5.1 The $O(St)$ analysis: drift due to particle inertia

The $O(St)$ correction to ${\it\bf\Omega}_{jeff}$ may be evaluated from (5.3) in a straightforward manner. We only note that ${\it\bf\Omega}_{jeff}$ in (5.3) includes the spin component, and the transverse components of ${\it\bf\Omega}_{1St}^{(1)}$ , that drive the $O(St)$ drift, therefore couple to the axial spin at leading order. For nearly spherical particles, Subramanian & Koch (Reference Subramanian and Koch2006b ) argued that the inertial drift may be interpreted as arising from an imbalance between the centrifugal and gyroscopic torques, both of which depend on the axial spin. Defining $\dot{{\it\phi}}_{jSt}=-{\it\Omega}_{1St}^{(1)x}/\sin {\it\theta}_{j}$ and $\dot{{\it\theta}}_{jSt}=St{\it\Omega}_{1St}^{(1)y}$ , the equations for the rates of change, at $O(St)$ , have the following general form:

(5.5) $$\begin{eqnarray}\displaystyle \left(\frac{\text{d}{\it\theta}_{j}}{\text{d}t}\right)_{St} & = & \displaystyle \sin {\it\theta}_{j}\cos {\it\theta}_{j} [F_{1}^{p}({\it\xi}_{0})+F_{2}^{p}({\it\xi}_{0})\cos 2{\it\phi}_{j}+F_{3}^{p}({\it\xi}_{0})\cos 2{\it\theta}_{j}+F_{4}^{p}({\it\xi}_{0})\cos 4{\it\phi}_{j}\nonumber\\ \displaystyle & & \displaystyle +\,F_{5}^{p}({\it\xi}_{0})\cos (2{\it\theta}_{j}-4{\it\phi}_{j})+F_{6}^{p}({\it\xi}_{0})\cos (2{\it\theta}_{j}+4{\it\phi}_{j})],\end{eqnarray}$$

(5.6) $$\begin{eqnarray}\displaystyle \left(\frac{\text{d}{\it\phi}_{j}}{\text{d}t}\right)_{St} & = & \displaystyle \sin {\it\phi}_{j}\cos {\it\phi}_{j} [G_{1}^{p}({\it\xi}_{0})+G_{2}^{p}({\it\xi}_{0})\cos 2{\it\theta}_{j}+G_{3}^{p}({\it\xi}_{0})\cos 2{\it\phi}_{j}\nonumber\\ \displaystyle & & \displaystyle +\,G_{4}^{p}({\it\xi}_{0})\cos (2{\it\theta}_{j})\cos (2{\it\phi}_{j})],\end{eqnarray}$$

for both prolate and oblate spheroids. The aspect-ratio-dependent functions are given by:

(5.7) $$\begin{eqnarray}\displaystyle & \displaystyle F_{1}^{p}({\it\xi}_{0})=\frac{(4{\it\xi}_{0}^{4}-7{\it\xi}_{0}^{2}+3)({\it\xi}_{0}^{2}\coth ^{-1}{\it\xi}_{0}+\coth ^{-1}{\it\xi}_{0}-{\it\xi}_{0})}{160{\it\xi}_{0}(1-2{\it\xi}_{0}^{2})^{2}}, & \displaystyle\end{eqnarray}$$

(5.8) $$\begin{eqnarray}\displaystyle & \displaystyle F_{2}^{p}({\it\xi}_{0})=\frac{\bar{{\it\xi}}_{0}^{4}(\coth ^{-1}{\it\xi}_{0}+{\it\xi}_{0}({\it\xi}_{0}\coth ^{-1}{\it\xi}_{0}-1))}{40{\it\xi}_{0}(1-2{\it\xi}_{0}^{2})^{2}}, & \displaystyle\end{eqnarray}$$

(5.9) $$\begin{eqnarray}\displaystyle & \displaystyle F_{3}^{p}({\it\xi}_{0})=F_{4}^{p}({\it\xi}_{0})=-\frac{\bar{{\it\xi}}_{0}^{2}({\it\xi}_{0}^{2}\coth ^{-1}{\it\xi}_{0}+\coth ^{-1}{\it\xi}_{0}-{\it\xi}_{0})}{160{\it\xi}_{0}(2{\it\xi}_{0}^{2}-1)^{2}}, & \displaystyle\end{eqnarray}$$

(5.10) $$\begin{eqnarray}\displaystyle & \displaystyle F_{5}^{p}({\it\xi}_{0})=F_{6}^{p}({\it\xi}_{0})=-\frac{F_{3}^{p}({\it\xi}_{0})}{2}, & \displaystyle\end{eqnarray}$$

(5.11) $$\begin{eqnarray}\displaystyle & \displaystyle G_{1}^{p}({\it\xi}_{0})=-\frac{(3{\it\xi}_{0}^{4}-5{\it\xi}_{0}^{2}+2)(({\it\xi}_{0}^{2}+1)\coth ^{-1}{\it\xi}_{0}-{\it\xi}_{0})}{40{\it\xi}_{0}(1-2{\it\xi}_{0}^{2})^{2}}, & \displaystyle\end{eqnarray}$$

(5.12) $$\begin{eqnarray}\displaystyle & \displaystyle G_{2}^{p}({\it\xi}_{0})={\it\xi}_{0}^{2}G_{3}^{p}({\it\xi}_{0})=-{\it\xi}_{0}^{2}G_{4}^{p}({\it\xi}_{0})=\frac{{\it\xi}_{0}(-{\it\xi}_{0}^{3}+{\it\xi}_{0}+({\it\xi}_{0}^{4}-1)\coth ^{-1}{\it\xi}_{0})}{40(1-2{\it\xi}_{0}^{2})^{2}}, & \displaystyle\end{eqnarray}$$

for a prolate spheroid, and for an oblate spheroid the functions can be obtained using the prolate–oblate transformation (note that the transformation ${\it\xi}_{0}\leftrightarrow \text{i}\bar{{\it\xi}}_{0}$ and $d\leftrightarrow -\text{i}d$ , has to be applied on the relevant dimensional angular velocity, having accounted for the aspect-ratio dependence that occurs in the relevant non-dimensional parameter – $St$ here and $Re$ in § 6.2). Although not needed for the calculations here, we note that the correction to the axial spin at $O(St)$ , ${\it\Omega}_{1St}^{(1)z}$ , is given by $H^{p}({\it\xi}_{0})\cos {\it\theta}_{j}\sin ^{2}{\it\theta}_{j}\sin 2{\it\phi}_{j}$ , with $H^{p}({\it\xi}_{0})=\bar{{\it\xi}}_{0}^{2}(\bar{{\it\xi}}_{0}^{2}\coth ^{-1}{\it\xi}_{0}-{\it\xi}_{0})/(40{\it\xi}_{0}(2{\it\xi}_{0}^{2}-1))$ , for a prolate spheroid. Thus, including particle inertia leads to an increased spin in the extensional quadrants of the simple shear flow and a reduction in the compressional quadrants, since the persistence of spin, at $O(St)$ , is proportional to the instantaneous rate of change of spin along a Jeffery orbit. The spin remains the same, to $O(St)$ , for orientations in the flow–vorticity ( ${\it\phi}_{j}=0,{\rm\pi}$ ) and gradient–vorticity planes ( ${\it\phi}_{j}={\rm\pi}/2,3{\rm\pi}/2$ ), since the rate of change of spin, at leading order, is zero for these orientations.

From the definition of $C$ for a prolate spheroid, the irreversible orbital drift, at $O(St)$ , may now be obtained as:

(5.13) $$\begin{eqnarray}\displaystyle \frac{\text{d}C}{\text{d}t} & = & \displaystyle \frac{C}{\sin {\it\theta}_{j}\cos {\it\theta}_{j}}\frac{\text{d}{\it\theta}_{j}}{\text{d}t}+\frac{C\cos {\it\phi}_{j}\sin {\it\phi}_{j}}{{\it\xi}_{0}^{2}\sin ^{2}{\it\phi}_{j}+\bar{{\it\xi}}_{0}^{2}\cos ^{2}{\it\phi}_{j}}\frac{\text{d}{\it\phi}_{j}}{\text{d}t},\end{eqnarray}$$

(5.14) $$\begin{eqnarray}\displaystyle & = & \displaystyle St\left[\frac{C}{\sin {\it\theta}_{j}\cos {\it\theta}_{j}}\left(\frac{\text{d}{\it\theta}_{j}}{\text{d}t}\right)_{St}+\frac{C\cos {\it\phi}_{j}\sin {\it\phi}_{j}}{{\it\xi}_{0}^{2}\sin ^{2}{\it\phi}_{j}+\bar{{\it\xi}}_{0}^{2}\cos ^{2}{\it\phi}_{j}}\left(\frac{\text{d}{\it\phi}_{j}}{\text{d}t}\right)_{St}\right],\end{eqnarray}$$

where $(\text{d}{\it\theta}_{j}/\text{d}t)_{St}$ and $(\text{d}{\it\phi}_{j}/\text{d}t)_{St}$ are given by (5.5) and (5.6). For $St\ll 1$ , the weak inertial drift leads to a tightly spiralling trajectory of the spheroidal axis, with each turn of this spiral closely approximating a Jeffery orbit. It is convenient to characterize the drift in terms of the pitch of this spiral measured in units of $C$ . The pitch equals the change in $C$ in a single Jeffery period, ${\rm\Delta}C$ , and $T_{jeff}^{-1}({\rm\Delta}C)$ may be regarded as the $C$ -component of an averaged inertial drift velocity. One obtains:

(5.15) $$\begin{eqnarray}\displaystyle {\rm\Delta}C_{p} & = & \displaystyle \int _{0}^{T_{jeff}}\frac{\text{d}C}{\text{d}t}\,\text{d}t,\end{eqnarray}$$

(5.16) $$\begin{eqnarray}\displaystyle & = & \displaystyle \left(\frac{\text{d}{\it\tau}}{\text{d}t}\right)^{-1}\int _{0}^{2{\rm\pi}}\frac{\text{d}C}{\text{d}t}\,\text{d}{\it\tau},\end{eqnarray}$$

(5.17) $$\begin{eqnarray}\displaystyle & = & \displaystyle St\frac{(2{\it\xi}_{0}^{2}-1)}{{\it\xi}_{0}\bar{{\it\xi}}_{0}}\int _{0}^{2{\rm\pi}}\left[\frac{C}{\sin {\it\theta}_{j}\cos {\it\theta}_{j}}\left(\frac{\text{d}{\it\theta}_{j}}{\text{d}t}\right)_{St}+\frac{C\cos {\it\phi}_{j}\sin {\it\phi}_{j}}{{\it\xi}_{0}^{2}\sin ^{2}{\it\phi}_{j}+\bar{{\it\xi}}_{0}^{2}\cos ^{2}{\it\phi}_{j}}\left(\frac{\text{d}{\it\phi}_{j}}{\text{d}t}\right)_{St}\right]\text{d}{\it\tau},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where the subscript ‘ $p$ ’ denotes particle inertia, and the ${\it\tau}$ -integrals are to be evaluated for $C$ fixed. Using (5.5) and (5.6):

(5.18) $$\begin{eqnarray}\displaystyle {\rm\Delta}C_{p} & = & \displaystyle St\,C\frac{(2{\it\xi}_{0}^{2}-1)}{{\it\xi}_{0}\bar{{\it\xi}}_{0}}\left\{\int _{0}^{2{\rm\pi}} [F_{1}^{p}({\it\xi}_{0})+F_{2}^{p}({\it\xi}_{0})\cos 2{\it\phi}_{j}+F_{3}^{p}({\it\xi}_{0})\cos 2{\it\theta}_{j}+F_{4}^{p}({\it\xi}_{0})\cos 4{\it\phi}_{j}\right.\nonumber\\ \displaystyle & & \displaystyle +\,F_{5}^{p}({\it\xi}_{0})\cos (2{\it\theta}_{j}-4{\it\phi}_{j})+F_{6}^{p}({\it\xi}_{0})\cos (2{\it\theta}_{j}+4{\it\phi}_{j})]\,\text{d}{\it\tau}\nonumber\\ \displaystyle & & \displaystyle +\,\int _{0}^{2{\rm\pi}}\frac{\sin ^{2}{\it\phi}_{j}\cos ^{2}{\it\phi}_{j}}{({\it\xi}_{0}^{2}\sin ^{2}{\it\phi}_{j}+\bar{{\it\xi}}_{0}^{2}\cos ^{2}{\it\phi}_{j})} [G_{1}^{p}({\it\xi}_{0})+G_{2}^{p}({\it\xi}_{0})\cos 2{\it\theta}_{j}\nonumber\\ \displaystyle & & \displaystyle +\left.G_{3}^{p}({\it\xi}_{0})\cos 2{\it\phi}_{j}+G_{4}^{p}({\it\xi}_{0})\cos 2{\it\theta}_{j}\cos 2{\it\phi}_{j} ]\,\text{d}{\it\tau}\vphantom{\displaystyle \int _{0}^{2{\rm\pi}}}\right\}.\end{eqnarray}$$

This may in turn be written as:

(5.19) $$\begin{eqnarray}\displaystyle {\rm\Delta}C_{p} & = & \displaystyle St\,C\frac{(2{\it\xi}_{0}^{2}-1)}{{\it\xi}_{0}\bar{{\it\xi}}_{0}}\{[I_{1}F_{1}^{p}({\it\xi}_{0})+I_{2}F_{2}^{p}({\it\xi}_{0})+I_{3}F_{3}^{p}({\it\xi}_{0})+I_{4}F_{4}^{p}({\it\xi}_{0})+I_{5}F_{5}^{p}({\it\xi}_{0})+I_{6}F_{6}^{p}({\it\xi}_{0})]\nonumber\\ \displaystyle & & \displaystyle +\,[J_{1}G_{1}^{p}({\it\xi}_{0})+J_{2}G_{2}^{p}({\it\xi}_{0})+J_{3}G_{3}^{p}({\it\xi}_{0})+J_{4}G_{4}^{p}({\it\xi}_{0})]\},\end{eqnarray}$$

where the values of $I_{i}$ and $J_{i}$ result from integrating the corresponding trigonometric functions in (5.18) over ${\it\tau}$ . To evaluate the drift to $O(St)$ , the trigonometric functions involved may be related to $C$ and ${\it\tau}$ using (4.11a,b ), the error in this approximation being $O(St^{2})$ . The expressions for $I_{1}$ to $I_{6}$ and $J_{1}$ to $J_{4}$ as functions of $C$ and ${\it\xi}_{0}$ are given in appendix C.

