{u}^t\boldsymbol{\cdot}\boldsymbol{f}_s\,{\rm d}V .\end{equation}

3.1.1. The $O(1)$ problem

For the $O(1)$ problem defined by (3.10), we choose $\int _{S_p} \boldsymbol {\sigma }^t\boldsymbol {\cdot } \boldsymbol {n}\,{\rm d}S=\boldsymbol {1}_1$, corresponding to the test spheroid translating due to a unit flow-aligned force. In the absence of boundaries, the induced velocity field is known in terms of vector spheroidal harmonics (Dabade, Marath & Subramanian Reference Dabade, Marath and Subramanian2015), and the surface force density to be used on the right-hand side is given by (Kushch & Sangani Reference Kushch and Sangani2003)

(3.12)\begin{equation} \boldsymbol{\sigma}^t\boldsymbol{\cdot} \boldsymbol{n} =-P^t \boldsymbol{1}_\xi+2\left(\frac{(\xi^2-1)^{1/2}}{\xi_0 (\xi^2-\eta^2)^{1/2}} \frac{\partial \boldsymbol{u}^t}{\partial \xi}+\frac{1}{2}\boldsymbol{1}_\xi\wedge\boldsymbol{\nabla}\wedge\boldsymbol{u}^t\right), \end{equation}

where $\xi \geq \xi _0>1$ is the coordinate characterizing a family of confocal spheroids with $\xi _0$ representing the spheroid surface and $\boldsymbol {1}_\xi$ is the unit normal to the spheroid surface; note that $\kappa =\xi _0/(\xi _0^2-1)^{1/2}$ and $\kappa =(\xi _0^2-1)^{1/2}/\xi _0$ for prolate and oblate spheroids, respectively. Evaluating the surface integral in (3.10), using (3.12), one obtains

(3.13)\begin{equation} \boldsymbol{U}_{p0}= \left[ \alpha+\frac{\gamma}{3\kappa^2}[\cos^2\phi_j+(\cos^2\theta_j+\kappa^2\sin^2\theta_j)\sin^2\phi_j] \right]\boldsymbol{1}_1, \end{equation}

which is Faxen's law for the translation of a force-free spheroid truncated to second order, the term proportional to $\gamma$ arising from the curvature of the ambient flow; higher-order terms in the expansion are zero for an ambient quadratic flow (Happel & Brenner Reference Happel and Brenner2012). (3.13) pertains to a spheroid of given orientation, and a time trajectory requires knowing $\theta _j$ and $\phi _j$ as functions of time. For this purpose, one may use Faxen's law relating the torque and angular velocity of a spheroid. This is again an infinite expansion in the general case, but with only the leading-order term, proportional to the ambient velocity gradient, surviving for a quadratic flow. Thus, spheroid rotation remains identical to that in an ambient linear flow, and application of the torque-free constraint yields

(3.14)\begin{align} \boldsymbol{\varOmega}_{p0}&=-\frac{\beta(\kappa^2-1)}{4(\kappa^2+1)}\cos\phi_j\sin 2\theta_j\boldsymbol{1}_1+ \frac{\beta(\kappa^2-1)}{4(\kappa^2+1)}\sin\phi_j\sin 2\theta_j\boldsymbol{1}_2\nonumber\\ &\quad +\frac{\beta}{2}\left[-1+\frac{(\kappa^2-1)}{(\kappa^2+1)}\cos 2\phi_j\sin^2 \theta_j\right] \boldsymbol{1}_3. \end{align}

The equations governing Jeffery orbits arise as components of (3.14) in the body-aligned coordinate system, and are given below in (3.18a,b). With $\theta _j$ and $\phi _j$ defined in this manner, $\boldsymbol {U}_{p0}$ as defined in (3.13) exhibits the time dependence described earlier in § 2. Substituting $\boldsymbol {U}_{p0}$ in (2.3), the ambient flow in the reference frame chosen for the reciprocal theorem is given by

(3.15)\begin{equation} \boldsymbol{u}^\infty=\left[ (\beta r_2 +\gamma r_2^2)-\frac{\gamma}{3\kappa^2}[\cos^2\phi_j+(\cos^2\theta_j+\kappa^2\sin^2\theta_j)\sin^2\phi_j] \right] \boldsymbol{1}_1. \end{equation}

Note that the $O(\lambda ^2)$ centre-of-mass oscillations mentioned in § 2 can only arise from $\boldsymbol {U}_{p0}$ having a component along the gradient direction, and this requires an expression for $\boldsymbol {\sigma }^t \boldsymbol {\cdot } \boldsymbol {n}$ that includes the influence of the plane boundaries.

3.1.2. The $O(Re_p)$ problem

At $O(Re_p)$, one chooses the test force as $\int _{S_p} \boldsymbol {\sigma }^t\boldsymbol {\cdot }\boldsymbol {n}\, {\rm d}S=-\boldsymbol {1}_2$, which leads to

(3.16)\begin{align} V_p=Re_p\,\boldsymbol{U}_{p1}\boldsymbol{\cdot}\boldsymbol{1}_2 &=-Re_p\int_{V^F}\boldsymbol{u}^{t2}\boldsymbol{\cdot}\left(\frac{\partial\boldsymbol{u}_s}{\partial t}+\boldsymbol{u}_s\boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{u}_s+\boldsymbol{u}_s \boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u}^\infty+\boldsymbol{u}^\infty \boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{u}_s\right) {\rm d}V\nonumber\\ &\quad -Re_p\,\boldsymbol{U}_p^{t2}\boldsymbol{\cdot} \frac{{\rm d}\boldsymbol{U}_{p0}}{{\rm d}t}-Re_p\,\boldsymbol{\varOmega}_p^{t2} \boldsymbol{\cdot}\frac{{\rm d}(\boldsymbol{I}_p\boldsymbol{\cdot}\boldsymbol{\varOmega}_{p0})}{{\rm d}t} \end{align}

for the instantaneous lift velocity of a neutrally buoyant spheroid in plane Poiseuille flow, where the volume integral is a function of the changing spheroid orientation via $\boldsymbol {u}^{t2}$ and $\boldsymbol {u}_s$. The additional superscript ‘$2$’ for the test problem quantities indicate the choice of the test force orientation above.

As already mentioned, for small confinement ratios, a spheroid in wall-bounded plane Poiseuille flow rotates along Jeffery orbits (Jeffery Reference Jeffery1922) in the Stokes limit. For small but finite $Re_p$, there is an additional $O(Re_p)$ drift across orbits that stabilizes the tumbling mode ($C=\infty$) for prolate spheroids, the spinning mode ($C=0$) for oblate spheroids with $0.14<\kappa <1$ and, depending on the initial orientation, either the spinning or tumbling mode for oblate spheroids with $\kappa <0.14$ (Einarsson et al. Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015; Dabade et al. Reference Dabade, Marath and Subramanian2016). The time scale for rotation along a Jeffery orbit is $T_{jeff}\sim O(H/V_{max})$ for $\kappa \sim O(1)$. The inertia-driven orbital drift above occurs on longer time scales $t_{drift} \sim O(Re_p^{-1}H/V_{max})$. Scaling arguments in § 3.2 below show that the volume integral in (3.16) is the dominant contribution to the cross-stream migration, being $O(1)$ for $\lambda \ll 1$, so the time scale for inertial migration is $t_{lift} \sim O(Re_p^{-1}\lambda ^{-1}H/V_{max})$. Since $t_{lift}/t_{drift}\sim \lambda ^{-1}\gg 1$, for purposes of the migration calculation, one may assume that the spheroid has settled into its stable Jeffery orbit. Moreover, since $t_{lift}/T_{jeff}\sim Re_p^{-1}\lambda ^{-1} \gg 1$, the leading-order migration is the result of an orientation-averaged lift velocity, the average being over the orientations sampled in the stabilized Jeffery orbit. As indicated below, this average may be approximated based on the Jeffery angular velocity for small $Re_p$. Superposed on this orientation-averaged migration trajectory would be small-amplitude oscillations of $O(Re_p\lambda )$, corresponding to the fluctuations in the instantaneous lift velocity arising from the rapidly changing spheroid orientation, as may be established formally using the method of multiple scales.

