3 Stable equilibrium configurations

3.1 Equilibrium configurations of a capsule with ${\it\lambda}=1$

3.1.1 Obvious equilibrium positions

Two obvious equilibrium configurations exist for an oblate capsule placed in a shear flow under Stokes flow conditions: when the capsule revolution axis is initially in the shear plane or perpendicular to it.

Figure 2.

Capsule shape evolution over one half-period at steady state when ${\it\xi}(0)=90^{\circ }$ ( ${\it\lambda}=1$ ). The grey scale corresponds to the normal component of the load $\boldsymbol{q}\boldsymbol{\cdot }\boldsymbol{n}$ on the membrane, with maximum values depending on $Ca$ . (a) $Ca=0.01$ : tumbling regime $(\max (\boldsymbol{q}\boldsymbol{\cdot }\boldsymbol{n}/G_{s})=0.15)$ ; (b) $Ca=0.3$ : swinging regime $(\max (\boldsymbol{q}\boldsymbol{\cdot }\boldsymbol{n}/G_{s})=2)$ ; (c)  $Ca=1.5$ : quasi-steady tank-treading regime $(\max (\boldsymbol{q}\boldsymbol{\cdot }\boldsymbol{n}/G_{s})=6)$ . The value of the non-dimensional time $\dot{{\it\gamma}}t$ is given below each shape. The points $A_{1}(t)$ (▪) and $A_{3}(t)$ (●) are initially on the long and short axes respectively in the shear plane. The point $M_{3}(t)$ (▵) represents the tip of the smallest capsule principal axis (see figure 1).

When the capsule revolution axis is initially in the shear plane ( ${\it\zeta}(0)={\it\xi}(0)=90^{\circ }$ , $A_{3}(0)$ and $M_{3}(0)$ in the shear plane), we have seen in the introduction that the capsule experiences tumbling at low $Ca$ followed by a transition towards swinging for higher values of $Ca$ . Figure 2 illustrates the characteristic dynamics of the capsule during half a period in both regimes (see Walter et al. (Reference Walter, Salsac and Barthès-Biesel2011) for more details). In this case, the points $A_{3}(t)$ and $M_{3}(t)$ remain in the shear plane for all the values of $Ca$ and time ( ${\it\zeta}(t)={\it\xi}(t)=90^{\circ }$ ).

When the capsule revolution axis is initially along the vorticity axis ( ${\it\zeta}(0)={\it\xi}(0)=0^{\circ }$ , $A_{3}(0)$ and $M_{3}(0)$ on the vorticity axis), the capsule cross-sections parallel to the shear plane are initially circular. They are deformed by the shear flow and the membrane rotates around the deformed shape. This capsule motion is called rolling, since the membrane rotates around the cross-section like a deformed wheel. At low $Ca$ (figure 3 a), the capsule cross-section is not much deformed so that the points $M_{3}$ and $A_{3}$ remain on the vorticity axis ( ${\it\zeta}(t)={\it\xi}(t)=0^{\circ }$ ). As the capillary number increases, the cross-section elongates in the straining direction (figure 3 b): the capsule small axis $\boldsymbol{u}_{3}$ and hence the point $M_{3}$ may eventually become located in the shear plane, while the point $A_{3}$ remains on the vorticity axis. In this case, the asymptotic values of the angles are ${\it\zeta}(t)=90^{\circ }$ and ${\it\xi}(t)=0^{\circ }$ .

Note that for large $Ca$ , the oscillation amplitude of the swinging regime observed when ${\it\zeta}(0)=90^{\circ }$  tends to zero so that the capsule experiences a quasi tank-treading motion (figure 2 c). This regime is visually the same as the rolling motion observed when ${\it\zeta}(0)=0^{\circ }$ at large $Ca$ (figure 3 b). The only way to distinguish between the two regimes is by monitoring the position of the point $A_{3}$ and the angle ${\it\xi}(t)$ . This shows that the best parameter to study the capsule motion is the angle ${\it\xi}(t)$ . Experimentally, it can be achieved by attaching markers to the membrane, but this is difficult to perform. Resorting to numerical simulations is thus useful to distinguish between those regimes.

Figure 3.

Capsule shape evolution over one half-period at steady state when ${\it\zeta}(0)=0^{\circ }$ ( ${\it\lambda}=1$ ): capsule in rolling regime. (a) $Ca=0.01$ ( $\max (\boldsymbol{q}\boldsymbol{\cdot }\boldsymbol{n}/G_{s})=0.1$ ); (b) $Ca=1.5$ ( $\max (\boldsymbol{q}\boldsymbol{\cdot }\boldsymbol{n}/G_{s})=5$ ). Same legend as in figure 2. The points $A_{3}(t)$ and $M_{3}(t)$ are superimposed in (a), whereas $A_{3}(t)$ is on the vorticity axis and $M_{3}(t)$ is in the shear plane in (b).