Figure 2. The drift due to the particle inertia, as characterized by the normalized change in the orbit constant in a single Jeffery period, $St^{-1}({\rm\Delta}C_{p}/(C^{2}+1))$ , plotted as a function of $C/(C+1)$ , for a prolate spheroid; $C/(C+1)=0$ and $C/(C+1)=1$ correspond to the log-rolling and tumbling modes, respectively. (a) Uses the additional normalization factor of ${\it\xi}_{0}^{2}$ , so the drift remains finite in the near-sphere limit ( ${\it\xi}_{0}\rightarrow \infty$ ). (b) Uses the normalization factor $(({\it\xi}_{0}-1)^{3/2}\log ({\it\xi}_{0}-1))^{-1}$ , to make the drift finite in the slender-fibre limit ( ${\it\xi}_{0}\rightarrow 1$ ).

The drift for a prolate spheroid that results from using (5.19) and (C 1)–(C 9) is plotted as a function of the normalized orbit constant, $C/(C+1)$ in figure 2 for various aspect ratios. The drift is evidently zero for $C=0$ and $C=\infty \,[C/(C+1)=1]$ on account of symmetry, but is positive for all other values of $C$ and for all aspect ratios. Thus, a massive prolate spheroid always drifts towards the tumbling mode. In the near-sphere limit ( ${\it\xi}_{0}\rightarrow \infty$ ), ${\rm\Delta}C_{p}\approx St\,(({\rm\pi}C)/30){\it\xi}_{0}^{-2}$ , as is expected on account of the drift being proportional to the square of the eccentricity ( $e=1/{\it\xi}_{0}$ ). In the limit of a slender fibre ( ${\it\xi}_{0}\rightarrow 1$ ), one finds ${\rm\Delta}C_{p}\approx -St\,(({\rm\pi}\sqrt{2}C)/5)({\it\xi}_{0}-1)^{3/2}\ln ({\it\xi}_{0}-1)$ . A leading-order estimate from the non-aligned phase of a rotating fibre comes out to be larger than $O[({\it\xi}_{0}-1)\ln ({\it\xi}_{0}-1)]$ . This estimate arises from transverse moments of inertia of $O({\it\xi}_{0}-1)$ driving a drift against a resistive torque of $O[\ln ({\it\xi}_{0}-1)]^{-1}$ predicted by slender body theory; the next correction in the non-aligned phase is $O[({\it\xi}_{0}-1)^{2}\ln ({\it\xi}_{0}-1)]$ . The actual estimate of $O[({\it\xi}_{0}-1)^{3/2}\ln ({\it\xi}_{0}-1)]$ above must therefore involve the dominant flow-aligned phase of the fibre. These near-sphere and slender-fibre scalings, together with a normalizing factor of $(C^{2}+1)^{-1}$ , are accounted for in figure 2 in order to render the ${\rm\Delta}C_{p}$ curve, in the relevant asymptotic limit, a finite one for all  $C$ .

The drift due to particle inertia for an oblate spheroid may be obtained in a similar manner with expressions for the analogues of the $I_{i}$ and $J_{i}$ now being obtained using (4.12a,b ). The normalized ${\rm\Delta}C_{p}$ for a massive oblate spheroid is plotted as a function of $C/(C+1)$ in figure 3, and is negative for all values of $C$ and aspect ratios, implying that the spheroid would asymptote to a steady spinning mode starting from any initial orientation. In the near-sphere limit, ${\rm\Delta}C_{p}$ is just the negative of that for a prolate spheroid. In contrast to a prolate spheroid, however, a normalization based on this near-sphere scaling of $O({\it\xi}_{0}^{-2})$ alone suffices for plotting ${\rm\Delta}C_{p}$ , since the inertial drift remains finite in the flat-disk limit – $\lim _{{\it\xi}_{0}\rightarrow 1}{\rm\Delta}C_{p}\approx -{\rm\pi}^{2}(((C^{2}+1)^{1/2}-1)/(20C))$ . The moments of inertia of a thin oblate spheroid are only $O({\it\xi}_{0}-1)^{1/2}$ , but their smallness appears to be compensated by the long, $O({\it\xi}_{0}-1)^{-1/2}$ , period available for inertia to act, leading to ${\rm\Delta}C_{p}$ being $O(1)$ for ${\it\xi}_{0}\rightarrow 1$ . Note that the ${\rm\Delta}C_{p}$ curve for any oblate spheroid crosses that of a near-sphere, with this cross-over point moving in from $C/(C+1)=1$ to a limiting value of approximately 0.75 for a flat disk. thus, for sufficiently thin spheroids, the inertial drift increases in magnitude below this cross-over $C$ , while decreasing for greater  $C$ values.

Figure 3. The drift due to the particle inertia, as characterized by the normalized change in the orbit constant in a single Jeffery period, $St^{-1}(({\rm\Delta}C_{p})/((C^{2}+1)))$ , plotted as a function of $C/(C+1)$ , for an oblate spheroid; $C/(C+1)=0$ and $C/(C+1)=1$ correspond to the spinning and the tumbling modes, respectively.

The expressions for $(\text{d}{\it\theta}_{j}/\text{d}t)_{St}$ and $(\text{d}{\it\phi}_{j}/\text{d}t)_{St}$ do not match those given in Subramanian & Koch (Reference Subramanian and Koch2006b ) for a near-sphere in the limit ${\it\xi}_{0}\rightarrow \infty$ . The expression for $(\text{d}{\it\theta}_{j}/\text{d}t)_{St}$ given in Subramanian & Koch (Reference Subramanian and Koch2006b , (3.15) therein) missed a $1/2$ in the term that multiplies $\cos (2{\it\phi}_{j})$ and the expression (3.16) for $(\text{d}{\it\phi}_{j}/\text{d}t)_{St}$ missed a factor of $(1+\sin ^{2}{\it\theta}_{j})/2$ . The expressions for the transverse components of the $O(St)$ angular velocity have been recently obtained by Einarsson et al. (Reference Einarsson, Angilella and Mehlig2014), for an arbitrary-aspect-ratio spheroid, directly from the equations of motion. On account of the spheroidal coordinates used here, and in order to facilitate a smooth transition to the $O(St^{2})$ results presented in the next subsection, we have rederived the equations governing the $O(St)$ drift. The $O(St)$ rates of change $\dot{{\it\phi}_{j}}$ and $\dot{{\it\theta}_{j}}$ in (5.5) and (5.6) match with those given in Einarsson et al. (Reference Einarsson, Angilella and Mehlig2014), although the results therein are not interpreted in terms of an orbital drift. The latter offers a convenient interpretation, and also a physically significant one. The influence of additional physical mechanisms on the orientation distribution, either the weak Brownian motion considered in § 7.2, or orientation fluctuations of a hydrodynamic origin that might arise at higher volume fractions from pair interactions, are best interpreted, in $(C,{\it\tau})$ coordinates, in terms of the distribution of Jeffery orbits arising from a balance of a deterministic inertial drift and diffusion along the $C$ coordinate.

Finally, it will be seen in §§ 6.1 and 6.2, where we determine the orbital drift ${\rm\Delta}C_{f}$ driven by fluid inertia, that, for extreme aspect ratios, fluid inertia always dominates particle inertia. This is because the moment of inertia of a slender fibre ( $O({\it\rho}L^{3}b^{2})$ ) or a flat disk ( $O({\it\rho}L^{4}b)$ ) is asymptotically smaller than the moment of inertia of the disturbed fluid sphere ( $O({\it\rho}L^{5})$ ) around the rotating particle that contributes to the fluid inertial forces and governs the $O(Re)$ drift (Subramanian & Koch Reference Subramanian and Koch2005).

5.2 The $O(St^{2})$ analysis: effect of particle inertia on the time period of rotation

There is no correction to the period of rotation at $O(St)$ for a Jeffery orbit with any  $C$ . This may be seen by writing the period $T$ as $T=\int _{{\rm\pi}}^{-{\rm\pi}}\,\text{d}{\it\phi}_{j}/\dot{{\it\phi}}_{j}$ , and using $\dot{{\it\phi}}=\dot{{\it\phi}}_{j0}+St(\text{d}{\it\phi}_{j}/\text{d}t)_{St}$ , with $\dot{{\it\phi}}_{j0}$ and $(\text{d}{\it\phi}_{j}/\text{d}t)_{St}$ being given by (4.10) and (5.6), respectively. The $O(St)$ correction to the Jeffery period then takes the form ${\rm\Delta}T_{St}=St\,\int _{{\rm\pi}}^{-{\rm\pi}}(\text{d}{\it\phi}_{j}/\dot{{\it\phi}}_{j0}^{2})(\text{d}{\it\phi}_{j}/\text{d}t)_{St}$ , which integrates to 0 for any fixed  $C$ . The correction to the time period for an inertial spheroid, comes at $O(St^{2})$ for all orbits except for the spinning mode ( $C=0$ ). As will be seen later in § 6, the rate of change of ${\it\phi}_{j}$ due to the fluid inertia at $O(Re)$ takes the same form as (5.6), and therefore there is no correction to the time period of rotation at $O(Re)$ either. For fluid inertia, as mentioned in earlier sections, the modification of the period arises at $O(Re^{3/2})$ and not at $O(Re^{2})$ , and the singular origin implies that such a modification would be present for both the tumbling and (steadily) spinning modes; the slowing down of a freely rotating sphere in simple shear has already been analysed (Lin, Peery & Schowalter Reference Lin, Peery and Schowalter1970; Stone, John & Lovalenti Reference Stone, John and Lovalenti2000).

In what follows, we determine the correction to the Jeffery period for a tumbling prolate spheroid. Using $\dot{{\it\phi}}_{j}=\dot{{\it\phi}}_{j0}+St(\text{d}{\it\phi}_{j}/\text{d}t)_{St}+St^{2}(\text{d}{\it\phi}_{j}/\text{d}t)_{St^{2}}$ , the $O(St^{2})$ correction to the period of rotation comes out to be:

(5.20) $$\begin{eqnarray}{\rm\Delta}T_{St^{2}}=St^{2}\int _{{\rm\pi}}^{-{\rm\pi}}\left[\frac{1}{\dot{{\it\phi}}_{j0}^{3}}\left(\frac{\text{d}{\it\phi}_{j}}{\text{d}t}\right)_{St}^{2}-\frac{1}{\dot{{\it\phi}}_{j0}^{2}}\left(\frac{\text{d}{\it\phi}_{j}}{\text{d}t}\right)_{St^{2}}\right],\end{eqnarray}$$

where $(\text{d}{\it\phi}_{j}/\text{d}t)_{St}$ is given by (5.6). Starting from (5.4), the final expression for $(\text{d}{\it\phi}_{j}/\text{d}t)_{St^{2}}$ valid for arbitrary $C$ is very lengthy. Owing to the sign of the $O(St)$ drift, for long times, one only needs the limiting form relevant for a tumbling mode ( ${\it\theta}_{j}={\rm\pi}/2,C=\infty$ ). This is significantly simpler and given by:

(5.21) $$\begin{eqnarray}\left(\frac{\text{d}{\it\phi}_{j}}{\text{d}t}\right)_{St^{2}}=H_{1}^{p}({\it\xi}_{0})\cos 2{\it\phi}_{j}+H_{2}^{p}({\it\xi}_{0})\cos 4{\it\phi}_{j}+H_{3}^{p}({\it\xi}_{0})\cos 6{\it\phi}_{j},\end{eqnarray}$$

where

(5.22) $$\begin{eqnarray}\displaystyle & \displaystyle H_{1}^{p}({\it\xi}_{0})=-\frac{(16{\it\xi}_{0}^{4}-16{\it\xi}_{0}^{2}+5)(\bar{{\it\xi}}_{0}^{2}{\it\xi}_{0}+(1-{\it\xi}_{0}^{4})\coth ^{-1}{\it\xi}_{0})^{2}}{3200{\it\xi}_{0}^{2}(2{\it\xi}_{0}^{2}-1)^{3}}, & \displaystyle\end{eqnarray}$$

(5.23) $$\begin{eqnarray}\displaystyle & \displaystyle H_{2}^{p}({\it\xi}_{0})=\frac{(\bar{{\it\xi}}_{0}^{2}{\it\xi}_{0}+(1-{\it\xi}_{0}^{4})\coth ^{-1}{\it\xi}_{0})^{2}}{400({\it\xi}_{0}-2{\it\xi}_{0}^{3})^{2}}, & \displaystyle\end{eqnarray}$$

(5.24) $$\begin{eqnarray}\displaystyle & \displaystyle H_{3}^{p}({\it\xi}_{0})=-\frac{3(\bar{{\it\xi}}_{0}^{2}{\it\xi}_{0}+(1-{\it\xi}_{0}^{4})\coth ^{-1}{\it\xi}_{0})^{2}}{3200{\it\xi}_{0}^{2}(2{\it\xi}_{0}^{2}-1)^{3}}. & \displaystyle\end{eqnarray}$$

Using (5.21)–(5.24) in (5.20), the change induced by particle inertia in the period of rotation of a tumbling prolate spheroid is given by:

(5.25) $$\begin{eqnarray}{\rm\Delta}T_{St^{2}}=St^{2}\frac{{\rm\pi}[2{\it\xi}_{0}(\bar{{\it\xi}}_{0}-{\it\xi}_{0})+1][(1-{\it\xi}_{0}^{4})\coth ^{-1}{\it\xi}_{0}+{\it\xi}_{0}^{3}-{\it\xi}_{0}]^{2}}{100{\it\xi}_{0}^{2}(2{\it\xi}_{0}^{2}-1)}.\end{eqnarray}$$

The analysis in the earlier section also shows that, at $O(St)$ , inertial persistence leads to a shift in the locations of the angular velocity extrema relative to those for the original Jeffery orbit. The angular displacements are of $O({\it\Omega}({\it\phi}_{m}){\it\tau}_{p})$ , ${\it\tau}_{p}=I/L^{(2)}$ being the inertial relaxation time that governs the persistence of the angular velocity. Here, ${\it\Omega}({\it\phi}_{m})$ is the angular speed at the extremum, $I$ is the (non-dimensional) equatorial moment of inertia relevant to the tumbling mode, while $L^{(2)}$ is the aspect-ratio scale for the test-torque coefficient. In the near-sphere limit, the angular displacements are $O(St)$ for both the maxima and minima, while for the slender-fibre limit, the test-torque coefficient is $O[\ln ({\it\xi}_{0}-1)]^{-1}$ , and these displacements are $O[St({\it\xi}_{0}-1)\ln ({\it\xi}_{0}-1)]$ and $O[St({\it\xi}_{0}-1)^{2}\ln ({\it\xi}_{0}-1)]$ for the maxima ( ${\it\phi}_{m}={\rm\pi}/2,3{\rm\pi}/2$ ) and minima ( ${\it\phi}_{m}=0,{\rm\pi}$ ), respectively. This $O(St)$ alteration in the angular velocity profile does not, however, change the period due to the antisymmetry of the angular acceleration profiles in the compressional and extensional quadrants of the Jeffery orbit. At $O(St^{2})$ , this antisymmetry is broken, with the result that the decrease in the traversal time of the extensional quadrants is greater than the corresponding increase in the compressional quadrants, leading to a net decrease in the period of rotation. For a near-sphere, this reduction in period is smaller than expected owing to a cancellation at $O(St^{2}{\it\xi}_{0}^{-2})$ ; (5.25) gives $-St^{2}({\rm\pi}/(450{\it\xi}_{0}^{4}))$ . For a slender fibre, the changes in angular velocity over the meridional portion of the trajectory (the non-aligned phase) govern the reduction in period. The $O(St)$ angular acceleration of a non-aligned slender fibre combines with a moment of inertia of $O({\it\xi}_{0}-1)$ , leading to a change in angular velocity of $St\,({\it\xi}_{0}-1)\ln ({\it\xi}_{0}-1)$ . An analogous argument implies that the change in angular velocity at $O(St^{2})$ is $St^{2}({\it\xi}_{0}-1)^{2}[\ln ({\it\xi}_{0}-1)]^{2}$ , which then gives the aspect-ratio scaling for the reduction in the period; the limiting form of (5.25) gives $-St^{2}({\rm\pi}/25)({\it\xi}_{0}-1)^{2}[\ln ({\it\xi}_{0}-1)]^{2}$ .