Owing to the aforementioned time scale separation in the limit $\lambda, Re_p \ll 1$, it is of interest to determine the Jeffery-averaged rather than the instantaneous lift velocity. Averaging both sides of (3.16) over a Jeffery period, one obtains

(3.17)\begin{equation} \langle V_p\rangle=-Re_p\int_{V^F}\left\langle\boldsymbol{u}^{t2}\boldsymbol{\cdot} \left(\frac{\partial\boldsymbol{u}_s}{\partial t}+\boldsymbol{u}_s\boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{u}_s+\boldsymbol{u}_s\boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}^\infty+\boldsymbol{u}^\infty \boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u}_s\right)\right\rangle {\rm d}V, \end{equation}

where the averaging operation is defined as $\langle.\rangle = ({1}/{T_{jeff}})\int _0^{T_{jeff}} (.)\,{\rm d}t$, and pertains to the inertially stabilized Jeffery orbit (that is, $C$ is either $0$ or $\infty$) for times longer than $O(Re_p^{-1}H/V_{max})$.

In writing down (3.17), we have neglected the terms on the right-hand side of (3.16) involving the translational and angular accelerations in the actual problem in the Stokesian limit. These depend on $\lambda$ at leading order, and are therefore asymptotically smaller than the Jeffery-averaged volume integral that, as mentioned above (see the paragraph following (3.16)), is independent of $\lambda$ for $\lambda \ll 1$. For a spinning spheroid, both acceleration terms are trivially zero because, independent of $\lambda$, a spinning spheroid (like a sphere) translates with a constant speed along an ambient streamline, while rotating at a uniform rate about the ambient vorticity direction. For a spheroid rotating in any other Jeffery orbit, both terms do lead to non-zero Jeffery-averaged contributions. The first of these terms is non-zero because of a correlation between the time-periodic variations of ${{\rm d}\boldsymbol {U}_{p0}}/{{\rm d}t}$ along the flow direction, and the analogous variation of the flow-directed component of $\boldsymbol {U}_p^{t2}$, the latter arising due to the periodically varying test spheroid orientation for a fixed force (along $-\boldsymbol {1}_2$). The acceleration in the actual problem is only $O(\lambda ^2)$, however, as may be seen from the Faxen's translation law given earlier. The smallness of the second term is because $\boldsymbol {\varOmega }_p^{t2}$, at leading order, is driven by the velocity gradient associated with the equal and oppositely directed image Stokeslet, and is again $O(\lambda ^2)$; the angular acceleration in the actual problem, ${{\rm d}(\boldsymbol {I}_p\boldsymbol {\cdot }\boldsymbol {\varOmega }_{p0})}/{{\rm d}t}$, is $O(1)$. Thus, both acceleration terms in (3.16) are $O(\lambda ^2)$ smaller than the volume integral.

The averaging operation in (3.17) corresponds to a fixed $C$, and is naturally evaluated in $(C,\tau )$ coordinates (Leal & Hinch Reference Leal and Hinch1971; Dabade et al. Reference Dabade, Marath and Subramanian2016). Here, $C={\tan \theta _j(\kappa ^2\sin ^2\phi _j+\cos ^2\phi _j)^{1/2}}/{\kappa }$ and $\tau =\tan ^{-1}[{1}/{\kappa \tan \phi _j}]$, where $\theta _j$ and $\phi _j$ are the polar and azimuthal angles characterizing the spheroid orientation. The latter is denoted by the unit vector $\boldsymbol {p}$ in figure 1(b) with $\boldsymbol {p}=\sin \theta _j\cos \phi _j\boldsymbol {1}_1+\sin \theta _j\sin \phi _j\boldsymbol {1}_2+\cos \theta _j\boldsymbol {1}_3$. The equations

(3.18a)$$\begin{gather} \frac{{\rm d}\phi_j}{{\rm d}t} =\beta \left(-\frac{1}{2}+\frac{\kappa^2-1}{2 (\kappa^2+1)}\cos 2\phi_j\right), \end{gather}$$

(3.18b)$$\begin{gather}\frac{{\rm d}\theta_j}{{\rm d}t} =\beta\frac{(\kappa^2-1)}{4(\kappa^2+1)}\sin 2\theta_j\sin 2\phi_j \end{gather}$$

describe rotation along Jeffery orbits, and are obtained as the individual Cartesian components of the Faxen angular velocity relation, (3.14), given above ($\boldsymbol {\varOmega }_{p0} \boldsymbol {\cdot } \boldsymbol {1}_{r_{b_2}} = {{\rm d}\theta _j}/{{\rm d}t}$, $\boldsymbol {\varOmega }_{p0} \boldsymbol {\cdot } \boldsymbol {1}_{r_{b_1}} = -\sin \theta _j({{\rm d}\phi _j}/{{\rm d}t})$). In terms of $C$ and $\tau$, (3.18a) and (3.18b) take the form ${{\rm d}C}/{{\rm d}t}=0$ and ${{\rm d}\tau }/{{\rm d}t}={\beta }/{\kappa +\kappa ^{-1}}$. The former must be the case by definition, while the latter may be used to transform the time-averaged integral in (3.17) into a $\tau$-averaged one. Inverting the definitions above, one may write $\theta _j$ and $\phi _j$ as

(3.19a)$$\begin{gather} \phi_j=\sin^{-1}\left[\frac{\cos\tau}{(\kappa^2-(\kappa^2-1) \cos^2\tau)^{1/2}}\right], \end{gather}$$

(3.19b)$$\begin{gather}\theta_j=\cos^{-1}\left[\frac{1}{(1+C^2\kappa^2-C^2(\kappa^2-1)\cos^2\tau)^{1/2}}\right], \end{gather}$$

which are used below in evaluating $\langle V_p \rangle$.

For $Re_p$ fixed, the Jeffery-averaged description of inertial migration, given by (3.17), breaks down for sufficiently small or large $\kappa$. This is because the leading-order Jeffery angular velocity becomes small in these limits, being $O(\kappa ^{-2})$ for fibres ($\kappa \gg 1$) close to flow alignment, and $O(\kappa ^2)$ for flat disks ($\kappa \ll 1$) close to alignment with the gradient–vorticity plane. As a result, the $O(Re_p)$ inertial correction becomes comparable in magnitude to the Jeffery contribution, leading to a slow down and eventual arrest of rotation (Marath & Subramanian Reference Marath and Subramanian2017), first shown by Subramanian & Koch (Reference Subramanian and Koch2005) for the case of a slender fibre. Herein, we confine ourselves to analysing the Jeffery-averaged approximation, only noting that it remains valid provided $Re_p\,\kappa /\ln \kappa \ll 1\ (Re_p/\kappa \ll 1)$ for $\kappa \gg 1 (\kappa \ll 1)$, an increasingly restrictive assumption for extreme aspect ratio particles (Subramanian & Koch Reference Subramanian and Koch2005; Marath & Subramanian Reference Marath and Subramanian2017). The consequence of an inertia-induced slow down at leading order, for the said particles, will be analysed in a later communication.

A final point worth mentioning is that, for thin oblate spheroids with $\kappa < 0.14$, the aforementioned inertial drift time scale of $O(Re_p^{-1}H/V_{max})$ only applies in the absence of stochastic orientation fluctuations. In the presence of such fluctuations, either of a thermal origin or otherwise (for instance, due to pair-hydrodynamic interactions or weak turbulence; see Subramanian & Marath Reference Subramanian and Marath2022), there is a barrier-hopping time associated with the eventual equilibration between the numbers of spinning and tumbling spheroids in a manner independent of the initial orientation distribution (Dabade et al. Reference Dabade, Marath and Subramanian2016; Marath, Dwivedi & Subramanian Reference Marath, Dwivedi and Subramanian2017; Marath & Subramanian Reference Marath and Subramanian2018). This time scale increases exponentially with decreasing amplitude of the fluctuations, becoming much longer than the nominal drift time scale, and likely comparable to the time scale for inertial migration. Under these conditions, the migration dynamics will have a probabilistic rather than deterministic character, being described by a kinetic equation for the probability density that is a function of both $C$ and the transverse channel coordinate ($s$) – we briefly revisit this issue when calculating the lift velocity for arbitrary $C$ later in this section.