3.1.2 Stable equilibrium of an initially off-plane capsule

In Stokes flow, the mechanical stability of an equilibrium configuration can only be tested by perturbing the capsule orientation, which corresponds here to positioning the revolution axis with an initial orientation ${\it\xi}(0)={\it\zeta}(0)\in \, ]\,0^{\circ },90^{\circ }[\,$ . We then follow the time evolution of the angles ${\it\zeta}(t)$ (to determine the overall capsule position) and ${\it\xi}(t)$ (to determine the type of capsule motion). Note that if ${\it\xi}(t)\rightarrow 90^{\circ }$ , the capsule tends towards the tumbling or the swinging regime (§ 3.1.1). On the other hand, if ${\it\xi}(t)\rightarrow 0^{\circ }$ , the capsule stable configuration is the rolling motion. If ${\it\xi}(t)$ and ${\it\zeta}(t)$ tend towards any other value in $]\,0,90^{\circ }[\,$ , the capsule takes at equilibrium a wobbling motion, where it precesses and oscillates about the vorticity axis.

We first investigate whether the initial orientation affects the final equilibrium configuration. We start by considering a weak flow strength ( $Ca=0.01$ ) and three different initial orientations ( ${\it\zeta}(0)={\it\xi}(0)=5^{\circ }$ , $45^{\circ }$ , $75^{\circ }$ ). As shown in figure 4(a), the angles ${\it\zeta}(t)$ and ${\it\xi}(t)$ both tend to $90^{\circ }$ independently of the initial orientation. The only effect of the initial position is the time it takes to reach equilibrium, which increases the further the capsule is initially from its final equilibrium position. For $Ca=0.01$ , the stable equilibrium regime is a quasi-solid tumbling motion (with no membrane rotation), which explains why the curves of ${\it\zeta}(t)$ and ${\it\xi}(t)$ are superimposed. For a larger flow strength $Ca=0.3$ , the angles ${\it\zeta}(t)$ and ${\it\xi}(t)$ both converge towards $90^{\circ }$ but do not follow the same time evolution during the transition phase (figure 4 b). During the transient phase, the capsule takes a global oscillating swinging motion (evidenced by the oscillations of ${\it\zeta}(t)$ ), during which the membrane rotates to bring the point $A_{3}$ in the shear plane. This result is independent of the initial orientation as illustrated in figure 4(c) for the angle ${\it\zeta}(t)$ .

When the capsule tends to position its small axis $OM_{3}$ and the point $A_{3}(t)$ in the shear plane, the resulting motion is identical to the one shown in figure 2 when the capsule revolution axis is initially in the shear plane ( ${\it\zeta}(0)={\it\xi}(0)=90^{\circ }$ ) (see supplementary movies 1 and 2 available at http://dx.doi.org/10.1017/jfm.2015.759). In conclusion, the tumbling and swinging motions are found to be the stable equilibrium configurations of an oblate capsule for low and medium-range capillary numbers up to  $Ca<0.9$ .

Figure 4.

Transient evolution of the angles ${\it\zeta}(t)$ (dashed line) and ${\it\xi}(t)$ (full line) for different initial orientations ( ${\it\lambda}=1$ ). The grey zones represent the convergence criteria: (a) tumbling motion; (b,c) swinging motion: (a)  $Ca=~0.01$ , ${\it\zeta}(0)={\it\xi}(0)=5^{\circ }$ , $45^{\circ }$ , $75^{\circ }$ , (b)  $Ca=0.3$ , ${\it\zeta}(0)={\it\xi}(0)=30^{\circ }$ (c)  $Ca=0.3$ , ${\it\zeta}(0)={\it\xi}(0)=30^{\circ }$ , $45^{\circ }$ , $75^{\circ }$ .