Figure 4. The $O(St^{2})$ corrections to the time period and the angular velocity extrema, as a function of ${\it\xi}_{0}$ , for a prolate spheroid in the tumbling mode.

It turns out that the minimum and maximum angular velocities in the tumbling mode are unaltered at $O(St)$ . Starting from (5.21), one obtains the following expressions, to $O(St^{2})$ , for the angular velocity extrema:

(5.26) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\Omega}_{max}=-\frac{{\it\xi}_{0}^{2}}{2{\it\xi}_{0}^{2}-1}+St^{2}\frac{{\it\xi}_{0}^{2}[-{\it\xi}_{0}^{3}+{\it\xi}_{0}+({\it\xi}_{0}^{4}-1)\coth ^{-1}{\it\xi}_{0}]^{2}}{400(2{\it\xi}_{0}^{2}-1)^{3}}, & \displaystyle\end{eqnarray}$$

(5.27) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\Omega}_{min}=-\frac{\bar{{\it\xi}}_{0}^{2}}{2{\it\xi}_{0}^{2}-1}-St^{2}\frac{\bar{{\it\xi}}_{0}^{8}[{\it\xi}_{0}-({\it\xi}_{0}^{2}+1)\coth ^{-1}{\it\xi}_{0}]^{2}}{400{\it\xi}_{0}^{2}(2{\it\xi}_{0}^{2}-1)^{3}}. & \displaystyle\end{eqnarray}$$

The corrections to the time period, maximum and minimum angular velocities are plotted as a function of the prolate spheroid eccentricity ( $e=1/{\it\xi}_{0}$ ) in figure 4. The effect of particle inertia is to make the angular velocity extrema approach each other, consistent with the general notion of an inertial resistance to angular acceleration (for large $St$ , the particle begins to rotate with a constant angular velocity of $4{\rm\pi}$ in the flow–gradient plane (Lundell Reference Lundell2011)). Although the changes in the maxima and minima are of opposite signs, the correction to time period is negative for all ${\it\xi}_{0}$ , implying that the time period decreases at $O(St^{2})$ . This decrease is consistent with recent simulations (Mao & Alexeev Reference Mao and Alexeev2014), although a quantitative comparison is not possible due to the simulations being carried out for much larger $St$ . A couple of points are in order before moving on to an analysis of fluid inertial effects. The expression for $(\text{d}{\it\phi}_{j}/\text{d}t)_{St^{2}}$ , valid for arbitrary $C$ , in (5.20), shows that particle inertia acts to decrease the period of rotation for arbitrary $C$ . Further, since the analysis does not explicitly depend on the disturbance velocity field, it can easily be generalized to an arbitrary axisymmetric particle, provided one uses the appropriate moment of inertia tensor, and the geometric aspect ratio is replaced by the effective aspect ratio (as obtained from the Jeffery rotation period; see Bretherton (Reference Bretherton1962)).

6.1 The $O(Re)$ orbital drift for prolate spheroids

The aspect-ratio-dependent functions appearing in the fluid inertial analogues of (5.5) and (5.6) are given by:

(6.1) $$\begin{eqnarray}\displaystyle F_{1}^{f}({\it\xi}_{0}) & = & \displaystyle ({\it\xi}_{0}^{2}(-648{\it\xi}_{0}^{12}+1350{\it\xi}_{0}^{10}-5571{\it\xi}_{0}^{8}+11\,841{\it\xi}_{0}^{6}-9269{\it\xi}_{0}^{4}+2263{\it\xi}_{0}^{2}+6)\nonumber\\ \displaystyle & & \displaystyle -\,27{\it\xi}_{0}^{2}(24{\it\xi}_{0}^{8}-14{\it\xi}_{0}^{6}-19{\it\xi}_{0}^{4}+16{\it\xi}_{0}^{2}-3)\bar{{\it\xi}}_{0}^{8}\coth ^{-1}({\it\xi}_{0})^{4}\nonumber\\ \displaystyle & & \displaystyle +\,9{\it\xi}_{0}(288{\it\xi}_{0}^{12}-564{\it\xi}_{0}^{10}-20{\it\xi}_{0}^{8}+799{\it\xi}_{0}^{6}-743{\it\xi}_{0}^{4}+261{\it\xi}_{0}^{2}-29)\bar{{\it\xi}}_{0}^{4}\coth ^{-1}({\it\xi}_{0})^{3}\nonumber\\ \displaystyle & & \displaystyle +\,{\it\xi}_{0} (2592{\it\xi}_{0}^{14}-7020{\it\xi}_{0}^{12}+13\,932{\it\xi}_{0}^{10}-21\,123{\it\xi}_{0}^{8}+14\,255{\it\xi}_{0}^{6}-577{\it\xi}_{0}^{4}\nonumber\\ \displaystyle & & \displaystyle -\,2711{\it\xi}_{0}^{2}+652)\coth ^{-1}{\it\xi}_{0}-3(1296{\it\xi}_{0}^{16}-4320{\it\xi}_{0}^{14}+5346{\it\xi}_{0}^{12}-1477{\it\xi}_{0}^{10}\nonumber\\ \displaystyle & & \displaystyle -\,4260{\it\xi}_{0}^{8}+6116{\it\xi}_{0}^{6}-3492{\it\xi}_{0}^{4}+849{\it\xi}_{0}^{2}-58)\coth ^{-1}({\it\xi}_{0})^{2} )\nonumber\\ \displaystyle & & \displaystyle \times (480{\it\xi}_{0}^{2}(-1+2{\it\xi}_{0}^{2})^{3}(-3{\it\xi}_{0}^{2}+3\bar{{\it\xi}}_{0}^{2}{\it\xi}_{0}\coth ^{-1}{\it\xi}_{0}+2)\nonumber\\ \displaystyle & & \displaystyle \times \,(-3{\it\xi}_{0}^{3}+5{\it\xi}_{0}+3\bar{{\it\xi}}_{0}^{4}\coth ^{-1}{\it\xi}_{0})((3{\it\xi}_{0}^{2}-1)\coth ^{-1}{\it\xi}_{0}-3{\it\xi}_{0}))^{-1}\end{eqnarray}$$

(6.2) $$\begin{eqnarray}\displaystyle F_{2}^{f}({\it\xi}_{0}) & = & \displaystyle -(\bar{{\it\xi}}_{0}^{2} (-9{\it\xi}_{0}^{9}+30{\it\xi}_{0}^{7}-115{\it\xi}_{0}^{5}+90{\it\xi}_{0}^{3}-12{\it\xi}_{0}\nonumber\\ \displaystyle & & \displaystyle +\,9\bar{{\it\xi}}_{0}^{8}({\it\xi}_{0}^{2}+1){\it\xi}_{0}^{2}\coth ^{-1}({\it\xi}_{0})^{3}-3\bar{{\it\xi}}_{0}^{4} (9{\it\xi}_{0}^{6}-10{\it\xi}_{0}^{4}-17{\it\xi}_{0}^{2}+14){\it\xi}_{0}\coth ^{-1}({\it\xi}_{0})^{2}\nonumber\\ \displaystyle & & \displaystyle +\,(27{\it\xi}_{0}^{10}-87{\it\xi}_{0}^{8}+133{\it\xi}_{0}^{6}-33{\it\xi}_{0}^{4}-52{\it\xi}_{0}^{2}+12)\coth ^{-1}{\it\xi}_{0} ))\nonumber\\ \displaystyle & & \displaystyle \times (40({\it\xi}_{0}-2{\it\xi}_{0}^{3})^{2}(-3{\it\xi}_{0}^{2}+3\bar{{\it\xi}}_{0}^{2}{\it\xi}_{0}\coth ^{-1}{\it\xi}_{0}+2)\nonumber\\ \displaystyle & & \displaystyle \times \,(-3{\it\xi}_{0}^{3}+5{\it\xi}_{0}+3\bar{{\it\xi}}_{0}^{4}\coth ^{-1}{\it\xi}_{0}))^{-1}\end{eqnarray}$$

(6.3) $$\begin{eqnarray}\displaystyle F_{3}^{f}({\it\xi}_{0}) & = & \displaystyle -({\it\xi}_{0}^{2}(378{\it\xi}_{0}^{10}+801{\it\xi}_{0}^{8}-4731{\it\xi}_{0}^{6}+5551{\it\xi}_{0}^{4}-2369{\it\xi}_{0}^{2}+342)\nonumber\\ \displaystyle & & \displaystyle -\,27{\it\xi}_{0}^{2}(6{\it\xi}_{0}^{6}+{\it\xi}_{0}^{4}-4{\it\xi}_{0}^{2}+1)\bar{{\it\xi}}_{0}^{8}\coth ^{-1}({\it\xi}_{0})^{4}+9{\it\xi}_{0} (12{\it\xi}_{0}^{10}+28{\it\xi}_{0}^{8}-201{\it\xi}_{0}^{6}\nonumber\\ \displaystyle & & \displaystyle +\,273{\it\xi}_{0}^{4}-147{\it\xi}_{0}^{2}+27)\bar{{\it\xi}}_{0}^{4}\coth ^{-1}({\it\xi}_{0})^{3}+ (-972{\it\xi}_{0}^{13}-324{\it\xi}_{0}^{11}+7365{\it\xi}_{0}^{9}\nonumber\\ \displaystyle & & \displaystyle -\,10\,409{\it\xi}_{0}^{7}+5143{\it\xi}_{0}^{5}-847{\it\xi}_{0}^{3}+44{\it\xi}_{0} )\coth ^{-1}{\it\xi}_{0}+3(216{\it\xi}_{0}^{14}-378{\it\xi}_{0}^{12}\nonumber\\ \displaystyle & & \displaystyle +\,109{\it\xi}_{0}^{10}-412{\it\xi}_{0}^{8}+1204{\it\xi}_{0}^{6}-1028{\it\xi}_{0}^{4}+311{\it\xi}_{0}^{2}-22)\coth ^{-1}({\it\xi}_{0})^{2} )\nonumber\\ \displaystyle & & \displaystyle \times (480{\it\xi}_{0}^{2}(-1+2{\it\xi}_{0}^{2})^{3}(-3{\it\xi}_{0}^{2}+3\bar{{\it\xi}}_{0}^{2}{\it\xi}_{0}\coth ^{-1}{\it\xi}_{0}+2)(-3{\it\xi}_{0}^{3}+5{\it\xi}_{0}\nonumber\\ \displaystyle & & \displaystyle +\,3\bar{{\it\xi}}_{0}^{4}\coth ^{-1}{\it\xi}_{0} )((3{\it\xi}_{0}^{2}-1)\coth ^{-1}{\it\xi}_{0}-3{\it\xi}_{0}))^{-1}\end{eqnarray}$$

(6.4) $$\begin{eqnarray}\displaystyle F_{4}^{f}({\it\xi}_{0}) & = & \displaystyle -2F_{5}^{f}({\it\xi}_{0})=-2F_{6}^{f}({\it\xi}_{0})=F_{3}^{f}({\it\xi}_{0})\end{eqnarray}$$

(6.5) $$\begin{eqnarray}\displaystyle G_{1}^{f}({\it\xi}_{0}) & = & \displaystyle ({\it\xi}_{0}^{2}(81{\it\xi}_{0}^{10}-414{\it\xi}_{0}^{8}+1074{\it\xi}_{0}^{6}-1162{\it\xi}_{0}^{4}+479{\it\xi}_{0}^{2}-54)\nonumber\\ \displaystyle & & \displaystyle +\,9{\it\xi}_{0}^{2}(9{\it\xi}_{0}^{6}-7{\it\xi}_{0}^{2}+2)\bar{{\it\xi}}_{0}^{8}\coth ^{-1}({\it\xi}_{0})^{4}-3{\it\xi}_{0} (108{\it\xi}_{0}^{10}-246{\it\xi}_{0}^{8}+69{\it\xi}_{0}^{6}\nonumber\\ \displaystyle & & \displaystyle +\,167{\it\xi}_{0}^{4}-129{\it\xi}_{0}^{2}+23)\bar{{\it\xi}}_{0}^{4}\coth ^{-1}({\it\xi}_{0})^{3}+ (-324{\it\xi}_{0}^{13}+1566{\it\xi}_{0}^{11}\nonumber\\ \displaystyle & & \displaystyle -\,3309{\it\xi}_{0}^{9}+3133{\it\xi}_{0}^{7}-1023{\it\xi}_{0}^{5}-79{\it\xi}_{0}^{3}+36{\it\xi}_{0} )\coth ^{-1}{\it\xi}_{0}\nonumber\\ \displaystyle & & \displaystyle +\,(486{\it\xi}_{0}^{14}-2214{\it\xi}_{0}^{12}+3819{\it\xi}_{0}^{10}-2568{\it\xi}_{0}^{8}-222{\it\xi}_{0}^{6}+1036{\it\xi}_{0}^{4}-355{\it\xi}_{0}^{2}\nonumber\\ \displaystyle & & \displaystyle +\,18)\coth ^{-1}({\it\xi}_{0})^{2} )(40({\it\xi}_{0}-2{\it\xi}_{0}^{3})^{2}(-3{\it\xi}_{0}^{2}+3\bar{{\it\xi}}_{0}^{2}{\it\xi}_{0}\coth ^{-1}{\it\xi}_{0}+2)\nonumber\\ \displaystyle & & \displaystyle \times \,(-3{\it\xi}_{0}^{3}+5{\it\xi}_{0}+3\bar{{\it\xi}}_{0}^{4}\coth ^{-1}{\it\xi}_{0})((3{\it\xi}_{0}^{2}-1)\coth ^{-1}{\it\xi}_{0}-3{\it\xi}_{0}))^{-1}\end{eqnarray}$$