3.2. Scaling analysis and the point-particle formulation

The dominant contribution to the volume integral in (3.17), for $\lambda \ll 1$, can arise from either scales of $O(L)$ or those of $O(H)$, the inertial screening length being irrelevant in the small-$Re_c$ limit. In order to assess the relative importance of these contributions, we consider the intermediate asymptotic interval $1\ll r\ll \lambda ^{-1}$ ($r$ is measured in units of $L$), in which case $\boldsymbol {u}^{t2}\sim 1/r$, $\boldsymbol {u}_s\sim \beta /r^2+\gamma /r^3$, corresponding to the Stokeslet scaling for the test velocity field, and the stresslet-cum-force-quadrupole scaling for the velocity field in the actual problem. Using these $r$ scalings along with $\boldsymbol {u}^\infty \sim \beta r + \gamma r^2$ for the ambient flow, and separating the estimates for the linear and nonlinear parts of the integrand in (3.17), one obtains

1. (i) $\boldsymbol {u}^{t2}\boldsymbol {\cdot }({\partial \boldsymbol {u}_s}/{\partial t}+ \boldsymbol {u}_s\boldsymbol {\cdot }\boldsymbol {\nabla }\boldsymbol {u}^\infty + \boldsymbol {u}^\infty \boldsymbol {\cdot }\boldsymbol {\nabla } \boldsymbol {u}_s)\sim {\beta ^2}/{r^3}+{\beta \gamma }/{r^2}+{\gamma \beta }/{r^4}+{\gamma ^2}/{r^3}$ (linear in $\boldsymbol {u}_s$),

2. (ii) $\boldsymbol {u}^{t2}\boldsymbol {\cdot }(\boldsymbol {u}_s\boldsymbol {\cdot }\boldsymbol {\nabla } \boldsymbol {u}_s)\sim {\beta ^2}/{r^6}+{\beta \gamma }/{r^7}+{\gamma ^2}/{r^8}$ (nonlinear in $\boldsymbol {u}_s$).

Note that, over the range of scales under consideration, wall effects only contribute at a smaller order in $\lambda$, and hence, use of the unbounded domain estimates above. Next, using ${\rm d}V\sim O(r^2\,{\rm d}r)$, one obtains the following estimates for the contributions of the different terms to the lift velocity integral:

(3.20a)$$\begin{gather} V_p^{linear}\sim Re_p\left(\beta^2\ln r+\beta\gamma r+\dfrac{\gamma\beta}{r}+\gamma^2\ln r\right), \end{gather}$$

(3.20b)$$\begin{gather}V_p^{nonlinear}\sim Re_p\left(\frac{\beta^2}{r^3}+\dfrac{\beta\gamma}{r^4}+\dfrac{\gamma^2}{r^5}\right). \end{gather}$$

The algebraically growing contribution in (3.20a) will be dominated by scales of $O(H)$ ($r\sim \lambda ^{-1}$, the outer region), while contributions in (3.20a,b) that decay with $r$ will be dominated by length scales of $O(L)$ ($r\sim O(1)$, the inner region). The algebraic growth with $r$ will be cutoff for $r\gtrsim \lambda ^{-1}$ from the more rapid decay of the disturbance velocity fields due to wall-induced screening (recall that this more rapid decay was used in neglecting the integral over $S_\infty$ in (3.5)). Likewise, the apparent divergence for $r\to 0$, for the algebraically decaying terms, will be cutoff at $r \sim O(1)$ by the finite size of the spheroid. The terms proportional to $\ln r$ in (3.20a) imply the dominance of the matching interval $1 \ll r \ll \lambda ^{-1}$, resulting in a leading-order contribution proportional to $\ln \lambda ^{-1}$, with logarithmically smaller contributions arising from the inner and outer regions. Use of these cutoffs leads to the estimates

(3.21a)$$\begin{gather} V_p^{linear}\sim Re_p\left[\beta^2(O(1)+\ln\lambda^{-1})+\beta\gamma\lambda^{-1}+\gamma\beta+ \gamma^2(O(1)+\ln\lambda^{-1})\right], \end{gather}$$

(3.21b)$$\begin{gather}V_p^{nonlinear}\sim Re_p(\beta^2+\beta\gamma+\gamma^2), \end{gather}$$

for the contributions from terms linear and nonlinear in $\boldsymbol {u}_s$. Using the definitions of $\beta$ and $\gamma$ given below (2.3), and reverting to the dimensional form, imply a leading-order lift velocity of the form $V_{max}^2 L^3/(\nu H^2)[\ln (H/L)+O(1)]$. Here, the contribution to the $O(1)$ term within brackets arises from the $\beta \gamma \lambda ^{-1}$ term in (3.21a) pertaining to the outer region, and from the $\beta ^2$ terms in both (3.21a) and (3.21b) pertaining, respectively, to the outer and inner regions; the logarithmically larger lift contribution arises from the $\beta ^2\ln \lambda ^{-1}$ term in (3.21a) that pertains to the matching region. However, the matching-region $\beta ^2\ln \lambda ^{-1}$ contribution and the inner-region $\beta ^2$ contribution must both vanish by symmetry because they cannot involve boundaries, and therefore, relate to the time-averaged lift on a neutrally buoyant spheroid in an unbounded simple shear flow. Since such a spheroid cannot exhibit a net lateral drift, only the outer-region $\beta ^2$ contribution survives. This and the outer-region $\beta \gamma$ contribution both yield a dimensional lift velocity of $V_{max}^2L^3/(\nu H^2)$; in dimensionless terms, the Jeffery-averaged volume integral in (3.17) is $O(1)$, with the scaled lift velocity being $O(Re_p)$. More detailed arguments along these lines given in Anand & Subramanian (Reference Anand and Subramanian2023) show that the next order contribution to the volume integral for a neutrally buoyant sphere arises from the inner region, involves an additional factor of $\lambda$, but leads to a qualitative alteration of the lift velocity profiles for large $Re_c$. Owing to the acceleration terms in (3.16) only being $O(\lambda ^2)$, the aforementioned inner-region contribution will also be relevant to a neutrally buoyant spheroid.

Owing to the dominant contribution arising from length scales of $O(H)$ implied by the above arguments, the spheroids in the actual and test problems can be replaced by the corresponding point singularities. Thus, $\boldsymbol {u}_s$ and $\boldsymbol {u}^{t2}$ may be approximated as being induced by a time-dependent stresslet ($\boldsymbol {u}_{str}$) and a Stokeslet due to a point force directed along the negative gradient direction ($\boldsymbol {u}_{St}$), respectively. The Jeffery-averaged lift velocity given by (3.17), at leading order in $\lambda$, may therefore be written as

(3.22)\begin{equation} \langle V_p\rangle =-Re_p\int_{V^F+V^P}\left\langle\boldsymbol{u}_{St}\boldsymbol{\cdot} \left(\dfrac{\partial\boldsymbol{u}_{str}}{\partial t}+ \boldsymbol{u}_{str} \boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u}^\infty+\boldsymbol{u}^\infty\boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{u}_{str} \right)\right\rangle {\rm d}\boldsymbol{r}, \end{equation}

where, on account of the subdominant nature of the length scales of $O(L)$, the domain of integration has now been extended to include the particle volume $V^P$. Thus, (3.22) is the leading-order point-particle approximation for the lift velocity for $Re_c\ll 1$.

In (3.22), $\boldsymbol {u}_{St}$ is time independent since the unit force in the test problem points in a fixed direction, despite the changing orientation of the test spheroid. Furthermore, $\langle \partial \boldsymbol {u}_{str}/\partial t\rangle =0$ for rotation along Jeffery orbits. Also noting that the ambient flow is steady in the chosen non-rotating reference frame, (3.22) reduces to

(3.23)\begin{equation} \langle V_p\rangle =-Re_p\int_{V^F+V^P}\boldsymbol{u}_{St} \boldsymbol{\cdot}(\langle\boldsymbol{u}_{str}\rangle\boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{u}^\infty+ \boldsymbol{u}^\infty\boldsymbol{\cdot}\boldsymbol{\nabla}\langle \boldsymbol{u}_{str}\rangle)\, {\rm d}\boldsymbol{r}. \end{equation}

Using $r_2 \sim O(\lambda ^{-1})$ in (3.15) on account of the outer-region dominance, the Faxen's correction to spheroid translation turns out to be $O(\lambda ^2)$ smaller than the terms linear and quadratic in $r_2$, and one may therefore approximate the ambient flow in (3.23) as $\boldsymbol {u}^\infty \approx (\beta r_2 +\gamma r_2^2)\boldsymbol {1}_1$.