For $Ca\geqslant 0.9$ , the principal short axis no longer remains within the shear plane: it exhibits an oscillation about the shear plane, which is superimposed onto the in-plane oscillation (see supplementary movie 3). The equilibrium state of the capsule thus evolves from a swinging to a wobbling (or oscillating–swinging) motion: the point $M_{3}(t)$ oscillates a little about the shear plane and the point $A_{3}(t)$ precesses around the vorticity axis with a constant mean inclination, as indicated in figure 5. As $Ca$ increases, the mean angle ${\it\xi}(t)=(OA_{3}(t),\boldsymbol{e}_{z})$ decreases until the capsule changes drastically its motion and starts rolling. For $Ca\geqslant 1.5$ , the capsule stable equilibrium configuration is rolling (see supplementary movie 4). This is illustrated in figure 6(a), which shows that, for $Ca=2$ , the angle ${\it\zeta}(t)$ tends towards $90^{\circ }$ whereas ${\it\xi}(t)$ converges towards $0^{\circ }$ , the smallest axis being in the shear plane because of the large profile elongation. It also shows that the capsule converges towards the rolling state whatever its initial orientation. In order to verify that the capsule does have a rolling motion, we have run the following test: once the capsule has reached its equilibrium configuration, we set the far-stream velocity of the external flow at zero and follow capsule reorientation during relaxation (see supplementary movie 5). The second part of the graph of figure 6(b) ( $\dot{{\it\gamma}}t\geqslant 1500$ ) shows that the angle ${\it\xi}(t)$ remains equal to $0^{\circ }$ , whereas ${\it\zeta}(t)$ decreases suddenly to $0^{\circ }$ when the flow is stopped. The capsule regains its oblate shape and has its revolution axis aligned with the vorticity axis. This indicates that the capsule had previously assumed, at equilibrium, a rolling motion that is identical to the one observed when the short axis is initially aligned with the vorticity axis ( ${\it\zeta}(0)={\it\xi}(0)=0^{\circ }$ ), as shown in figure 3(b).

Figure 5.

Wobbling motion for three initial orientations ${\it\xi}(0)=30^{\circ }$ , $45^{\circ }$ and $60^{\circ }$ at $Ca=0.9$ ( ${\it\lambda}=1$ ). The small axis oscillates a little about the shear plane, while $A_{3}(t)$ precesses and oscillates around the vorticity axis with an inclination which tends to $44^{\circ }\pm 12^{\circ }$ for all three initial orientations.

Figure 6.

Time evolution of the angles ${\it\zeta}(t)$ and ${\it\xi}(t)$ at $Ca=2$ ( ${\it\lambda}=1$ ). (a) Time response ( $\dot{{\it\gamma}}t\in [0,600]$ ) of the capsule for different initial orientations showing the convergence towards a rolling motion; (b) Long-time response (for ${\it\xi}(0)=15^{\circ }$ ) followed by a relaxation phase ( $\boldsymbol{v}^{\infty }=0$ for $\dot{{\it\gamma}}t\geqslant 1500$ ), during which both ${\it\xi}$ and ${\it\zeta}$ go to zero, thus showing that the capsule revolution axis is aligned with the vorticity axis.

To summarize the cases for ${\it\lambda}=1$ and $a/b=1/2$ , we have shown that, for $Ca<0.9$ , the situation is identical to the one considered by Walter et al. (Reference Walter, Salsac and Barthès-Biesel2011), where the capsule revolution axis is initially positioned in the shear plane ( ${\it\zeta}(0)={\it\xi}(0)=90^{\circ }$ ). Correspondingly, the capsule assumes a quasi-solid tumbling motion for $Ca<0.02$ and a swinging motion for $0.05<Ca<0.9$ . The two regimes are separated by a transition motion characterised by the transient occurrence of a quasi-circular profile within the shear plane ( $Ca\in \,[0.02,0.05]$ ). For $0.9\leqslant Ca\leqslant 1.5$ , the capsule goes through a wobbling motion (also called oscillating precession), where its small axis oscillates about the shear plane while the point $A_{3}(t)$ precesses about the vorticity axis and gets nearer to it as $Ca$ increases. Ultimately, for large values of $Ca$ ( $Ca\geqslant 1.5$ ), the capsule has a rolling motion. All these equilibrium states are independent of the initial orientation.