(6.6) $$\begin{eqnarray}\displaystyle G_{2}^{f}({\it\xi}_{0}) & = & \displaystyle (-{\it\xi}_{0}^{2}(27{\it\xi}_{0}^{10}-180{\it\xi}_{0}^{8}+204{\it\xi}_{0}^{6}+68{\it\xi}_{0}^{4}-133{\it\xi}_{0}^{2}+18)\nonumber\\ \displaystyle & & \displaystyle -\,9{\it\xi}_{0}^{4}(3{\it\xi}_{0}^{4}+2{\it\xi}_{0}^{2}-1)\bar{{\it\xi}}_{0}^{8}\coth ^{-1}({\it\xi}_{0})^{4}+3{\it\xi}_{0} (36{\it\xi}_{0}^{10}-78{\it\xi}_{0}^{8}+73{\it\xi}_{0}^{6}-69{\it\xi}_{0}^{4}\nonumber\\ \displaystyle & & \displaystyle +\,35{\it\xi}_{0}^{2}-5)\bar{{\it\xi}}_{0}^{4}\coth ^{-1}({\it\xi}_{0})^{3}+{\it\xi}_{0} (108{\it\xi}_{0}^{12}-630{\it\xi}_{0}^{10}+1041{\it\xi}_{0}^{8}-617{\it\xi}_{0}^{6}\nonumber\\ \displaystyle & & \displaystyle +\,115{\it\xi}_{0}^{4}-29{\it\xi}_{0}^{2}+12)\coth ^{-1}{\it\xi}_{0}+ (-162{\it\xi}_{0}^{14}+810{\it\xi}_{0}^{12}-1551{\it\xi}_{0}^{10}\nonumber\\ \displaystyle & & \displaystyle +\,1600{\it\xi}_{0}^{8}-1054{\it\xi}_{0}^{6}+448{\it\xi}_{0}^{4}-97{\it\xi}_{0}^{2}+6)\coth ^{-1}({\it\xi}_{0})^{2} )\nonumber\\ \displaystyle & & \displaystyle \times (40({\it\xi}_{0}-2{\it\xi}_{0}^{3})^{2}(-3{\it\xi}_{0}^{2}+3\bar{{\it\xi}}_{0}^{2}{\it\xi}_{0}\coth ^{-1}{\it\xi}_{0}+2)(-3{\it\xi}_{0}^{3}\nonumber\\ \displaystyle & & \displaystyle +\,5{\it\xi}_{0}+3\bar{{\it\xi}}_{0}^{4}\coth ^{-1}{\it\xi}_{0} )((3{\it\xi}_{0}^{2}-1)\coth ^{-1}{\it\xi}_{0}-3{\it\xi}_{0}))^{-1}\end{eqnarray}$$

(6.7) $$\begin{eqnarray}\displaystyle G_{3}^{f}({\it\xi}_{0}) & = & \displaystyle ({\it\xi}_{0}^{2}(378{\it\xi}_{0}^{10}+801{\it\xi}_{0}^{8}-4731{\it\xi}_{0}^{6}+5551{\it\xi}_{0}^{4}-2369{\it\xi}_{0}^{2}+342)\nonumber\\ \displaystyle & & \displaystyle -\,27{\it\xi}_{0}^{2}(6{\it\xi}_{0}^{6}+{\it\xi}_{0}^{4}-4{\it\xi}_{0}^{2}+1)\bar{{\it\xi}}_{0}^{8}\coth ^{-1}({\it\xi}_{0})^{4}+9{\it\xi}_{0} (12{\it\xi}_{0}^{10}+28{\it\xi}_{0}^{8}\nonumber\\ \displaystyle & & \displaystyle -\,201{\it\xi}_{0}^{6}+273{\it\xi}_{0}^{4}-147{\it\xi}_{0}^{2}+27)\bar{{\it\xi}}_{0}^{4}\coth ^{-1}({\it\xi}_{0})^{3}+ (-972{\it\xi}_{0}^{13}\nonumber\\ \displaystyle & & \displaystyle -\,324{\it\xi}_{0}^{11}+7365{\it\xi}_{0}^{9}-10\,409{\it\xi}_{0}^{7}+5143{\it\xi}_{0}^{5}-847{\it\xi}_{0}^{3}+44{\it\xi}_{0} )\coth ^{-1}{\it\xi}_{0}\nonumber\\ \displaystyle & & \displaystyle +\,3(216{\it\xi}_{0}^{14}-378{\it\xi}_{0}^{12}+109{\it\xi}_{0}^{10}-412{\it\xi}_{0}^{8}+1204{\it\xi}_{0}^{6}-1028{\it\xi}_{0}^{4}\nonumber\\ \displaystyle & & \displaystyle +\,311{\it\xi}_{0}^{2}-22){\coth ^{-1}({\it\xi}_{0})}^{2} )(120{\it\xi}_{0}^{2}(2{\it\xi}_{0}^{2}-1)^{3}\nonumber\\ \displaystyle & & \displaystyle \times \,(-3{\it\xi}_{0}^{2}+3\bar{{\it\xi}}_{0}^{2}{\it\xi}_{0}\coth ^{-1}{\it\xi}_{0}+2)(-3{\it\xi}_{0}^{3}+5{\it\xi}_{0}+3\bar{{\it\xi}}_{0}^{4}\coth ^{-1}{\it\xi}_{0})\nonumber\\ \displaystyle & & \displaystyle \times \,((3{\it\xi}_{0}^{2}-1)\coth ^{-1}{\it\xi}_{0}-3{\it\xi}_{0}))^{-1}\end{eqnarray}$$

(6.8) $$\begin{eqnarray}\displaystyle G_{4}^{f}({\it\xi}_{0}) & = & \displaystyle -G_{3}^{f}({\it\xi}_{0}).\end{eqnarray}$$

The near-sphere limits of the above functions ( ${\it\xi}_{0}\rightarrow \infty$ ) are given by:

(6.9a−c ) $$\begin{eqnarray}\displaystyle & \displaystyle F_{1}^{f}({\it\xi}_{0})\approx \frac{11}{280{\it\xi}_{0}^{2}},\quad F_{2}^{f}({\it\xi}_{0})\approx -\frac{37}{840{\it\xi}_{0}^{2}},\quad F_{3}^{f}({\it\xi}_{0})\approx -\frac{163}{31\,360{\it\xi}_{0}^{4}} & \displaystyle\end{eqnarray}$$

(6.10a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{71}G_{1}^{f}({\it\xi}_{0})=\frac{1}{3}G_{2}^{f}({\it\xi}_{0})\approx \frac{1}{840{\it\xi}_{0}^{2}},\quad G_{3}^{f}({\it\xi}_{0})\approx \frac{163}{7840{\it\xi}_{0}^{4}}, & \displaystyle\end{eqnarray}$$

and the slender-fibre limits ( ${\it\xi}_{0}\rightarrow 1$ ) are given by:

(6.11) $$\begin{eqnarray}\displaystyle & \displaystyle F_{1}^{f}({\it\xi}_{0})=-F_{3}^{f}({\it\xi}_{0})\approx -\frac{7}{240[\log ({\it\xi}_{0}-1)-\log 2+3]}, & \displaystyle\end{eqnarray}$$

(6.12) $$\begin{eqnarray}\displaystyle & \displaystyle F_{2}^{f}({\it\xi}_{0})\approx -{\textstyle \frac{2}{5}}({\it\xi}_{0}-1), & \displaystyle\end{eqnarray}$$

(6.13) $$\begin{eqnarray}\displaystyle & \displaystyle G_{1}^{f}({\it\xi}_{0})=-G_{2}^{f}({\it\xi}_{0})=-\frac{3}{7}G_{3}^{f}({\it\xi}_{0})\approx \frac{1}{20[\log ({\it\xi}_{0}-1)-\log 2+3]}. & \displaystyle\end{eqnarray}$$

The inertial drift evaluated using the above near-sphere and slender-fibre asymptotes above are compared with the earlier analytical results of Subramanian & Koch (Reference Subramanian and Koch2005) and Subramanian & Koch (Reference Subramanian and Koch2006b ) in appendix D.

The normalized change in the drift constant ( ${\rm\Delta}C_{f}$ ) is plotted as a function of $C$ in figure 5 for various aspect ratios. ${\rm\Delta}C_{f}$ is positive for all aspect ratios and for all values of $C$ . Fluid inertia therefore causes a prolate spheroid to drift towards the tumbling mode starting from an arbitrary initial orientation. For a near-sphere, ${\rm\Delta}C_{f}$ reduces to:

(6.14) $$\begin{eqnarray}\displaystyle \lim _{{\it\xi}_{0}\rightarrow \infty }{\rm\Delta}C_{f}=\frac{11{\rm\pi}C}{70{\it\xi}_{0}^{2}}, & & \displaystyle\end{eqnarray}$$

at leading order and, for a slender fibre, one obtains:

(6.15) $$\begin{eqnarray}\displaystyle \lim _{{\it\xi}_{0}\rightarrow 1}{\rm\Delta}C_{f}=-\frac{\sqrt{2}{\rm\pi}C}{15({\it\xi}_{0}-1)^{1/2}\ln ({\it\xi}_{0}-1)}. & & \displaystyle\end{eqnarray}$$

These expressions motivate the normalizations used in figure 5. The factor ${\it\xi}_{0}^{2}$ in the near-sphere limit is identical to that for particle inertia, as would be expected since an inertial drift in either case would scale with the square of the eccentricity. In the limit of a slender fibre, the inertial terms may be linearized at leading order, being proportional to the leading-order Stokes disturbance field of $O[\ln ({\it\xi}_{0}-1)]^{-1}$ associated with the axisymmetric extensional component of the simple shear (given by (3.7) in § 3). The resulting inertial angular velocity is $Re\cdot O[\ln ({\it\xi}_{0}-1)]^{-1}$ . Over the $O({\it\xi}_{0}-1)^{-1/2}$ Jeffery period, this leads to an angular displacement and a ${\rm\Delta}C_{f}$ of $O[({\it\xi}_{0}-1)^{-1/2}/\ln ({\it\xi}_{0}-1)]$ , as in (6.15). Recall from § 5.1 that ${\rm\Delta}C_{p}\sim ({\it\xi}_{0}-1)^{3/2}\ln ({\it\xi}_{0}-1)$ , so ${\rm\Delta}C_{f}\gg {\rm\Delta}C_{p}$ , and fluid inertia is dominant for ${\it\xi}_{0}\rightarrow 1$ . In other words, one requires an asymptotically large density ratio of $St/Re\sim O[1/(({\it\xi}_{0}-1)\ln ({\it\xi}_{0}-1))]^{2}$ for particle inertia to influence the inertial drift of a slender fibre.

Figure 5. The drift due to fluid inertia, as characterized by the normalized change in the orbit constant in a single Jeffery period, $Re^{-1}({\rm\Delta}C_{f}/(C^{2}+1))$ , plotted as a function of $C/(C+1)$ , for a prolate spheroid; $C/(C+1)=0$ and $C/(C+1)=1$ correspond to the log-rolling and tumbling modes, respectively. (a) Uses the additional normalization factor of ${\it\xi}_{0}^{2}$ , so the drift remains finite in the near-sphere limit ( ${\it\xi}_{0}\rightarrow \infty$ ). (b) Uses the normalization factor $({\it\xi}_{0}-1)^{1/2}\ln ({\it\xi}_{0}-1)$ , to render the drift finite in the slender-fibre limit ( ${\it\xi}_{0}\rightarrow 1$ ).

Figure 6. The drift due to fluid inertia, as characterized by the normalized change in the orbit constant in a single Jeffery period, $Re^{-1}({\rm\Delta}C_{f}/(C^{2}+1))$ , plotted as a function of $C/(C+1)$ , for an oblate spheroid; $C/(C+1)=0$ and $C/(C+1)=1$ correspond to the spinning and the tumbling modes, respectively. (a) Uses the additional normalization factor of ${\it\xi}_{0}^{2}$ , so the drift remains finite in the near-sphere limit ( ${\it\xi}_{0}\rightarrow \infty$ ). (b) Uses the normalization factor $({\it\xi}_{0}-1)^{1/2}$ , to render the drift finite in the flat-disk limit ( ${\it\xi}_{0}\rightarrow 1$ ). (c) Magnified view highlighting the shift in the repeller location with changing ${\it\xi}_{0}$ .