With the spheroid volume neglected, the volume integral in (3.23) is most easily evaluated by Fourier transforming the flow ($r_1$) and vorticity ($r_3$) coordinates, the partial Fourier transform being defined as

(3.24)\begin{equation} \hat{f}(k_1,r_2,k_3) =\int_{-\infty}^\infty \int_{-\infty}^\infty {\rm d}r_1 \,{\rm d}r_3\exp({\iota(k_1 r_1+k_3 r_3)})f(r_1,r_2,r_3). \end{equation}

Applying the convolution theorem along these coordinates (Arfken, Weber & Harris Reference Arfken, Weber and Harris2011), and substituting the above approximate form of $\boldsymbol {u}^\infty$ in (3.23), one obtains

(3.25)\begin{align} \langle V_p\rangle &=-\frac{Re_p}{4{\rm \pi}^2}\int_{-s\lambda^{-1}}^{(1-s)\lambda^{-1}} \,{\rm d}r_2\int {\rm d}\boldsymbol{k}_\perp\,\hat{\boldsymbol{u}}_{St}(-\boldsymbol{k}_\perp,r_2;y_2) \boldsymbol{\cdot} \left[\langle\hat{\boldsymbol{u}}_{str}\rangle(\boldsymbol{k}_\perp,r_2;y_2)\boldsymbol{\cdot} \boldsymbol{1}_2(\beta+2\gamma r_2)\boldsymbol{1}_1\right.\nonumber\\ &\quad \left.-\iota k_1(\beta r_2 +\gamma r_2^2) \langle\hat{\boldsymbol{u}}_{str}\rangle(\boldsymbol{k}_\perp,r_2;y_2)\right], \end{align}

in terms of the partial Fourier transforms of the Stokeslet and stresslet velocity fields; here, $\boldsymbol {k}_\perp \equiv (k_1,k_3)$ and $y_2 =s\lambda ^{-1}$ is the transverse distance of the stresslet from the lower wall. The expression for $\hat {\boldsymbol {u}}_{St}$ is derived in Appendix A, while that for $\langle \hat {\boldsymbol {u}}_{str}\rangle$ is developed in the next subsection.

3.3. Solution for $(\langle \boldsymbol{u}_{str}\rangle, \langle \,p_{str}\rangle )$

The Jeffery-averaged disturbance fields ($\langle \boldsymbol {u}_{str}\rangle, \langle \,p_{str}\rangle$), that appear in the point-particle approximation of the inertial lift velocity given by (3.23), satisfy

(3.26a)$$\begin{gather} \nabla^2 \langle\boldsymbol{u}_{str}\rangle-\boldsymbol{\nabla}\langle \,p_{str}\rangle= \beta\langle\boldsymbol{S}(\boldsymbol{p})\rangle\boldsymbol{\cdot} \boldsymbol{\nabla}\delta(\boldsymbol{r}), \end{gather}$$

(3.26b)$$\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot}\langle\boldsymbol{u}_{str}\rangle=0, \end{gather}$$

with the boundary conditions

(3.27a)$$\begin{gather} \langle\boldsymbol{u}_{str}\rangle=0 \quad\text{at } r_2=-s\lambda^{-1}, (1-s)\lambda^{-1}, \end{gather}$$

(3.27b)$$\begin{gather}\langle\boldsymbol{u}_{str}\rangle\rightarrow 0\quad \text{for } r_1,r_3\rightarrow\infty (r_2\ \text{fixed}), \end{gather}$$

where the stresslet singularity on the right-hand side of (3.26a) approximates the torque-free neutrally buoyant spheroid, in an unbounded simple shear flow, on scales much larger than $O(L)$. The stresslet coefficient $\boldsymbol {S}$ is a function of the spheroid orientation $\boldsymbol {p}$, being given by (Marath & Subramanian Reference Marath and Subramanian2017)

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

where $E_{ij}={\beta }/{2}(\delta _{i1}\delta _{j2}+\delta _{i2}\delta _{j1})$ is the rate of strain tensor associated with the local simple shear. The simple shear flow may be resolved into an axisymmetric extension aligned with $\boldsymbol {p}$, longitudinal planar extensions in a pair of orthogonal planes containing $\boldsymbol {p}$, and a pair of transverse planar extensions in the plane perpendicular to $\boldsymbol {p}$ (Subramanian & Koch Reference Subramanian and Koch2006; Dabade et al. Reference Dabade, Marath and Subramanian2016). The $A_i$'s in (3.28) are the $\kappa$-dependent stresslet amplitudes corresponding to the aforesaid component flows, there being only three of these (in contrast to the five component flows) owing to the axisymmetry of the spheroidal geometry (Kim & Karrila Reference Kim and Karrila1991; Marath & Subramanian Reference Marath and Subramanian2017). For $\kappa >1$, these amplitudes are given by

(3.29)$$\begin{gather} A_1=-\frac{16{\rm \pi}(\kappa^2-1)^{5/2}}{9\kappa^3[-3(\kappa^2-1)^{1/2}\kappa+2\kappa^2 \cosh^{-1}(\kappa)+\cosh ^{-1}(\kappa)]}, \end{gather}$$

(3.30)$$\begin{gather}A_2=-\frac{16{\rm \pi}(\kappa^2-1)^3}{3\kappa^2(\kappa^2+1)(\kappa^4+\kappa^2-3(\kappa^2-1)^{1/2}\kappa\cosh^{-1}(\kappa)-2)}, \end{gather}$$

(3.31)$$\begin{gather}A_3=-\frac{32{\rm \pi}(\kappa^2-1)^3}{3\kappa^3(2\kappa^5-7\kappa^3+3(\kappa^2-1)^{1/2}\cosh^{-1}(\kappa)+5\kappa)}. \end{gather}$$

The corresponding expressions for an oblate spheroid ($\kappa <1$) may be obtained by first substituting $\kappa =\xi _0/(\xi _0^2-1)^{1/2}$ in terms of the coordinate $\xi _0\ ({>}1)$ labelling the surface of the spheroid, and then using the transformation $d\to -\iota d, \xi _0\to \iota (\xi _0^2-1)^{1/2}$ in the dimensional form of the stresslet obtained from multiplying $\boldsymbol {S}$ above by $\mu L^3 V_{max}/H$ with $L= d\xi _0$ (Marath & Subramanian Reference Marath and Subramanian2017). Note that $\boldsymbol {p}$, and thence $\boldsymbol {S}$, is a function of time owing to rotation along Jeffery orbits. In the limit of a sphere, $A_1 = A_2 = A_3 = -{20{\rm \pi} }/{3}$, and $\boldsymbol {S}$ reduces to $-20{\rm \pi} \boldsymbol {E}/3$, independent of time, corresponding to the stresslet induced by a freely rotating sphere that leads to the well-known Einstein coefficient in the suspension viscosity.

While one may, in principle, solve for ($\langle \boldsymbol {u}_{str}\rangle, \langle \,p_{str}\rangle$) in a manner similar to that for the bounded domain Stokeslet fields (see Appendix A) by defining the disturbance fields as the sum of an unbounded domain contribution, and a contribution that accounts for the confinement induced by channel walls, one may also obtain them directly as a gradient of the bounded domain Stokeslet fields, derived in Appendix A; the gradient is with respect to the location $\boldsymbol {y}$ of the Stokeslet (Swan & Brady Reference Swan and Brady2010). Therefore,

(3.32)\begin{equation} \langle\boldsymbol{u}_{str}\rangle=\beta\langle\boldsymbol{S}\rangle:\frac{\partial\boldsymbol{J}}{ \partial\boldsymbol{y}}, \end{equation}

where the bounded domain Stokeslet field corresponding to a point force $\boldsymbol {F}$ is given by $\boldsymbol {J}\boldsymbol {\cdot } \boldsymbol {F}$; the Fourier transform ($\hat {\boldsymbol {J}}$) of the tensor $\boldsymbol {J}$ is defined in Appendix A (see text below (A25)). Since it is $\langle \hat {\boldsymbol {u}}_{str} \rangle$ that appears in the final expression for the point-particle approximation of the lift velocity, (3.25), we Fourier transform (3.32) to obtain

(3.33)\begin{equation} \langle\hat{u}_{{str},i}\rangle=\beta \langle S_{jm}\rangle\hat{N}_{ijm}, \end{equation}

where the third-order tensor $\hat {N}_{ijm}$ is defined as

(3.34)\begin{equation} \hat{N}_{ijm}=\iota\hat{J}_{im}(k_1 \delta_{j1}+k_3 \delta_{j3})+\dfrac{\partial\hat{J}_{im}}{\partial y_2}\delta_{j2}. \end{equation}

In (3.33), $\langle \boldsymbol {S}\rangle ={(1}/{2{\rm \pi} })\int _0^{2{\rm \pi} } \boldsymbol {S}\,{\rm d}\tau$ denotes the time-averaged stresslet corresponding to a neutrally buoyant spheroid rotating along a given Jeffery orbit in an unbounded simple shear flow (with unit velocity gradient). The expression for $\langle \boldsymbol {S}\rangle$ is given below in the next subsection.

3.4. The time-averaged lift velocity $\langle V_p \rangle$

Using (3.33) for the stresslet velocity field in (3.25), the Jeffery-averaged lift velocity takes the form

(3.35)\begin{align} \langle V_p\rangle &=-\frac{Re_p}{4{\rm \pi}^2}\int_{-s\lambda^{-1}}^{(1-s)\lambda^{-1}} \,{\rm d}r_2\int {\rm d}\boldsymbol{k}_\perp\,\hat{\boldsymbol{u}}_{St}(-\boldsymbol{k}_\perp,r_2;y_2)\nonumber\\ &\quad \boldsymbol{\cdot} \left[\beta\boldsymbol{1}_2\boldsymbol{\cdot}\hat{\boldsymbol{N}}: \langle\boldsymbol{S}\rangle(\boldsymbol{k}_\perp,r_2;y_2)(\beta+2\gamma r_2)\boldsymbol{1}_1\right.\nonumber\\ &\quad \left.-\iota k_1(\beta r_2 +\gamma r_2^2) \beta\hat{\boldsymbol{N}}:\langle\boldsymbol{S}\rangle(\boldsymbol{k}_\perp,r_2;y_2)\right]. \end{align}

Note that the averaging operation ($\langle.\rangle$) in (3.35) is independent of the shear rate since it involves the scaled Jeffery angular velocity that is only a function of $\kappa$ and $\phi _j$ for the inertially stabilized orbits; the shear-rate dependence only comes in at $O(Re_p^{{3}/{2}}$) (Marath & Subramanian Reference Marath and Subramanian2017). Thus, $\langle \boldsymbol {S}\rangle$ must be linear in $\boldsymbol {E}$, with a prefactor that is a function of $C$ and $\kappa$. For simple shear flow, this implies that $\langle S_{12}\rangle =\langle S_{21}\rangle$ are the only non-zero components of the Jeffery-averaged stresslet. The following expressions for these components may be obtained by starting from (3.28), with $\boldsymbol {p}$ expressed in terms of $C$ and $\tau$ using (3.19a,b), followed by an integration over $\tau$:

(3.36)\begin{align} \langle S_{12}\rangle&=\frac{1}{4(\kappa^2-1)^2[(C^2+1)(C^2\kappa^2+1)]^{1/2}}\left\{3 A_1 \kappa^2\left(2+C^2(\kappa^2+1)\right.\right.\nonumber\\ &\quad \left.-2 [(C^2+1) (C^2 \kappa^2+1)]^{1/2}\right)\nonumber\\ &\quad +2 A_2(\kappa^2+1)\left(\kappa^2\left([(C^2+1) (C^2 \kappa^2+1)]^{1/2}-2 C^2-1\right)\right.\nonumber\\ &\quad\left.+[(C^2+1) (C^2 \kappa^2+1)]^{1/2}-1\right)\nonumber\\ &\quad +A_3 \left(\kappa^4(C^2+2)+\kappa^2(-2 [(C^2+1) (C^2 \kappa^2+1)]^{1/2}\right.\nonumber\\ &\quad \left.\left.+C^2-2)+2\right)\right\} \end{align}

for prolate spheroids and

(3.37)\begin{align} \langle S_{12}\rangle &= \frac{1}{4(\kappa^2-1)^2(C^2+1)(C^2\kappa^2+1)}\left\{3 A_1 \kappa^2\left(-2C^4\kappa^2 +C^2(\kappa^2+1)\right.\right.(-2\nonumber\\ &\quad\left.+[(C^2+1)(C^2\kappa^2+1)]^{1/2})+2 (-1+[(C^2+1)(C^2\kappa^2+1)]^{1/2})\right)\nonumber\\ &\quad +2 A_2(\kappa^2+1)\left(C^4 (\kappa^2+\kappa^4)-(1+\kappa^2)(-1+[(C^2+1)(C^2\kappa^2+1)]^{1/2})\right.\nonumber\\ &\quad\left.+C^2(1+\kappa^2(2+\kappa^2-2[(C^2+1)(C^2\kappa^2+1)]^{1/2}))\right)\nonumber\\ &\quad +A_3 \left(2[(C^2+1)(C^2\kappa^2+1)]^{1/2}+\kappa^2(-2-2C^4\kappa^2\right.\nonumber\\ &\quad+2(\kappa^2-1)[(C^2+1)(C^2\kappa^2+1)]^{1/2})\nonumber\\ &\quad\left.\left.+C^2(1+\kappa^2)(-2+[(C^2+1)(C^2\kappa^2+1)]^{1/2})\right) \right\} \end{align}

for oblate spheroids. For both (3.36) and (3.37), $\lim _{\kappa \rightarrow 1} \langle S_{12}\rangle =-10{\rm \pi} /3$ independent of $C$, which yields the Einstein coefficient ($5/2$) in the $O(\phi )$ contribution to the suspension viscosity, $\phi$ being the sphere volume fraction; the $C$ independence arises owing to the sphere orientation being a degenerate degree of freedom. At the other extreme, one has $\lim _{\kappa \to \infty } \langle S_{12}\rangle |_{C=\infty }=-2{\rm \pi} /(3\kappa \ln \kappa )$ in (3.36). Here, the factor $1/\ln \kappa$ arises from viscous slender body theory (Subramanian & Koch Reference Subramanian and Koch2005), while the additional $\kappa ^{-1}$ factor reflects the probability of the occurrence of non-aligned orientations that contribute dominantly to the viscosity of a dilute non-interacting suspension of slender fibres (Leal & Hinch Reference Leal and Hinch1971; Dabade et al. Reference Dabade, Marath and Subramanian2016).

For times much longer than $O(Re_p^{-1}H/V_{max})$, only the inertially stabilized orbits are relevant to cross-stream migration. As mentioned earlier, for prolate spheroids, this is the tumbling orbit ($C=\infty$), in which case, $\langle S_{12}\rangle$ in (3.36) reduces to

(3.38)\begin{equation} \langle S_{12}\rangle|_{C=\infty} = \frac{ (3 A_1+A_3)\kappa+2 A_2 (\kappa ^2+1)}{4 (\kappa +1)^2}. \end{equation}

For oblate spheroids, one can have either the spinning ($C=0$) or tumbling ($C=\infty$) orbits depending on $\kappa$, and accordingly, $\langle S_{12}\rangle$ in (3.37) reduces to

(3.39)$$\begin{gather} \langle S_{12}\rangle|_{C=\infty} = \frac{ (3 A_1+A_3)\kappa+2 A_2 (\kappa ^2+1)}{4 (\kappa +1)^2}, \end{gather}$$

(3.40)$$\begin{gather}\langle S_{12}\rangle|_{C=0} = \frac{A_3}{2}. \end{gather}$$

Equations (3.38)–(3.40) will be used to determine $\langle V_p \rangle$ below. In light of the above, one may write (3.33) in the form

(3.41)\begin{equation} \langle\hat{u}_{{str},i}\rangle=\beta \langle S_{12}\rangle\,(\hat{N}_{i12}+\hat{N}_{i21}), \end{equation}

with the Jeffery-averaged lift velocity given by

(3.42)\begin{align} \langle V_p\rangle &=-\frac{Re_p\langle S_{12}\rangle} {4{\rm \pi}^2} \int_{-s\lambda^{-1}}^{(1-s)\lambda^{-1}} {\rm d}r_2\int {\rm d}\boldsymbol{k}_\perp\,\hat{u}_{{St},i} (-\boldsymbol{k}_\perp,r_2;y_2) \nonumber\\ &\quad \times\big[\beta(\beta+2\gamma r_2)(\hat{N}_{212}+\hat{N}_{221})(\boldsymbol{k}_\perp,r_2;y_2)\delta_{i1}\nonumber\\ &\quad -\iota \beta k_1(\beta r_2 +\gamma r_2^2) (\hat{N}_{i12}+\hat{N}_{i21}) (\boldsymbol{k}_\perp,r_2;y_2)\big]. \end{align}

Since the dominant contributions to the integral in (3.42) come from scales of $O(H)$ (or, equivalently, $k_\perp \sim O(H^{-1})$), it is natural to transform to rescaled variables $\boldsymbol {k}_\perp =\boldsymbol {k}_\perp ''\lambda, r_2=r_2''/\lambda$. The original disturbance fields may be written in terms of the rescaled ones as $\hat {\boldsymbol {u}}_{St} =\hat {\boldsymbol {u}}_{St}^{\prime \prime } /\lambda$, $\hat {\boldsymbol {N}}=\hat {\boldsymbol {N}}''$, and furthermore, noting that $y_2''= s$ and ${\rm d}\boldsymbol {k}_\perp \,{\rm d}r_2\equiv \lambda \, {\rm d}\boldsymbol {k}_\perp ''\,{\rm d}r_2''$, one obtains