6.2 The $O(Re)$ orbital drift for oblate spheroids

The functions, $F_{i}^{f}({\it\xi}_{0})$ and $G_{i}^{f}({\it\xi}_{0})$ , for an oblate spheroid are obtained from those for a prolate spheroid in the usual manner. The flat-disk limits of the resulting functions are given by

(6.16a−c ) $$\begin{eqnarray}\displaystyle & \displaystyle F_{1}^{f}({\it\xi}_{0})\approx -{\textstyle \frac{29}{480}},\quad F_{2}^{f}({\it\xi}_{0})\approx -{\textstyle \frac{1}{20}},\quad F_{3}^{f}({\it\xi}_{0})\approx -{\textstyle \frac{11}{480}}, & \displaystyle\end{eqnarray}$$

(6.17a−c ) $$\begin{eqnarray}\displaystyle G_{1}^{f}({\it\xi}_{0})\approx {\textstyle \frac{3}{40}},\quad G_{2}^{f}({\it\xi}_{0})\approx {\textstyle \frac{1}{40}},\quad G_{3}^{f}({\it\xi}_{0})\approx {\textstyle \frac{11}{120}}. & & \displaystyle\end{eqnarray}$$

To calculate ${\rm\Delta}C_{f}$ , aside from the aspect-ratio-dependent functions given above, one also needs to replace the integrals $I_{i}$ and $J_{i}$ by their oblate analogues. The ${\rm\Delta}C_{f}$ for an oblate spheroid is plotted against $C$ in figure 6. In contrast to the particle inertia case (see figure 3), the near-sphere normalization alone is not sufficient for an oblate spheroid. The inertial angular velocity approaches a finite $O(Re)$ value independent of aspect ratio in the flat-disk limit, and this leads to ${\rm\Delta}C_{f}\sim Re\cdot O({\it\xi}_{0}-1)^{-1/2}$ for ${\it\xi}_{0}\rightarrow 1$ owing to the diverging Jeffery period. The differing scalings for particle and fluid inertia imply that the density ratio $St/Re$ must become asymptotically large, of $O({\it\xi}_{0}-1)^{-1/2}$ (although still far smaller than the corresponding prolate estimate), before particle inertia can begin to exert an influence on the orientation distribution of flat disks. Figure 6 includes separate plots of ${\rm\Delta}C_{f}$ with the near-sphere and the flat-disk normalizations.

From the plots in figure 6, we note that, unlike a prolate spheroid, ${\rm\Delta}C_{f}$ is no longer single-signed. Now, ${\rm\Delta}C_{f}$ must certainly be negative, for all values of $C$ , for a near-sphere, as is implied by the prolate–oblate transformation and a ${\rm\Delta}C_{f}$ of $O({\it\xi}_{0}^{-2})$ . This remains true for aspect ratios greater than approximately $0.142\,({\it\xi}_{0}\approx 1.01)$ , and such oblate spheroids drift towards a steady spinning mode starting from any initial orientation. For aspect ratios smaller than this critical value, the ${\rm\Delta}C_{f}$ curves cross the $C$ -axis, so the drift becomes positive beyond a critical value of $C$ (say,  $C^{\ast }$ ). $C^{\ast }$ is a function of ${\it\xi}_{0}$ , and equals $\infty$ for an oblate spheroid with the critical aspect ratio (0.142), decreasing to $\sqrt{35}$ in the limit ${\it\xi}_{0}\rightarrow 1$ . For a flat disk, ${\rm\Delta}C_{f}$ is given by:

(6.18) $$\begin{eqnarray}\displaystyle \lim _{{\it\xi}_{0}\rightarrow 1}{\rm\Delta}C_{f}=\frac{{\rm\pi}(7+C^{2}-7(1+C^{2})^{1/2})}{15\sqrt{2}C({\it\xi}_{0}-1)^{1/2}}=\frac{{\rm\pi}(\sqrt{1+C^{2}}-1)(\sqrt{1+C^{2}}-\sqrt{1+{C^{\ast }}^{2}})}{15\sqrt{2}C({\it\xi}_{0}-1)^{1/2}}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The movement of $C^{\ast }$ towards its limiting value in the range $1<{\it\xi}_{0}<1.01$ is highlighted by the magnified view in figure 6. As the aspect ratio decreases below 0.142, a pair of repellers (which are Jeffery orbits at this order of approximation) bifurcate from the tumbling orbit, separating the unit sphere into distinct basins of attraction. The region on the unit sphere between the repelling orbits, extending across the plane of symmetry of the ambient shear, corresponds now to orientations that asymptote towards a tumbling mode, while the region around the vorticity axis enclosed by each of the repellers corresponds to orientations that asymptote towards a steady spinning mode. With decreasing aspect ratio, the two repelling orbits flatten out into increasingly thin ellipses centred about the gradient–vorticity plane. Eventually, in the flat-disk limit, the repellers reduce to a pair of arcs (with an angular extent of approximately 161°) on the great circle in which the gradient–vorticity plane intersects the unit sphere. As a result, a flat disk, like a prolate spheroid, approaches a tumbling mode from almost any initial orientation (except for a set of measure zero corresponding to the repellers). This happens despite the movement of $C^{\ast }$ towards a finite limiting value (from $\infty$ to $\sqrt{35}$ ) for ${\it\xi}_{0}\rightarrow 1$ . This is because, as noted earlier in § 4, for sufficiently thin oblate spheroids, the orbit constant is appropriately defined as $C=({\it\xi}_{0}^{2}-1)^{-1/2}\tan {\it\theta}_{j}\cos {\it\phi}_{j}$ , so that for a given Jeffery orbit to remain bounded away from the gradient–vorticity plane as ${\it\xi}_{0}\rightarrow 1$ , $C$ must diverge as $({\it\xi}_{0}-1)^{-1/2}$ ; in other words, Jeffery orbits corresponding to any finite $C$ in the flat-disk limit (including the $C^{\ast }=\sqrt{35}$ , the zero crossing of the ${\rm\Delta}C_{f}$ curve for a flat disk) must collapse onto the gradient–vorticity plane (the angular extent of the resulting arc is $C/\sqrt{1+C^{2}}$ , and with $C=C^{\ast }$ , this gives 161° as mentioned above). To illustrate the approach of the ${\rm\Delta}C_{f}$ zero crossing towards $C=0$ for a flat disk, the fluid inertial drift is plotted against the rescaled orbital coordinate $C({\it\xi}_{0}^{2}-1)^{1/2}/(1+C({\it\xi}_{0}^{2}-1)^{1/2})$ in figure 7. Figure 8 illustrates the repelling (Jeffery) orbits starting from the equator of the unit sphere ( $C=\infty$ ), and moving towards smaller $C$ values with decreasing aspect ratio. Figure 9 shows the nature of the finite- $Re$ spiralling trajectories of the orientation vector on either side of these repellers. The emergence of the repellers may also be interpreted as a subcritical pitchfork bifurcation (figure 10), with $1/C$ being the relevant coordinate, and the spheroid aspect ratio being the relevant parameter. For aspect ratios less than 0.142, the tumbling orbit is a local attractor bounded by a pair of unstable solutions (repeller). The coalescence of the three (two unstable and one stable) solutions, at the critical aspect ratio, renders the tumbling orbit unstable, and the system migrates to a distant equilibrium (the spinning mode). In the vicinity of the critical aspect ratio, the repeller locations are given by $1/C^{\ast }=\pm 1.48({\it\xi}_{0}-0.142)^{1/2}$ .

Figure 7. (a) Magnified view of the drift for the oblate spheroid plotted against a rescaled orbital coordinate shown in (b). The cross-over point shifts towards zero with decreasing aspect ratio.

Figure 8. The repeller orbit location for various aspect ratios. As the aspect ratios decreases below 0.142 ( ${\it\xi}_{0}=1.01$ ) a repeller orbit (denoted by thick black lines) emerges from $C/(C+1)=1$ (the tumbling orbit). With further decrease in the aspect ratio, the repeller progressively shrinks, collapsing into the vicinity of the gradient–vorticity plane in the flat-disk limit.

Figure 9. The nature of the spiralling trajectories on either side of the repeller is shown for a pair of aspect ratios. For the purpose of illustration, a large Stokes number ( $St=0.8$ ) is chosen.

Figure 10. The oblate spheroid orientation dynamics from a dynamical systems perspective: a subcritical pitchfork bifurcation at an aspect ratio of $0.142\,({\it\xi}_{0}=1.01)$ .

6.3 Drift due to fluid and particle inertia: role of the particle-to-fluid density ratio ( $St/Re={\it\rho}_{p}/{\it\rho}_{f}$ )

For a neutrally buoyant spheroid, the inertial drift is given by (5.19), but with the aspect-ratio-dependent functions now being the sums of the $F_{i}^{(f)}({\it\xi}_{0})\,[G_{i}^{(f)}({\it\xi}_{0})]$ and the $F_{i}^{(p)}({\it\xi}_{0})\,[G_{i}^{(p)}({\it\xi}_{0})]$ . The drifts for neutrally buoyant prolate and oblate spheroids are plotted against the normalized orbit constant in figures 11 and 12, respectively. These curves closely resemble those in figures 5 and 6, showing that fluid inertia dominates the inertia of the particle for most aspect ratios. Accordingly, the critical aspect ratio at which $({\rm\Delta}C_{f}+{\rm\Delta}C_{p})$ first changes sign (for $C\rightarrow \infty$ ) is only slightly altered from the original value, 0.142, for hollow oblate spheroids (fluid inertia acting alone), to 0.137 for a neutrally buoyant oblate spheroid.

The effect of density ratio ( $St/Re$ ) is best illustrated for small aspect-ratio oblate spheroids, since the particle inertial (single signed) and fluid inertial (zero crossing for aspect ratios below a critical value) drift curves are distinct in character in this case. As $St/Re$ increases from 0, for a fixed aspect ratio below 0.142, one expects the location of the repeller to move towards the tumbling orbit, and ${\rm\Delta}C$ to eventually become single-signed. This is shown in figure 13 for an oblate spheroid with an aspect ratio of $0.032\,({\it\xi}_{0}=1.001)$ ; the $St$ and $Re$ here vary from 0 to 0.15 ( $St^{\ast }$ ) and 0.0033 ( $Re^{\ast }$ ), respectively, the maximum values being chosen so as to make the particle and the fluid inertial drifts of the same order. The actual drift is proportional to $(1-{\it\alpha})Re^{\ast }{\rm\Delta}C_{f}+{\it\alpha}St^{\ast }{\rm\Delta}C_{p}$ , with the value of ${\it\alpha}$ fixing the ratio $St/Re$ . It is worth noting that, even for the chosen small aspect ratio, the drift becomes single-signed only at $St/Re\approx 75$ . Since both particle and fluid inertia induce a drift in the same direction for a prolate spheroid, one expects no qualitative changes in the drift curve with varying $St/Re$ . Hence, in figure 14, we have plotted separately the normalized fluid and particle inertial drifts for a prolate spheroid of aspect ratio 4 to again reinforce the dominance of fluid inertia. The peak drift values are 0.238 and 0.0058 for the particle and fluid inertia cases (attained at $C$ values of 1.20 and 1.10 respectively), implying that particle inertia begins to control the inertial drift of the prolate spheroid of aspect ratio 4 only when the density ratio exceeds approximately 41. Thus, for a prolate spheroid of any reasonable aspect ratio and a density ratio of order unity, the drift is controlled by fluid inertial forces, and it is only for gas–solid systems that particle inertia becomes significant.

The above fact needs particular emphasis in light of the recent comments of Aidun and co-workers (Rosen et al. Reference Rosen, Lundell and Aidun2014), which might in turn might be due to the incorrect fluid inertial analysis of Subramanian & Koch (Reference Subramanian and Koch2006b ) that predicted a nearly spherical prolate particle to drift towards the vorticity axis (see appendix D). For instance, it has been emphasized on more than one occasion by Rosen et al. (Reference Rosen, Lundell and Aidun2014) that the initial tumbling phase (for small $Re$ ) arises due to centrifuging effects associated with particle inertia, and fluid inertial effects act in opposition to this tendency, leading to subsequent bifurcations at higher $Re$ to log-rolling and kayaking modes. The authors also attempt to rationalize the results of some early experiments by Mason and co-workers (Karnis, Goldsmith & Mason Reference Karnis, Goldsmith and Mason1963), who found cylindrical particles to apparently conform to the minimum dissipation hypothesis for $Re<10^{-6}$ . The present analysis, reinforced by figure 14, shows that this is clearly not true, and it is almost entirely due to fluid inertia alone that a prolate spheroid of any reasonable aspect ratio, suspended in a liquid, approaches the tumbling mode for small  $Re$ .

Figure 11. The drift due to fluid and particle inertia for a prolate (neutrally buoyant) spheroid, as characterized by the normalized change in the orbit constant in a single Jeffery period, $Re^{-1}(({\rm\Delta}C_{f}+{\rm\Delta}C_{p})/(C^{2}+1))$ , plotted as a function of $C/(C+1)$ ; $C/(C+1)=0$ and $C/(C+1)=1$ correspond to the log-rolling and tumbling modes, respectively. (a) Uses the additional normalization factor of ${\it\xi}_{0}^{2}$ , so that the drift remains finite in the near-sphere limit ( ${\it\xi}_{0}\rightarrow \infty$ ). (b) Uses the normalization factor $({\it\xi}_{0}-1)^{1/2}\log ({\it\xi}_{0}-1)$ such that the drift remains finite in the slender rod limit ( ${\it\xi}_{0}\rightarrow 1$ ). The contribution due to fluid inertia dominates particle inertia for all aspect ratios.

Figure 12. The drift due to fluid and particle inertia for an oblate (neutrally buoyant) spheroid, as characterized by the normalized change in the orbit constant in a single Jeffery period, $Re^{-1}(({\rm\Delta}C_{p}+{\rm\Delta}C_{f})/(C^{2}+1))$ , plotted as a function of $C/(C+1)$ ; $C/(C+1)=0$ and $C/(C+1)=1$ correspond to the spinning and tumbling modes, respectively. The plot uses the additional normalization factor of ${\it\xi}_{0}^{2}$ , so the drift remains finite in the near-sphere limit ( ${\it\xi}_{0}\rightarrow \infty$ ).

Figure 13. (it a) Magnified view close to the tumbling orbit of the combined drift due to particle and fluid inertia plotted for various values of the ratio $St/Re$ with $0\leqslant St\leqslant St^{\ast }=0.15$ and $0\leqslant Re\leqslant Re^{\ast }=0.0033$ for ${\it\xi}_{0}=1.001$ shown in (b).

Figure 14. The drifts due to fluid and particle inertia are plotted for a prolate spheroid of aspect ratio 4.