(3.43)\begin{align} \langle V_p\rangle &=-\frac{Re_p\langle S_{12}\rangle} {4{\rm \pi}^2} \int_{-s}^{1-s} {\rm d}r_2''\int {\rm d}\boldsymbol{k}''_\perp\,\hat{u}_{{St},i}^{\prime\prime} (-\boldsymbol{k}''_\perp,r_2'';s) \nonumber\\ &\quad\times \big[\beta(\beta+2\gamma'' r_2'')(\hat{N}_{212}''+\hat{N}_{221}'') (\boldsymbol{k}''_\perp,r_2'';s)\delta_{i1}\nonumber\\ &\quad -\iota \beta k_1''(\beta r_2'' +\gamma'' r_2''^2) (\hat{N}_{i12}''+\hat{N}_{i21}'') (\boldsymbol{k}''_\perp,r_2'';s)\big], \end{align}

where $\gamma ''=\gamma \lambda ^{-1}=-4$, and $\langle S_{12} \rangle$ is given by (3.38), (3.39) or (3.40) depending on $\kappa$. The essential consequence of the rescaling above is to show that the volume integral in (3.43) is independent of $\lambda$. In fact, the integral is only a function of $s$ (the particle location in the channel; see figure 1), with the additional dependence on $\kappa$ entirely contained in $\langle S_{12} \rangle$. Accounting for the scaling $V_{max}\lambda$ used for $V_p$, (3.43) shows that the point-particle framework leads to a lift velocity of $O(V_{max}\lambda Re_p)$, which was used earlier in § 3.1 to estimate the inertial migration time scale ($t_{lift}$).

A consequence of the integral in (3.43) being independent of $\kappa$ is that, within the Jeffery-averaged framework used, the inertial lift velocity of a sphere, and the time-averaged lift velocity of a spheroid, only differ by a multiplicative function of $C$ and $\kappa$ in general, and only by a function of $\kappa$ if one assumes the spheroid to have settled onto the inertially stabilized Jeffery orbit – this multiplicative function is the ratio of the Jeffery-averaged spheroid stresslet to the sphere stresslet, being equal to $-{3\langle S_{12} \rangle }/{10{\rm \pi} }$. An immediate outcome of this proportionality relation is that the shapes of the lift velocity profiles for the two cases must be identical, with the zero crossings in particular (that correspond to the Segre–Silberberg equilibria for a sphere) being identical. In other words, a change in $\kappa$ only affects the magnitude of the inertial lift, the equilibrium positions being unaffected. In light of the obvious importance of the above conclusion for shape-sorting applications, it is worth documenting the reasons that lead to the simple proportionality relation.

1. (i) The dominant contribution from the linearized inertial terms on scales of $O(H)$. This outer-region dominance led to the point-particle approximation, and thence, to the time independence of the test velocity field in the reciprocal theorem volume integral.

2. (ii) The asymptotic smallness, for small confinement ratios, of the additional contributions arising from the translational and angular accelerations associated with the motion of the neutrally buoyant spheroid in the Stokes limit,

3. (iii) The approximation of the time averaging based on the $Re_p-$independent Jeffery angular velocity, which leads to the time-averaged spheroid stresslet being proportional to $\boldsymbol {E}$ (just as the sphere stresslet).

The Jeffery-averaged lift velocity in (3.43) may be written in the compact form

(3.44)\begin{equation} \langle V_p\rangle=Re_p\langle S_{12}\rangle(\kappa)(\beta^2 F(s)+\beta\gamma''G(s)), \end{equation}

where the functions $F(s)$ and $G(s)$ are given by

(3.45)\begin{align} F(s)&=-\frac{1} {4{\rm \pi}^2} \int_{-s}^{1-s} {\rm d}r_2''\int {\rm d}\boldsymbol{k}''_\perp\, \hat{u}_{{St},i}^{\prime\prime} (-\boldsymbol{k}''_\perp,r_2'';s) \nonumber\\ &\quad \times\big[(\hat{N}_{212}''+\hat{N}_{221}'') (\boldsymbol{k}''_\perp,r_2'';s)\delta_{i1}-\iota k_1'' r_2'' (\hat{N}_{i12}''+\hat{N}_{i21}'') (\boldsymbol{k}''_\perp,r_2'';s)\big], \end{align}

(3.46)\begin{align} G(s)&=-\frac{1} {4{\rm \pi}^2} \int_{-s}^{1-s} {\rm d}r_2''\int {\rm d}\boldsymbol{k}''_\perp\, \hat{u}_{{St},i}^{\prime\prime} (-\boldsymbol{k}''_\perp,r_2'';s) \nonumber\\ &\quad\times\big[2 r_2''(\hat{N}_{212}''+\hat{N}_{221}'') (\boldsymbol{k}''_\perp,r_2'';s)\delta_{i1}-\iota k_1''r_2''^2 (\hat{N}_{i12}''+\hat{N}_{i21}'') (\boldsymbol{k}''_\perp,r_2'';s)\big], \end{align}

and satisfy $F(s)=-F(1-s)$ and $G(s)=G(1-s)$, consistent with the antisymmetry of the lift velocity profile about the channel centreline.

In (3.44) the $\beta ^2$ contribution characterizes the effect of the asymmetrically located walls on the disturbance stresslet, and leads to migration away from the walls. In contrast, the $\beta \gamma ''$ contribution, which characterizes the interaction of the stresslet with the ambient profile curvature, causes migration away from the channel centreline; note that this contribution is still influenced by the walls, given that the dominant scales are $O(H)$. The Segre–Silberberg equilibria emerge from a balance between these two opposing effects. Substituting $\beta =4(1-2s)$ and $\gamma ''=-4$ in (3.44), the final expression for the time-averaged lift velocity of a neutrally buoyant spheroid of an arbitrary aspect ratio $\kappa$, for $Re_c\ll 1$, is given by

(3.47)\begin{equation} \langle V_p\rangle=Re_p\langle S_{12}\rangle(\kappa)\left[16(1-2s)^2 F(s)-16(1-2s)G(s)\right]. \end{equation}

The Fourier integrals in (3.45) and (3.46) can be calculated using plane polar coordinates, $k_1''=k_\perp ''\cos \phi$, $k_3''=k_\perp ''\sin \phi$, with ${\rm d}\boldsymbol {k}_\perp ''\equiv k_\perp ''\,{\rm d}k_\perp ''\,{\rm d}\phi$; here $k_\perp ''\in [0,\infty )$ and $\phi \in [0,2{\rm \pi} ]$. The integrals over the transverse coordinate $r_2''$ and the azimuthal angle $\phi$ may be done analytically, reducing (3.45) and (3.46) to the one-dimensional integrals

(3.48)$$\begin{gather} F(s)=\int_0^\infty {\rm d}k_\perp'' \dfrac{k_\perp''{\rm e}^{-k_\perp'' (27 s+16)} I(k_\perp'',s)}{48 {\rm \pi}\big({\rm e}^{2 k_\perp''}-1\big) \left[-2 {\rm e}^{2 k_\perp''} \big(2 k_\perp''^2+1\big)+ {\rm e}^{4 k_\perp''}+1\right]^2}, \end{gather}$$

(3.49)$$\begin{gather}G(s)=\int_0^\infty {\rm d}k_\perp'' \dfrac{{\rm e}^{-k_\perp'' (27 s+16)} J(k_\perp'',s)}{192 {\rm \pi}k_\perp''^2 \big({\rm e}^{2 k_\perp''}-1\big) \left[-2 {\rm e}^{2 k_\perp''} \big(2 k_\perp''^2+1\big)+{\rm e}^{4 k_\perp''}+1\right]^2}, \end{gather}$$

with $I(k_\perp '',s)$ and $J(k_\perp '',s)$ defined in Appendix B. The $k_\perp ''$ integrals above are evaluated numerically for various $s$, using Gauss–Legendre quadrature, after replacing the infinite interval with a finite one – $(0,K_{max})$. Numerical convergence, especially near the channel walls, depends sensitively on $K_{max}$. As the spheroid approaches either wall, the contribution to the $k_\perp ''$ integrals comes from progressively smaller scales (compared with $H$) or, equivalently, from larger and larger $k_\perp ''$. In fact, the relevant scale changes from $H$ ($k_\perp ''\sim O(1)$) to the distance from the wall – either $s$ (lower wall) or $1-s$ (upper wall). As a result, the choice of any finite $K_{max}$, however large, will only lead to converged lift velocities down to a certain non-zero $s$.