7.1 The shear viscosity of a suspension of non-Brownian spheroids

While the distribution across Jeffery orbits is indeterminate in the Stokes limit, accounting for the drift due to either particle or fluid inertia, and in the absence of stochastic de-correlation mechanisms (of either a thermal or hydrodynamic origin), must lead to a singular steady-state orientation distribution. For sufficiently long times, the spheroids are confined to the tumbling and/or spinning modes, and ${\it\Omega}(\boldsymbol{p})$ must therefore involve delta functions at either or both of ${\it\theta}_{j}=0$ and ${\it\theta}_{j}={\rm\pi}/2$ . Evidently, only one of these delta functions is present when the drift is single-signed over the entire unit sphere of orientations. As already seen, fluid inertia alone led to a change in sign of the drift for oblate spheroids with aspect ratios smaller than 0.142. For neutrally buoyant spheroids with $Re=St$ , as seen in § 6.3, the drift arises from the combined effects of particle and fluid inertia, and the single-signed drift regime for oblate spheroids is then found to extend down to a slightly smaller aspect ratio of 0.137 owing to the $O(St)$ component of the drift stabilizing the spinning mode. Note that the drift is virtually unaltered close to the flat-disk limit, where the fluid inertial contribution is greater than that due to particle inertia by a factor of $O({\it\xi}_{0}-1)^{-1/2}$ ; in particular, the change in sign of the drift continues to occur at $C^{\ast }\approx \sqrt{35}$ for ${\it\xi}_{0}\rightarrow 1$ .

As a result, for prolate spheroids of any aspect ratio:

(7.12) $$\begin{eqnarray}\displaystyle {\it\Omega}(\boldsymbol{p})=\frac{\bar{{\it\xi}}_{0}{\it\xi}_{0}{\it\delta}\left({\it\theta}_{j}-\displaystyle \frac{{\rm\pi}}{2}\right)}{2{\rm\pi}\sin {\it\theta}_{j}({\it\xi}_{0}^{2}\sin ^{2}{\it\phi}_{j}+\bar{{\it\xi}}_{0}^{2}\cos ^{2}{\it\phi}_{j})}, & & \displaystyle\end{eqnarray}$$

and for neutrally buoyant oblate spheroids with aspect ratios greater than 0.137:

(7.13) $$\begin{eqnarray}\displaystyle {\it\Omega}(\boldsymbol{p})=\frac{{\it\delta}({\it\theta}_{j})}{2{\rm\pi}\sin {\it\theta}_{j}}. & & \displaystyle\end{eqnarray}$$

For neutrally buoyant oblate spheroids with aspect ratios less than 0.137:

(7.14) $$\begin{eqnarray}\displaystyle {\it\Omega}(\boldsymbol{p})=\frac{{\mathcal{A}}_{1}(C^{\ast }){\it\delta}({\it\theta}_{j})}{2{\rm\pi}\sin {\it\theta}_{j}}+\frac{\bar{{\it\xi}}_{0}{\it\xi}_{0}\left[{\mathcal{A}}_{2}(C^{\ast }){\it\delta}\left({\it\theta}_{j}-\displaystyle \frac{{\rm\pi}}{2}\right)\right]}{2{\rm\pi}\sin {\it\theta}_{j}({\it\xi}_{0}^{2}\cos ^{2}{\it\phi}_{j}+\bar{{\it\xi}}_{0}^{2}\sin ^{2}{\it\phi}_{j})}, & & \displaystyle\end{eqnarray}$$

where the constants ${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$ , in addition to depending on $C^{\ast }$ (and thereby on the aspect ratio), are functions of the initial orientation distribution. This dependence arises because the relative proportions of oblate spheroids asymptoting to the tumbling and spinning modes depend on the number of particles located on either side of the repeller (the Jeffery orbit with $C=C^{\ast }({\it\xi}_{0})$ ) on the unit sphere at the initial instant. For an initial isotropic distribution, for instance, ${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$ are, respectively, proportional to the areas enclosed within and in between the pair of Jeffery-orbit repellers. Thus, for non-Brownian neutrally buoyant oblate spheroids in the range of aspect ratios $(0,0.137)$ , the inertial suspension rheology, at leading order, still depends on the initial state of the dilute suspension. While inclusion of weak inertial effects does make the functional form of the steady-state orientation distribution determinate, in that ${\it\Omega}(\boldsymbol{p})$ always consists of a pair of delta functions, the relative amplitudes of these delta functions continues to depend on the initial distribution of orientations. In what follows, for oblate spheroids in the above range of aspect ratios, we evaluate the shear viscosity in the dilute regime assuming an initially isotropic orientation distribution.

Figure 15. The intrinsic viscosity coefficient as a function of eccentricity for a dilute suspension of neutrally buoyant prolate spheroids: scaled with $nd^{3}{\it\xi}_{0}^{3}$ (a); scaled with $nd^{3}{\it\xi}_{0}^{2}({\it\xi}_{0}^{2}-1)^{1/2}/\text{log}\,{\it\kappa}$ (b).

For the prolate case, using (7.12) in (7.11) with $\boldsymbol{p}=\cos {\it\phi}_{j}\mathbf{1}_{x}^{\prime }+\sin {\it\phi}_{j}\mathbf{1}_{y}^{\prime }$ , one obtains:

(7.15) $$\begin{eqnarray}\displaystyle \langle {\it\sigma}_{ij}^{p}\rangle =n\left[D_{2}^{(p)}({\it\xi}_{0})+\left(\frac{3}{2}D_{1}^{(p)}({\it\xi}_{0})-2D_{2}^{(p)}({\it\xi}_{0})+\frac{D_{3}^{(p)}({\it\xi}_{0})}{2}\right)\frac{{\it\xi}_{0}\,\bar{{\it\xi}}_{0}}{({\it\xi}_{0}+\bar{{\it\xi}}_{0})^{2}}+\right]\unicode[STIX]{x1D60C}_{ij}. & & \displaystyle\end{eqnarray}$$

The rheology is evidently Newtonian. In terms of an effective viscosity ${\it\mu}_{eff}$ , we have from (7.15) that $({\it\mu}_{eff}-{\it\mu})/[(nL^{3}){\it\mu}]=(1/2)(D_{2}^{(p)}({\it\xi}_{0})+((3/2)D_{1}^{(p)}({\it\xi}_{0})-2D_{2}^{(p)}({\it\xi}_{0})+(D_{3}^{(p)}({\it\xi}_{0}))/2)(({\it\xi}_{0}\,\bar{{\it\xi}}_{0})/({\it\xi}_{0}+\bar{{\it\xi}}_{0})^{2}))$ for a suspension of prolate spheroids at small but finite $Re$ . Figure 15(a) shows this intrinsic viscosity coefficient as a function of the spheroid eccentricity. It is seen to vary from a value corresponding to the Einstein coefficient (the value near 10 comes from the factor $4{\rm\pi}/3$ involved in the spherical volume fraction) for a near-sphere to a vanishingly small contribution for a slender spheroid. This happens due to a change in the scaling of $({\it\mu}_{eff}-{\it\mu})/{\it\mu}$ from $O(nL^{3})$ in the near-sphere limit to $O(nL^{2}b)/\ln {\it\kappa}$ in the slender-fibre limit; from (7.15), $\lim _{{\it\xi}_{0}\rightarrow \infty }({\it\mu}_{eff}-{\it\mu})/{\it\mu}=(10{\rm\pi}/3)nL^{3}$ and $\lim _{{\it\xi}_{0}\rightarrow 1}({\it\mu}_{eff}-{\it\mu})/{\it\mu}=(2{\rm\pi}/(3\ln {\it\kappa}))nL^{2}b$ . For large aspect ratios ( ${\it\kappa}\rightarrow \infty$ ), the dominant contribution to the averaged stresslet is from non-aligned fibres, with the probability of such orientations only being $O({\it\kappa}^{-1})$ . From viscous slender body theory, the stresslet for a non-aligned fibre arises from a linear force density of $O[{\it\mu}\dot{{\it\gamma}}L/\ln {\it\kappa}]$ acting with a moment arm of $O(L)$ over the length of the spheroid, and is $O[{\it\mu}\dot{{\it\gamma}}L^{3}/\ln {\it\kappa}]$ . The resulting (dimensional) stress is $O(n)\cdot O({\it\kappa}^{-1})\cdot O[{\it\mu}\dot{{\it\gamma}}L^{3}/\ln {\it\kappa}]$ , leading to an effective viscosity $({\it\mu}_{eff}-{\it\mu})/{\it\mu}$ of $O[nL^{2}b/\ln {\it\kappa}]$ . The stresslet for flow-aligned fibres is smaller than the non-aligned contribution by $O({\it\kappa}^{-2}\ln {\it\kappa})$ , and only contributes to a small correction of $O(nLb^{2})$ to the above estimate. Figure 15(b) uses the slender-fibre scaling above to obtain a finite viscosity coefficient in the slender-fibre limit; note that the near-sphere asymptote is zero in these units owing to the additional logarithmic factor involved. The scaling of the intrinsic viscosity with aspect ratio, in the slender-fibre limit, is controlled by the anisotropy of orientations within a given Jeffery orbit, and thus, Leal and Hinch’s (Leal & Hinch Reference Leal and Hinch1971) original calculation, involving the effect of weak Brownian diffusion in setting up a steady smooth distribution across Jeffery orbits, also leads to a viscosity coefficient of $O[nL^{2}b/\ln {\it\kappa}]$ for large aspect ratios, albeit with a different $O(1)$ coefficient.

For oblate spheroids with aspect ratios greater than 0.137, $\boldsymbol{p}=\mathbf{1}_{z}^{\prime }$ , and only the two transverse planar extensions in (3.2) contribute to the rheology. Using (7.13), (7.11) reduces to:

(7.16) $$\begin{eqnarray}\displaystyle \langle {\it\sigma}_{ij}^{p}\rangle =nD_{3}^{(o)}({\it\xi}_{0})\unicode[STIX]{x1D60C}_{ij}\quad 1.01<{\it\xi}_{0}<\infty . & & \displaystyle\end{eqnarray}$$

For oblate spheroids with aspect ratios smaller than 0.137, (7.14) is used in (7.11) to obtain:

(7.17) $$\begin{eqnarray}\displaystyle \langle {\it\sigma}_{ij}^{p}\rangle & = & \displaystyle n\left[D_{3}^{(o)}({\it\xi}_{0})\,{\mathcal{A}}_{1}+\left(D_{2}^{(o)}({\it\xi}_{0})+\left(\frac{3}{2}D_{1}^{(o)}({\it\xi}_{0})-2D_{2}^{(o)}({\it\xi}_{0})+\frac{D_{3}^{(o)}({\it\xi}_{0})}{2}\right)\frac{{\it\xi}_{0}\,\bar{{\it\xi}}_{0}}{({\it\xi}_{0}+\bar{{\it\xi}}_{0})^{2}}\right)\right.\nonumber\\ \displaystyle & & \displaystyle \times \left.(1-{\mathcal{A}}_{1})\vphantom{\left(\frac{{\it\xi}_{0}\,\bar{{\it\xi}}_{0}}{({\it\xi}_{0}+\bar{{\it\xi}}_{0})^{2}}\right)}\right]\unicode[STIX]{x1D60C}_{ij}\quad 1<{\it\xi}_{0}<1.01.\end{eqnarray}$$

The area within a Jeffery orbit, with an orbit constant $C$ , is given by $2{\rm\pi}-4{\it\kappa}(1+C^{2})^{-1/2}{\it\Pi}[1-{\it\kappa}^{2},-C^{2}({\it\kappa}^{2}-1)/(1+C^{2})]$ , and normalizing by the area of the unit hemisphere, one obtains ${\mathcal{A}}_{1}=1-(2{\it\kappa}/{\rm\pi})(1+C^{2})^{-1/2}{\it\Pi}[1-{\it\kappa}^{2},-C^{2}({\it\kappa}^{2}-1)/(1+C^{2})]$ and ${\mathcal{A}}_{2}=1-{\mathcal{A}}_{1}$ for an initially isotropic orientation distribution, ${\it\Pi}(x,y)$ being the complete elliptic function of the third kind (Gradshteyn & Ryzhik Reference Gradshteyn and Ryzhik2007). The intrinsic viscosity coefficient, $({\it\mu}_{eff}-{\it\mu})/[(nL^{3}){\it\mu}]$ for a suspension of oblate spheroids, over the entire range of eccentricities is plotted in figure 16(a). There is a kink (a discontinuity in slope) in the curve at $e\approx 0.99$ due to the oblate spheroids transitioning from a pure spinning mode to a weighted combination of spinning and tumbling modes. The part of the viscosity curve for $e>0.9905$ (aspect ratios smaller than 0.137) is plotted as a discrete sequence of points because the relative proportions of spinning and tumbling spheroids in this range of aspect ratios is a function of the repeller location $C^{\ast }$ , and this is found numerically from the zero crossing in a plot of ${\rm\Delta}C$ against $C/(C+1)$ for a particular aspect ratio (similar to figure 6). Figure 16(b) shows a magnified view of the aforementioned kink. Here, the curve corresponding to the pure spinning mode is continued until $e=1$ to emphasize the transition from spinning oblate spheroids, of aspect ratio 0.137, at $e=0.9905$ , to tumbling flat disks at $e=1$ . Note that the spinning-mode curve would terminate in a finite coefficient at $e=1$ , since the viscosity coefficient for spinning disks is $O(nL^{3})$ as for spheres. The bifurcation at $e=0.9905$ , however, implies that the viscosity coefficient at $e=1$ , arises almost entirely from tumbling flat disks, and is asymptotically smaller than $O(nL^{3})$ . The appropriate scale in the flat-disk limit may be obtained by noting that the averaged stresslet arises from the combination of an $O({\it\mu}\dot{{\it\gamma}}L^{3})$ stresslet associated with an $O({\it\kappa})$ fraction of spinning disks, and a comparable $O({\it\mu}\dot{{\it\gamma}}bL^{2})$ stresslet associated with disks that tumble in the flow–gradient plane (unlike the prolate case, both aligned and non-aligned flat disks end up contributing, at the same order, to the tumbling stress component). This leads to an effective viscosity ${\it\mu}_{eff}-{\it\mu}\sim O(nbL^{2}){\it\mu}$ in the flat-disk limit. Figure 17 plots $({\it\mu}_{eff}-{\it\mu})/[(nbL^{2}){\it\mu}]$ as a function of the spheroid eccentricity, which leads to a finite value in the flat-disk limit; the spinning-mode coefficient diverges as $O({\it\xi}_{0}-1)^{-1/2}$ with this normalization. The viscosity coefficient for sufficiently thin oblate spheroids is again controlled by the anisotropy of orientations within a given Jeffery orbit, and the effects of weak Brownian motion, although resulting in a different distribution across Jeffery orbits, lead to a similar scale in the flat-disk limit (Leal & Hinch Reference Leal and Hinch1971).