To analyse the lift close to the lower wall, for example, one defines a rescaled wavenumber $k_\perp ''=k_w/s$. Next, expanding the integrands in (3.48) and (3.49) for $s\to 0$ with $k_w$ fixed, the near-wall lift velocity takes the form

(3.50)\begin{equation} \lim_{s\to 0}\,\langle V_p\rangle=\langle V_p\rangle^{wall}=-\frac{Re_p\langle S_{12}\rangle(\kappa)}{3{\rm \pi}} \int_0^\infty {\rm d}k_w\,{\rm e}^{-2 k_w} k_w (3 k_w^2-2 k_w+3). \end{equation}

The limiting value near the upper wall is equal in magnitude, but of an opposite sign, as must be the case by symmetry. The integral in (3.50) can be evaluated analytically, and yields

(3.51)\begin{equation} \langle V_p\rangle^{wall}={\pm} \dfrac{11 Re_p\langle S_{12}\rangle(\kappa)}{24{\rm \pi}}, \end{equation}

with the upper and lower signs pertaining to the corresponding channel wall. For a sphere, (3.51) reduces to $55 Re_p/36$, a value originally given by Vasseur & Cox (Reference Vasseur and Cox1976), albeit without an accompanying explanation. In the results shown in the next subsection, we have chosen $K_{max}=10^4$ for the numerical evaluation of (3.48) and (3.49), to ensure accuracy down to $s=0.001$ without the aid of a large-$k_\perp ''$ asymptote. This choice also enables a close approach to the near-wall limiting value above, and the sufficiency of $200$ quadrature points in obtaining converged results for arbitrary $\kappa$.

The sign of the limiting value in (3.51) points to a wall-induced repulsion. Furthermore, (3.51) arises entirely from the $\beta ^2$ contribution, with the $\beta \gamma ''$ contribution being asymptotically small, implying that the leading-order repulsion may be determined independently by considering a sphere or spheroid moving parallel to a single plane wall subject to a linear shearing flow. The near-wall estimates of the said contributions may be obtained by considering a spheroid, regarded as the relevant point singularity, close to either wall. For the lower wall, this would mean $s\ll 1$, with an inner region characterized by distances of the order of the spheroid size ($\lambda < r''\ll s$), and an outer region characterized by distances of the order of the spheroid-wall separation ($r'' \gtrsim s$); recall that $r''$ was measured in units of $H$. In accordance with (3.20), the lift velocity integral grows on scales pertaining to the matching interval, $\lambda \ll r'' \ll s$, with the dominant contributions being of the form $O(\beta ^2)[\ln r'' + O(1)]$ and $O(\beta \gamma '' r'')$. The logarithmic term is again zero due to symmetry, and the linear growth is cutoff for distances larger than $O(s)$ owing to the more rapid decay arising from wall-induced cancellation of the leading-order singularity. For $s \ll r'' \ll 1$, the actual velocity field exhibits an $O(1/r''^3)$ quadrupolar decay owing to wall-induced stresslet cancellation; the test velocity field is also $O(1/r''^3)$ over this range of distances owing to the wall-induced cancellation of both the point force and force dipole for a Stokeslet of a perpendicular orientation. This more rapid decay ensures convergence of the volume integral, and using $r'' \sim s$ in the growing terms above yields the limiting forms of $O(\beta ^2)$ and $O(\beta \gamma '' s)$ close to the lower wall.

A final important point concerns the approach of the near-wall lift velocity to a finite value, rather than zero, as one approaches the channel walls. The latter must be the case on account of the eventual diverging lubrication resistance associated with the thin intervening fluid layer between the particle and the wall, and the discrepancy is on account of the regime of validity of the present calculation. As stated in the problem definition in § 2, the near-wall limit above pertains to spheroid-wall separations that, although much smaller than $O(H)$, are nevertheless much larger than $L$. We comment further on the nature of the lift profile for small separations in § 4.

4.1. The finite-$Re_c$ formulation

The Stokesian disturbance field due to a freely suspended particle in an ambient linear flow only decays algebraically, as $1/r^2$, at large distances, a fact already used in the scaling arguments in § 3.2. The algebraic decay implies that distinct outer-region expansions for both the velocity and pressure fields become necessary on scales of $O(HRe_c^{-1/2})$, with the leading terms satisfying the linearized Navier–Stokes and continuity equations. To write down these equations, one transforms to outer coordinates using $\boldsymbol {r}=Re_p^{-1/2}\boldsymbol {R}$, which corresponds to using $HRe_c^{-{1}/{2}}$ (rather than $L$ as in § 2) as the relevant length scale. The Stokesian rates of decay in the inner region suggest the scalings $\boldsymbol {u}'=Re_p\boldsymbol {U}$ and $p'=Re_p^{3/2} P$ for the leading-order terms in the outer expansions, where $\boldsymbol {U}$ and $P$ satisfy

(4.1a)\begin{gather} \frac{\partial^2 U_i}{\partial R_m^2}-\frac{\partial P}{\partial R_i}-\frac{\partial U_i}{\partial t}-U_2(\beta+2\gamma''R_2 Re_c^{-1/2})\delta_{i1}-(\beta R_2+\gamma''R_2^2 Re_c^{-1/2})\frac{\partial U_i}{\partial R_1}\nonumber\\ =\beta S_{im}\frac{\partial\delta(\boldsymbol{R})}{\partial R_m}, \end{gather}

(4.1b)\begin{gather} \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{U}=0. \end{gather}

The Faxen correction contributes at a higher order in $\lambda$ and, therefore, $u^\infty _i\approx Re_p^{-1/2}(\beta R_2+\gamma ''\,Re_c^{-1/2} R_2^2) \delta _{i1}$ has been used in (4.1a). Equations (4.1a,b) must be supplemented by the conditions

(4.2a)$$\begin{gather} U_i \sim \frac{3\beta R_i R_j S_{jm}R_m}{4{\rm \pi} R^5} \quad \text{for } \boldsymbol{R}\rightarrow 0, \end{gather}$$

(4.2b)$$\begin{gather}\boldsymbol{U}=0\quad \text{at } R_2=-s\,Re_c^{1/2}, (1-s)\,Re_c^{1/2}, \end{gather}$$

where (4.2b) denotes the no-slip conditions on the channel walls, while (4.2a) is the requirement of matching to the stresslet velocity field that arises as the far-field form of the inner-region Stokesian field. In (4.1a) and (4.2a), $\boldsymbol {S}$ is the tensorial amplitude defined in (3.28).

As explained in § 3.2, the separation between the migration and drift time scales implies one needs only solve for the Jeffery-averaged lift velocity, and towards this end, we average (4.1a,b) and (4.2a,b) over a single period of the stable Jeffery orbit. One obtains

(4.3a)\begin{gather} \frac{\partial^2\langle U_i\rangle}{\partial R_m^2} -\frac{\partial\langle P\rangle}{\partial R_i}-\langle U_2\rangle(\beta+2\gamma''R_2 Re_c^{-1/2})\delta_{i1}-(\beta R_2+\gamma''R_2^2 Re_c^{-1/2})\frac{\partial\langle U_i\rangle}{\partial R_1}\nonumber\\ =\beta\langle S_{12}\rangle\left[\delta_{i1}\frac{\partial\delta(\boldsymbol{R})}{\partial R_2}+\delta_{i2} \frac{\partial\delta(\boldsymbol{R})}{\partial R_1}\right], \end{gather}

(4.3b)\begin{gather} \boldsymbol{\nabla}\boldsymbol{\cdot} \langle\boldsymbol{U}\rangle=0, \end{gather}

where $\langle \boldsymbol {U}\rangle$ satisfies

(4.4a)$$\begin{gather} \langle U_i\rangle \sim \frac{3 \beta \langle S_{12}\rangle R_1 R_2 R_i}{4{\rm \pi} R^5}\quad\text{for } \boldsymbol{R}\rightarrow 0, \end{gather}$$

(4.4b)$$\begin{gather}\langle\boldsymbol{U}\rangle=0\quad \text{at } R_2=-s\,Re_c^{1/2}, (1-s)\,Re_c^{1/2}. \end{gather}$$