Figure 16. (a) Intrinsic viscosity coefficient as a function of eccentricity for a dilute suspension of neutrally buoyant oblate spheroids (scaled with $nd^{3}{\it\xi}_{0}^{3}$ ). An initial isotropic orientation distribution is assumed while calculating the viscosity for aspect ratios less than the critical aspect ratio (0.137). The dotted line denotes the viscosity coefficient for these viscosity ratios. (b) Magnified view of the viscosity coefficient transitioning from the spinning to the tumbling asymptote close to the flat-disk limit.

Figure 17. (a) Intrinsic viscosity coefficient as a function of eccentricity for a dilute suspension of neutrally buoyant oblate spheroids presented in figure 16, is now appropriately rescaled in the flat-disk limit (scaled with $nd^{3}{\it\xi}_{0}^{2}({\it\xi}_{0}^{2}-1)^{1/2}$ ). (b) Magnified view of (a).

Apart from the kink at $e=0.9905$ discussed above, there are two points worth noting in figure 17. The first is that the intrinsic viscosity curve corresponds to a steady-state orientation distribution. Within the framework of an orbital drift, the time required to attain such a steady state diverges in the flat-disk limit owing to the diverging Jeffery period; recall that the Jeffery period is $2{\rm\pi}(({\it\kappa}^{2}+1)/{\it\kappa})$ and is $O({\it\kappa}^{-1})$ for ${\it\kappa}\rightarrow 0$ . Note, however, that the orbital drift interpretation, that assumes the inertial trajectory to be a tightly wound spiral, becomes increasingly restrictive for both large and small aspect ratios (see § 8). Notwithstanding this restriction, the viscosity coefficient, plotted for any finite time, will deviate from the steady-state plot for sufficiently thin oblate spheroids, asymptoting to a (frozen-in-time) isotropic orientation distribution at $e=1$ . For longer times, this deviation from the steady-state curve will occur at progressively smaller aspect ratios. The second feature is the difference between the value of the intrinsic viscosity coefficient at $e=1$ , and that corresponding to a suspension consisting only of tumbling flat disks (indicated by the horizontal dashed line in figure 17(b)). This jump comes from the implicit assumption of an infinite suspension for which a statistical description, in terms of an orientation probability density, is appropriate. For any finite-sized system, there will be a small enough aspect ratio when the area within the Jeffery repeller, corresponding to $C^{\ast }=\sqrt{35}$ , is small enough that the number of spheroid orientations in this tiny region of the unit sphere is of order unity, and a probabilistic description is no longer valid. Below such an aspect ratio, the viscosity coefficient will approach the lower value corresponding to the pure tumbling mode (again of $O(nbL^{2})$ ). Said differently, the jump in the viscosity coefficient at $e=1$ is an artefact of the thermodynamic limit.

7.2 The rheology of a suspension of oblate spheroids with aspect ratio less than 0.137: the role of weak Brownian motion

In this section, in an effort to resolve the indeterminacy in the steady-state rheology that arose in § 7.1 for oblate spheroids with aspect ratios less than 0.137, we include the effects of weak Brownian motion (large but finite rotary Péclet number $Pe_{r}$ ). As will be seen below, Brownian motion together with the inertial drift does lead to a steady state independent of initial conditions. To examine this steady state, we write down the governing drift–diffusion equation for the orientation probability density:

(7.18) $$\begin{eqnarray}\boldsymbol{{\rm\nabla}}_{\boldsymbol{p}}\boldsymbol{\cdot }(\dot{\boldsymbol{p}}{\it\Omega})=\frac{1}{Pe_{r}}{\rm\nabla}_{\boldsymbol{p}}^{2}{\it\Omega}.\end{eqnarray}$$

The left-hand side denotes the convection of ${\it\Omega}$ with $\dot{\boldsymbol{p}}=\dot{\boldsymbol{p}}_{jeff}+Re\,\dot{\boldsymbol{p}}_{i}$ , where $\dot{\boldsymbol{p}}_{jeff}$ is the Stokesian rate of change of orientation along a Jeffery orbit, and $\dot{\boldsymbol{p}}_{i}$ denotes the inertial drift with components both along and perpendicular to the Jeffery orbits. The orientational Laplacian on the right-hand side denotes the Brownian motion of $\boldsymbol{p}$ on the unit sphere. The $(C,{\it\tau})$ coordinate system is most convenient for solving (7.18), since in this coordinate system, $\dot{\boldsymbol{p}}_{jeff}$ has only a ${\it\tau}$ component, and determines only the orientation distribution along any Jeffery orbit ( $g(C,{\it\tau})$ as given by (7.10)), while the distribution across orbits ( $f(C)$ in (7.11)) emerges from a one-dimensional drift–diffusion balance, in the $C$ coordinate, between inertial convection and Brownian diffusion. The nature of this reduced distribution, $f(C)$ , is controlled by the parameter $Re\,Pe_{r}$ , which determines the relative magnitudes of inertial drift and thermal diffusion. While § 7.1 dealt with the case $Re\,Pe_{r}=\infty$ , the focus here is on the (singular) nearly athermal limit $Re\,Pe_{r}\gg 1$ . The additional contribution to the suspension stress (the so-called Brownian or diffusion stress) is asymptotically small in this limit, and the suspension stress is still given by $n\langle \unicode[STIX]{x1D61A}_{ij}\rangle$ , as in § 7.1.

To obtain the governing equation for $f(C)$ , we again follow Leal & Hinch (Reference Leal and Hinch1971), and integrate (7.18) over the area enclosed by a single Jeffery orbit ( $2{\rm\pi}{\mathcal{A}}_{1}$ , as defined in § 7.1):

(7.19) $$\begin{eqnarray}\int _{{\mathcal{A}}_{1}}Re\boldsymbol{{\rm\nabla}}_{\boldsymbol{p}}\boldsymbol{\cdot }(\dot{\boldsymbol{p}}{\it\Omega})\,\text{d}S=\frac{1}{Pe_{r}}\int _{{\mathcal{A}}_{1}}{\rm\nabla}_{\boldsymbol{p}}^{2}{\it\Omega}\,\text{d}S.\end{eqnarray}$$

Application of the divergence theorem leads to:

(7.20) $$\begin{eqnarray}\int _{C_{1}}Re(\dot{\boldsymbol{p}}_{i}\boldsymbol{\cdot }\boldsymbol{n}_{j}){\it\Omega}\,\text{d}l=\frac{1}{Pe_{r}}\int _{C_{1}}\boldsymbol{n}_{j}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{\boldsymbol{p}}{\it\Omega}\,\text{d}l,\end{eqnarray}$$

where the term involving the Jeffery convection integrates to zero. Here, $C_{1}$ is the Jeffery orbit with orbit constant $C$ , $\text{d}l=h_{{\it\tau}}\,\text{d}{\it\tau}$ being the arclength element along this orbit, and $\boldsymbol{n}_{j}$ is the unit vector, in the tangent plane of the unit sphere, and normal to this Jeffery orbit. We now write the inertial drift in $(C,{\it\tau})$ coordinates as $\dot{\boldsymbol{p}}_{i}=u_{ic}\hat{\boldsymbol{C}}+u_{i{\it\tau}}\hat{{\bf\tau}}$ , with $\hat{\boldsymbol{C}}=\mathbf{1}_{{\it\theta}}$ and $\hat{{\bf\tau}}$ tangent to the Jeffery orbit. Using $\hat{{\bf\tau}}\boldsymbol{\cdot }\boldsymbol{n}_{j}=0$ and $\hat{\boldsymbol{C}}\boldsymbol{\cdot }\boldsymbol{n}_{j}=\sin {\it\alpha}$ , and the definition of the gradient operator (see appendix B 6), (7.20) becomes

(7.21) $$\begin{eqnarray}Re\,Pe_{r}\int _{0}^{2{\rm\pi}}u_{ic}\sin {\it\alpha}\,{\it\Omega}\,h_{{\it\tau}}\,\text{d}{\it\tau}=\int _{0}^{2{\rm\pi}}\left(\frac{h_{{\it\tau}}}{h_{C}\sin {\it\alpha}}\frac{\partial {\it\Omega}}{\partial C}-\cot {\it\alpha}\frac{\partial {\it\Omega}}{\partial {\it\tau}}\right)\text{d}{\it\tau}.\end{eqnarray}$$

Using ${\it\Omega}(C,{\it\tau})=f(C)g(C,{\it\tau})$ , with $g(C,{\it\tau})$ given by (7.10), (7.21) becomes:

(7.22) $$\begin{eqnarray}\displaystyle & & \displaystyle \left[Re\,Pe_{r}\int _{0}^{2{\rm\pi}}\frac{u_{ic}}{h_{C}}\,\text{d}{\it\tau}-\int _{0}^{2{\rm\pi}}\left(\frac{h_{{\it\tau}}}{h_{C}\sin {\it\alpha}}\frac{\partial g}{\partial C}-\cot {\it\alpha}\frac{\partial g}{\partial {\it\tau}}\right)\text{d}{\it\tau}\right]f(C)\nonumber\\ \displaystyle & & \displaystyle \quad =\left(\int _{0}^{2{\rm\pi}}\frac{1}{h_{C}^{2}\sin ^{2}{\it\alpha}}\,\text{d}{\it\tau}\right)\frac{\text{d}f}{\text{d}C}.\end{eqnarray}$$

Expectedly, (7.22) is a ${\it\tau}$ -averaged drift–diffusion equation where the drift includes both the inertial contribution and a Brownian contribution that arises due to the curvature and non-orthogonality of the $(C,{\it\tau})$ coordinates. Now, $u_{ic}/h_{C}=\text{d}C/\text{d}t$ , and the inertial drift term in (7.22) may be written as $\int _{0}^{2{\rm\pi}}(\text{d}C/\text{d}t)\,\text{d}{\it\tau}$ , which, from (5.17), equals $({\rm\Delta}C_{p}+{\rm\Delta}C_{f}){\it\kappa}/({\it\kappa}^{2}+1)$ for neutrally buoyant spheroids. This is just the dimensionless Jeffery-orbit-averaged drift velocity, of a neutrally buoyant oblate spheroid, with the distance (across Jeffery orbits) measured in units of $C$ and time measured in units of the inverse shear rate. The expressions for ${\rm\Delta}C_{p}$ and ${\rm\Delta}C_{f}$ , as functions of the aspect ratio (or ${\it\xi}_{0}$ ), have already been given in §§ 5.1 and 6.2, respectively. Defining ${\rm\Delta}C_{o}(C;{\it\kappa})=({\rm\Delta}C_{p}+{\rm\Delta}C_{f}){\it\kappa}/({\it\kappa}^{2}+1)$ , (7.22) takes the form:

(7.23) $$\begin{eqnarray}{\it\chi}_{1}\frac{\text{d}f}{\text{d}C}=\left(\frac{Re\,Pe_{r}{\rm\Delta}C_{o}}{{\rm\pi}}-\frac{{\it\chi}_{2}}{C}\right)f,\end{eqnarray}$$

where ${\it\chi}_{1}(C;{\it\kappa})=(({\it\kappa}^{2}+1/{\it\kappa}^{2})+C^{2}(7/2+1/(4{\it\kappa}^{2})+{\it\kappa}^{2}/4)+C^{4}({\it\kappa}^{2}+1))$ and ${\it\chi}_{2}=(-({\it\kappa}^{2}+1)/{\it\kappa}^{2}+C^{2}(6-(7/2+1/(4{\it\kappa}^{2})+{\it\kappa}^{2}/4))+2C^{4}({\it\kappa}^{2}+1))$ are obtained by evaluating the ${\it\tau}$ -integrals in (7.22). Equation (7.23) can readily be integrated formally to obtain:

(7.24) $$\begin{eqnarray}f(C)=N\exp \left[\int ^{C}\left(\frac{Re\,Pe_{r}{\rm\Delta}C_{o}(C^{\prime };{\it\kappa})}{{\rm\pi}}-\frac{{\it\chi}_{2}(C^{\prime };{\it\kappa})}{C^{\prime }}\right)\frac{1}{{\it\chi}_{1}(C^{\prime };{\it\kappa})}\,\text{d}C^{\prime }\right],\end{eqnarray}$$

where $N$ is a normalization constant determined from the constraint $\int _{0}^{\infty }f(C)\,\text{d}C=1/4{\rm\pi}$ . Since (7.24) is the solution of a drift–diffusion balance in one dimension, it is an equilibrium distribution with the flux being zero at every point along the $C$ -axis, the no-flux conditions at the boundary conditions (one of them at infinity) being automatically satisfied. In this sense, the solution is analogous to the much simpler Gaussian equilibrium for a Brownian particle in a bounded harmonic potential (Wax Reference Wax2013). Interpreting (7.24) within the framework of the harmonic potential problem, the equivalent $C$ -dependent potential for the present case is given by $U(C)=Re\,Pe_{r}\int ^{C}({\rm\Delta}C_{o}(C^{\prime };{\it\kappa})/{\rm\pi}-{\it\chi}_{2}(C^{\prime };{\it\kappa})/(Re\,Pe_{r}C^{\prime }))(1/{\it\chi}_{1}(C^{\prime };{\it\kappa}))\,\text{d}C^{\prime }$ where, unlike the harmonic oscillator case, $U(C)$ also incorporates the $C$ -dependence of the diffusivity in (7.23); $(Re\,Pe_{r})^{-1}$ is the dimensionless temperature. Figure 18 shows a plot of $U(C)$ against $C/(C+1)$ , for different $Re\,Pe_{r}$ , for an oblate spheroid of aspect ratio 0.0135. In the limit $Re\,Pe_{r}=0$ , the potential has only one minimum, corresponding to Brownian motion alone, and this is located close to the tumbling mode. But, as the value of $Re\,Pe_{r}$ increases, the inertial drift towards the spinning mode (for $C<C^{\ast }$ with $C^{\ast }\approx 6.07$ for an aspect ratio of 0.0135) causes the potential to become a bistable one with a second shallow minimum developing in the vicinity of the spinning mode. Note that the minima of the bistable potential, for any finite $Re\,Pe_{r}$ , are not exactly at $C=0$ and $C=\infty$ due to the presence of an additional Brownian drift component; such a coincidence occurs only at $Re\,Pe_{r}=\infty$ . The bistable nature of $U(C)$ implies that $f(C)$ will have a pair of peaks corresponding to the two minima, and this is borne out in the plots that follow. It is worth noting that, although the $U(C)$ minimum and the $f(C)$ maximum (see below), in the vicinity of $C=0$ , appear quite shallow, they would be much sharper in terms of the actual angular coordinates since, as already mentioned, close to the flat-disk limit, all the finite- $C$ Jeffery orbits are squished to within an asymptotically small interval in the vicinity of the gradient–vorticity plane (figure 8).

Figure 18. The $C$ -dependent potential for an oblate spheroid of aspect ratio 0.0135 is plotted against $C/(C+1)$ for various values of $Re\,Pe_{r}$ .

For $Re\,Pe_{r}=0$ , (7.23) reduces to the equation solved by Leal & Hinch (Reference Leal and Hinch1971), who analysed the effect of Brownian motion alone in determining $f(C)$ . The deviation in $f(C)$ from the ‘Leal–Hinch distribution’ due to the emergence of an inertial drift, for moderate $Re\,Pe_{r}$ , is shown in figure 19, again for an oblate spheroid with aspect ratio 0.0135. Figure 20 shows the normalized $f(C)$ for much larger values of $Re\,Pe_{r}$ for four different aspect ratios in the range 0.011–0.0135 (the purely Brownian Leal–Hinch curve is also shown for comparison in each figure). One expects to obtain bimodal distributions over some range of large $Re\,Pe_{r}$ values owing to the bidirectional inertial drift for aspect ratios less than 0.137; the original delta functions used in § 7.1, for the rheology calculation, will now be smoothed by Brownian motion into finite peaks with a small spread that varies in some inverse proportion with $Re\,Pe_{r}$ . The precise scaling of the spinning and tumbling peak widths may be found from the asymptotic forms of the drift and diffusivity coefficients for small and large $C$ . For $C\rightarrow 0$ , ${\rm\Delta}C_{o}\approx {\rm\pi}C/30$ and (7.23) takes the form $(1/f)(\text{d}f/\text{d}C)=C((Re\,Pe_{r}{\it\kappa}^{2})/30)+1/C$ , implying a spinning peak width of $O(1/\sqrt{Re\,Pe_{r}{\it\kappa}^{2}})$ ; for $C\rightarrow \infty$ , ${\rm\Delta}C_{o}\approx {\rm\pi}C/6$ and (7.23) takes the form $(1/f)(\text{d}f/\text{d}C)=(1/C^{3})((Re\,Pe_{r})/6)-2/C$ , implying a tumbling peak width of $O(1/\sqrt{Re\,Pe_{r}})$ . The smoothing effect of Brownian motion is, of course, expected, and is of a relatively minor significance in the rheological context, since the regularization of the delta functions only leads to a vanishingly small correction to the steady-state shear viscosity for $Pe_{r}\rightarrow \infty$ . This has, for instance, been done in the context of particle inertia by Einarsson et al. (Reference Einarsson, Angilella and Mehlig2014) with $St\,Pe_{r}$ being the relevant parameter.

Figure 19. The $f(C)$ for an oblate spheroid of aspect ratio 0.0135 is plotted against $C/(C+1)$ for moderate values of $Re\,Pe_{r}$ . The Leal–Hinch distribution corresponds to $Re\,Pe_{r}=0$ .

The non-trivial effect of Brownian motion is in determining the ratio of the tumbling-to-spinning peak amplitude, which, in sharp contrast to § 7.1, is now independent of initial conditions. This independence is evident from (7.24), where the ratio of tumbling-to-spinning peak amplitudes is proportional to the exponential of the difference in the corresponding potentials ( $U_{min}(C/(C+1)\rightarrow 1)-U_{min}(C/(C+1)\rightarrow 0)$ ). Figure 20 shows that this ratio crosses unity, as the aspect ratio decreases from 0.137 and approaches zero, implying a transition from a dominance of spinning oblate spheroids to tumbling oblate spheroids. For really large values of $Re\,Pe_{r}$ , this transition occurs at approximately an aspect ratio of 0.0126 (notice the shift in dominant peak from the spinning to the tumbling one at $Re\,Pe_{r}=200\,000$ , as one changes the aspect ratio from 0.0127 to 0.0124); this may also be seen from the relative depths of the two minima of $U(C)$ for $Re\,Pe_{r}\rightarrow \infty$ . In the athermal limit of interest here, the difference between potential minima is enormously magnified by the factor $Re\,Pe_{r}$ , and a change in sign of $U(\infty )-U(0)$ implies, for long enough times, a transition from a suspension of spinning spheroids to one of tumbling spheroids. The interval of aspect ratios over which such a transition (spinning $\rightarrow$ tumbling) occurs (with decreasing aspect ratio) must scale inversely with $Re\,Pe_{r}$ , implying that the intrinsic viscosity coefficient, in the limit $Re\,Pe_{r}\rightarrow \infty$ , must undergo a jump at the critical aspect ratio of 0.0126 from a large $O(nL^{3})$ value to a much smaller $O(nL^{2}b)$ value. Figure 21 evaluates the viscosity coefficient with an $O(nL^{2}b)$ normalization using an orientation distribution given by (7.13) above this critical aspect ratio, and the oblate analogue of (7.12) below it. Note that the aforementioned critical aspect ratio differs from the critical value of 0.137 found in the absence of Brownian motion ( $Re\,Pe_{r}=\infty$ ), implying the singular role of Brownian motion; the viscosity coefficient now varies smoothly across the earlier critical value. This singular effect of thermal forces is contingent on taking the limit $t\rightarrow \infty$ before $Re\,Pe_{r}\rightarrow \infty$ , thereby allowing even weak Brownian motion to act over an asymptotically large period of time (this ordering of limits is in the same sense as that in the original Leal–Hinch analysis (Leal & Hinch Reference Leal and Hinch1971), except that $Pe_{r}$ is now replaced by $Re\,Pe_{r}$ ).

Figure 20. The $f(C)$ for different aspect ratios (a) 0.0124; (b) 0.0127; (c) 0.011 and (d) 0.0135 is plotted against the orbital coordinate for various values of $Re\,Pe_{r}$ (0–2000 000). As $ReP_{r}$ increases, for aspect ratios less than 0.0126, the probability leaks towards the tumbling mode, and for aspect ratios greater than 0.0126, the leak is towards the spinning mode.

The aforementioned critical aspect ratio can also be calculated from the asymptotic forms that the distribution, $f(C)$ , takes as $C\rightarrow \infty$ and $C\rightarrow 0$ . In the latter case, $f(C)$ takes the form $C\text{e}^{[{\it\alpha}_{4}C^{2}+Re\,Pe_{r}({\it\alpha}_{5}C^{2}+{\it\alpha}_{6})]}$ , and for $C\rightarrow \infty$ , $f(C)$ takes the form $(1/C^{2})\,\text{e}^{[{\it\alpha}_{1}/C^{2}+Re\,Pe_{r}({\it\alpha}_{2}/C^{2}+{\it\alpha}_{3})]}$ , where the ${\it\alpha}_{i}$ are functions of the spheroid aspect ratio. The ratio of the maxima of these two asymptotes must be equated to unity to obtain the critical aspect ratio corresponding to the spinning–tumbling transition. This gives ${\it\alpha}_{3}={\it\alpha}_{6}$ at the leading order with corrections of $O(1/(Re\,Pe_{r}))$ . The aspect-ratio functions, ${\it\alpha}_{3}$ and ${\it\alpha}_{6}$ , are plotted in figure 22 for aspect ratios ranging from 0 to 0.136, and the curves cross at the aspect ratio of 0.0126.

Figure 21. (a) Intrinsic viscosity for a suspension of Brownian oblate spheroids is plotted as a function of the aspect ratio (scaled with $nd^{3}{\it\xi}_{0}^{2}({\it\xi}_{0}^{2}-1)^{1/2}$ ). (b) Shows the jump from the spinning to the tumbling value at the critical aspect ratio of 0.0126.

Figure 22. (a) The functions ${\it\alpha}_{3}$ and ${\it\alpha}_{6}$ , in the large $C$ and the small $C$ asymptotic forms of distribution are plotted against the aspect ratio in (a). The cross-over of the two functions gives the critical aspect ratio. (b) Zoomed view of (a) which shows the crossing of the functions at 0.0126.

Starting from an arbitrary initial condition, the orientation probability density evolves in a non-trivial fashion for $Re\,Pe_{r}\gg 1$ . The inertial drift must, on a shorter time scale of $O(Re^{-1}\dot{{\it\gamma}})$ (the time required for a spheroid orientation to drift across the unit sphere), lead to the establishment of the tumbling and spinning peaks whose amplitudes depend on the initial orientation distribution as found in § 7.1. The establishment of these peaks must then be followed by a process of ‘equilibration’ that involves the slow transfer of probability (of spheroid orientations) from the tumbling to the spinning peak, or vice versa, depending on whether one is above or below the aforementioned critical aspect ratio, leading to the peak ratio eventually approaching an initial-condition-independent value. This second (slower) process may be interpreted as an activated (barrier-hopping) process that requires an activation energy of $O(Re\,Pe_{r})$ corresponding to the potential maximum in figure 18, and the classical analysis of Kramer in the context of reaction-rate theory implies that the time scale for equilibration between the peak amplitudes would scale as $\exp [Re\,Pe_{r}]$ (Wax Reference Wax2013). This is, of course, much longer than the $O(Re^{-1})$ scale associated with the inertial drift setting up the initial-condition-dependent peaks for large $Re\,Pe_{r}$ . An analogy may also be drawn between the tumbling–spinning transition analysed here and the coil-stretch transition in the polymer physics context, with the strain rate replacing the aspect ratio as the relevant parameter (De Gennes Reference De Gennes1974; Hinch Reference Hinch1974), and the tumbling and spinning modes corresponding, respectively, to the coiled and stretched configurations, respectively. In the thermodynamic limit, and for sufficiently long times, there must be a complete transition of all polymer molecules from the coiled to the stretched configuration when a dilute polymer solution is subjected to a supercritical extension rate (greater than a Deborah number of $1/2$ ; Hsieh & Larson Reference Hsieh and Larson2005). It is also, however, true that the importance of the so-called coil-stretch hysteresis is precisely due to the long-lived metastable states (either the coiled or the stretched configurations, depending on whether the extension rate is above or below the critical value). For instance, as pointed out by Hinch (Reference Hinch1974), a weak subcritical flow can indeed maintain a stretched polymer molecule, although the stretching would have originally required the imposition of a supercritical extension. Based on this analogy with the coil-stretch transition, one expects a similar hysteresis with regard to the tumbling and spinning modes in the present context. In other words, for time scales much shorter than the peak-equilibration time above, both tumbling and spinning modes are allowed configurations for oblate spheroids with aspect ratios less than 0.137. The simpler spheroidal geometry also allows us to identify the basins of attraction corresponding to the tumbling and spinning modes on the unit sphere; the corresponding basins of attraction for the coiled and stretched states of a polymer molecule in the absence of Brownian motion, although unphysical, would necessitate examination of a much more complicated configuration space.

It again needs emphasis that, for any finite $Re\,Pe_{r}$ , there will be a non-trivial distribution of spheroid orientations spread across the entire unit sphere. In the athermal limit, however, the preponderance of spheroid orientations would be in the vicinity of either the tumbling or spinning modes, whichever is stable in the sense of corresponding to a lower potential. Thus, the primary effect of Brownian motion, at leading order, is only in determining the relative stability of these modes; the smoothing of the peaks, and the resulting finite spread of orientations, are second-order effects that become vanishingly small for $Pe_{r}\rightarrow \infty$ . It is especially worth noting that, although the specific case of Brownian motion has been analysed above in some detail, an alternate (weak) stochastic mechanism, in the presence of a bidirectional inertial drift, should also lead to a unique steady-state orientation distribution in a similar manner, although the existence of, and the actual value of a critical aspect ratio, depends on the details of the mechanism. For instance, outside the infinitely dilute limit, pair-hydrodynamic interactions would have the same qualitative effect. For oblate spheroids, successive uncorrelated pair interactions would lead to a relaxation process in orientation space similar to Brownian motion, although the process is non-local owing to each pair interaction leading to, in general, a large jump in orientation (these jumps become small, of $O(\ln {\it\kappa})^{-1}$ , and the associated pair interactions are describable in terms of a hydrodynamic rotary diffusivity, only for slender prolate spheroids (Rahnama et al. Reference Rahnama, Koch and Shaqfeh1995). Thus, if hydrodynamically induced orientation de-correlations were responsible for setting up a steady-state orientation distribution, $f(C)$ would be governed by a Boltzmann-like integro-differential equation in the $C$ -coordinate of the form:

(7.25) $$\begin{eqnarray}\frac{\partial f}{\partial t}+\frac{\partial }{\partial C}\left(\frac{Re\,Pe_{r}{\rm\Delta}C_{o}}{{\rm\pi}}f\right)+\frac{f}{{\it\tau}[f(C)]}-\int \text{d}C_{1}^{\prime }{\mathcal{K}}(C,C^{\prime }\mid C_{1},C_{1}^{\prime })[f(C_{1})f(C_{1}^{\prime })]=0\end{eqnarray}$$

where both the relaxation time (which depends on $f(C)$ , and is $O((nL^{3}\dot{{\it\gamma}})^{-1})$ in the dilute limit of interest) and the kernel ${\mathcal{K}}(C,C^{\prime }\mid C_{1},C_{1}^{\prime })$ depends on the details of the pair-hydrodynamic interactions between spheroids (an unsolved problem). In summary, although we have analysed specifically the role of weak Brownian motion in eliminating the rheological indeterminacy associated with a dilute suspension of oblate spheroids, with aspect ratios less than 0.137, one expects the proposed scenario to remain qualitatively valid even outside the restricted regime considered. A similar situation would prevail even at finite $Re$ , and at higher volume fractions until the inertial trajectory topology begins to differ qualitatively from that implied by the $O(Re)$ analysis here (due to, for instance, the inertial drift no longer being bidirectional or from hydrodynamic fluctuations affecting the distribution of orientation within a Jeffery orbit). The role of Brownian motion above will now be replaced by hydrodynamic orientation fluctuations (at either zero or finite $Re$ ) with the single dimensionless parameter governing the orientation distribution (and the rheology) now being $Re/nL^{3}$ ( $1/nL^{3}$ playing the role of $Pe_{r}$ , since the Brownian diffusion time of $O(D_{r}^{-1})$ is now replaced by the time between successive pair interactions of $O[(nL^{3}\dot{{\it\gamma}})^{-1}]$ ).