Here, $\langle S_{12}\rangle (\kappa )$ corresponds to the stresslet averaged over the relevant stable Jeffery orbit, and has been given in (3.38)–(3.40). Due to the linearity of (4.3a,b) and (4.4a,b), and the fact that the Jeffery-averaged spheroid stresslet tensor differs from that for a sphere only by a scalar multiplicative factor, $\langle \boldsymbol {U} \rangle$ and $\langle P \rangle$ differ from their spherical analogs only by $-{3\langle S_{12}\rangle (\kappa )}/{10{\rm \pi} }$, corresponding to the ratio of the aforementioned stresslets. This proportionality relation must hold for any linear functional of the disturbance fields, and in particular, for the lift velocity that is a linear functional of $\langle U_2 \rangle$; see (4.8) below. Thus, similar to the case of $Re_c\ll 1$, the Jeffery-averaged lift profiles at a given finite $Re_c$, for an arbitrary aspect ratio spheroid, have the same shape as those for a sphere at the same $Re_c$. It follows that the associated pair of equilibria are identical to those for a sphere regardless of $Re_c$, and as for a sphere (Schonberg & Hinch Reference Schonberg and Hinch1989), must migrate wallward with increasing $Re_c$. It is worth reiterating that the requirement for a Jeffery-averaged analysis to remain valid becomes restrictive for extreme aspect ratio spheroids. The regime of validity, originally stated after (3.19) and expressed in terms of $Re_c$, is given by $Re_c\lambda ^2 \kappa /\ln \kappa \ll 1$ and $Re_c\lambda ^2/\kappa \ll 1$ for $\kappa \gg 1$ and $\kappa \ll 1$, respectively; these point to the restriction becoming more severe with increasing $Re_c$. The implications of the finite-$Re_c$ Jeffery-averaged analysis above, for shape sorting, remain the same as those discussed in § 3.5.

For purposes of completeness, we now follow along the lines of Schonberg & Hinch (Reference Schonberg and Hinch1989), and briefly present the manner in which inertial lift is determined for $Re_c \gtrsim O(1)$. After implementing the partial Fourier transform defined in (3.24), followed by some algebraic manipulation, one obtains the coupled ODEs for $\widehat {\langle P \rangle}$ and $\widehat {\langle U_2 \rangle}$,

(4.5a)$$\begin{gather} \frac{{\rm d}^2 \widehat{\langle P\rangle}}{{\rm d}R_2^2}-k_\perp^2\widehat{\langle P\rangle}= 2\iota k_1 \widehat{\langle U_2\rangle}(\beta+2\gamma''R_2 Re_c^{-1/2}), \end{gather}$$

(4.5b)$$\begin{gather}\frac{{\rm d}^2\widehat{\langle U_2\rangle}}{{\rm d}R_2^2}-k_\perp^2\widehat{\langle U_2\rangle}= \frac{{\rm d}\widehat{\langle P\rangle}}{{\rm d}R_2}-\iota k_1 \widehat{\langle U_2\rangle}(\beta R_2+\gamma''R_2^2 Re_c^{-1/2}), \end{gather}$$

with the conditions,

(4.6a)$$\begin{gather} \widehat{\langle U_2\rangle} \sim\frac{\iota \beta\langle S_{12}\rangle k_1 |R_2|\,{\rm e}^{-k_\perp |R_2|}}{2}\quad \text{for } k_1,k_3\to\infty \text{ and } R_2\rightarrow 0, \end{gather}$$

(4.6b)$$\begin{gather}\widehat{\langle U_2\rangle}=\frac{{\rm d}\widehat{\langle U_2\rangle}}{{\rm d}R_2}=0\quad \text{at } R_2=-s\,Re_c^{1/2},R_2=(1-s)\,Re_c^{1/2}, \end{gather}$$

where $k_\perp ^2=k_1^2+k_3^2$ as before. The delta-function forcing in (4.3a) leads to the following jump conditions across the particle location ($R_2=0$):

(4.7a)$$\begin{gather} \widehat{\langle P\rangle}^+(k_1,0^+,k_3)-\widehat{\langle P\rangle}^-(k_1,0^-,k_3) =2\iota k_1 \beta\langle S_{12}\rangle, \end{gather}$$

(4.7b)$$\begin{gather}\frac{{\rm d}\widehat{\langle P\rangle}^+}{{\rm d}R_2}(k_1,0^+,k_3)- \frac{{\rm d}\widehat{\langle P\rangle}^-}{{\rm d}R_2}(k_1,0^-,k_3) =0, \end{gather}$$

(4.7c)$$\begin{gather}\widehat{\langle U_2\rangle}^+(k_1,0^+,k_3)-\widehat{\langle U_2\rangle}^-(k_1,0^-,k_3) =0, \end{gather}$$

(4.7d)$$\begin{gather}\frac{{\rm d}\widehat{\langle U_2\rangle}^+}{{\rm d}R_2}(k_1,0^+,k_3)- \frac{{\rm d}\widehat{\langle U_2\rangle}^-}{{\rm d}R_2}(k_1,0^-,k_3) =\iota k_1 \beta \langle S_{12}\rangle. \end{gather}$$

Here the superscripts ‘$+$’ and ‘$-$’ denote the limiting values attained on approaching $R_2 = 0$ from the regions $0< R_2\leq (1-s)\,Re_c^{1/2}$ and $-s\,Re_c^{1/2}\leq R_2<0$, respectively; these conditions are derived in Appendix C. The limiting form of $\langle \boldsymbol {U} \rangle$ in the matching region ($\boldsymbol {R}\ll 1$) is the sum of the singular stresslet contribution given in (4.4a), and a uniform flow along the gradient (cross-stream) direction that is a consequence of fluid inertia. The neutrally buoyant spheroid being force free is convected by this uniform flow that therefore equals the inertial lift velocity, and may be determined from the limit of the inverse transform for $\boldsymbol {R}\to 0$,

(4.8)\begin{equation} \langle V_p\rangle=\frac{Re_p}{4{\rm \pi}^2}{\rm Re}\left\{\int_{-\infty}^\infty\int_{-\infty}^\infty \,\widehat{\langle U_2\rangle}^\pm(k_1,0^\pm,k_3)\,{\rm d}k_1\, {\rm d}k_3\right\}. \end{equation}

Here, ${\rm Re}\{.\}$ denotes taking the real part that eliminates the purely imaginary stresslet contribution. As indicated, one may use either $\widehat {\langle U_2\rangle }^-$ or $\widehat {\langle U_2\rangle }^+$ on account of continuity; see (4.7c). The governing ODEs (4.5a,b), the boundary conditions (4.6a,b) and the jump conditions (4.7a–d) can be solved using the shooting technique described in Appendix A of Schmid, Henningson & Jankowski (Reference Schmid, Henningson and Jankowski2002), a brief description of which is given in Appendix D.

The integral in (4.8) is evaluated numerically using a two-dimensional Gauss-Legendre quadrature over a circle of a sufficiently large radius $K_m$ in the $k_1\unicode{x2013}k_3$ plane. An analytical large-$k_\perp$ asymptote was calculated using the steps outlined in Hogg (Reference Hogg1994), and added to the numerical integral, to improve convergence. The analysis involves expanding $\widehat {\langle U_2\rangle }$ and $\widehat {\langle P\rangle }$ in inverse powers of $k_\perp$, an ansatz valid only in an $O(k_\perp ^{-1})$ neighbourhood of $R_2 = 0$. As a result, satisfaction of (4.6a,b) is replaced by a far-field decay requirement for ${R_2k_\perp \gg 1}$. Including only the exponentially decaying solutions on either side of $R_2 = 0$, satisfying the jump conditions (4.7a–d), and then performing the inverse Fourier transform, one obtains

(4.9)\begin{equation} \langle V_p\rangle^{far\ field}\approx-\frac{9\beta\gamma''\langle S_{21}\rangle Re_p}{64{\rm \pi} K_m Re_c^{1/2}}, \end{equation}

at leading order. As in § 3, the choice of $K_m$ is dictated by the distance of the particle from the walls. The relevant scale sufficiently close to either wall is still the spheroid-wall separation ($s$ or $1-s$) – the near-wall lift remains the same as that for $Re_c\ll 1$, being the far-field lift experienced by a particle in the presence of a single plane boundary. While one needs to keep increasing $K_m$ with approach to either wall (that is, for sufficiently small $s$ or $1-s$), to obtain a converged result, this increase is only necessary once $s$ or $(1-s)$ becomes less than $O(Re_c^{-1/2})$. Thus, to capture the wall-induced repulsion for finite $Re_c$, one needs to ensure $K_m Re_c^{1/2} s_{min}, K_m Re_c^{1/2} (1-s)_{min}\gg 1$. In our calculations we chose $K_m=200$ for $0.1\leq Re_c\leq 10$ to ensure accurate lift velocities down to $s, 1-s\approx 0.03$. For $Re_c>10$, accurate lift profiles were obtained down to $s, 1-s\approx 0.05$ for $K_m=40$.