3.2 Influence of ${\it\lambda}$ on the equilibrium configurations

We now study the influence of the viscosity ratio on the stable equilibrium configurations of the oblate capsule ( $a/b=1/2$ ). We consider only viscosity contrasts larger than unity, as it was shown for initially spherical capsules that ${\it\lambda}<1$ had little influence on the dynamics (Foessel et al. Reference Foessel, Walter, Salsac and Barthès-Biesel2011). As the mechanical equilibrium configuration is independent of the initial orientation when ${\it\zeta}(0)\in \, ]\,0^{\circ },90^{\circ }[\,$ , we position the capsule with an initial angle ${\it\zeta}(0)=45^{\circ }$ and increase the viscosity of the internal fluid. Figure 7 shows the temporal evolution of the angles ${\it\zeta}(t)$ and ${\it\xi}(t)$ for ${\it\lambda}=1$ , 2, 4 and 8.5 in the case of a capillary number $Ca=0.2$ . For ${\it\lambda}=1$ , the capsule exhibits a swinging motion, as previously discussed. However, for ${\it\lambda}=2$ , the angle ${\it\zeta}(t)$ converges towards a value close to 90°  while ${\it\xi}(t)$ oscillates about 77°: this indicates that a wobbling motion has been set-up. When the viscosity ratio is further increased ( ${\it\lambda}=4$ ), a transition to rolling occurs: the angle ${\it\xi}(t)$ tends to $0^{\circ }$ while the capsule short axis exhibits dampened oscillations about the shear plane. For still higher values ( ${\it\lambda}=8.5$ ), the flow strength is not high enough to deform the capsule much: the smallest semi-diameter is on the vorticity axis ( ${\it\zeta}\rightarrow 0^{\circ }$ ) and the profile in the shear plane is only slightly deformed. This shows that the viscosity ratio has a strong influence on the capsule equilibrium state for medium-range capillary numbers.

Figure 7.

Time evolution of the angles (a) ${\it\zeta}(t)$ and (b) ${\it\xi}(t)$ for different viscosity ratios ${\it\lambda}$ ( ${\it\zeta}(0)=45^{\circ }\,Ca=0.2$ ).

The combined effect of the viscosity ratio and capillary number is summarized in figure 8. For ${\it\lambda}<3$ , the mechanical equilibrium configurations correspond to the ones observed for ${\it\lambda}=1$ . The viscosity ratio, however, influences the capillary number at which the tumbling-to-swinging and swinging-to-rolling transitions occur. The transition between tumbling and swinging takes place at higher $Ca$ , since the increase in internal viscosity reduces the capsule deformability. For example, the transition is delayed from $0.02\leqslant Ca\leqslant 0.05$ when ${\it\lambda}=1$ to $0.06\leqslant Ca\leqslant 0.09$ when ${\it\lambda}=2$ . On the contrary, the transition between swinging and rolling rather tends to occur for lower values of $Ca$ , as ${\it\lambda}$ increases. For example, at $Ca=0.5$ , the capsule converges towards the swinging regime at ${\it\lambda}=1$ and towards the rolling regime at ${\it\lambda}=2$ .

The direct consequence of these two observations on the regime transitions is the disappearance of the swinging motion for ${\it\lambda}\sim 3$ . For ${\it\lambda}\gtrsim 3$ , the stable mechanical equilibrium states are then only the tumbling and the rolling regimes. The capillary number of transition between tumbling and rolling further decreases with ${\it\lambda}$ : at high ${\it\lambda}$ , the rolling regime thus becomes the main mechanical equilibrium configuration that is likely to be observed (figure 8).

Figure 8.

Sketch of the mechanical equilibrium configurations of an oblate capsule as a function of the capillary number $Ca$ and viscosity ratio ${\it\lambda}$ . The grey zones represent the tumbling-to-swinging and swinging-to-rolling transitions.

Figure 9.

(a) Shear plane deformation and (b) semi-axis length along the vorticity axis. Open symbols: tumbling/swinging regime; full symbols: rolling regime.

It is of interest to compute the capsule deformation within the shear plane $D_{xy}=(L-B)/(L+B)$ , where $L$ and $B$ are the longest and shortest semi-axes of the ellipsoid of inertia in the shear plane, respectively. The deformation $D_{xy}$ is a quantity which is easily measured in the swinging or rolling regimes. We also provide the capsule semi-axis length $L_{z}$ along the vorticity axis, as it enables us to completely specify the geometry of the converged capsule shape together with $D_{xy}$ and the constant volume constraint. Since both $D_{xy}$ and $L_{z}$ oscillate in the swinging regime, we give values that are averaged over ten periods. For ${\it\lambda}=1$ , we recover the results of Walter et al. (Reference Walter, Salsac and Barthès-Biesel2011), which show a small decrease of $D_{xy}$ and $L_{z}$ during the tumbling-to-swinging transition, followed by a steady increase of both quantities with increasing $Ca$ (figure 9). A very similar behaviour is found for ${\it\lambda}=2$ . For $Ca\geqslant 0.2$ and ${\it\lambda}\geqslant 3$ , the rolling regime prevails, where $D_{xy}$ steadily increases with $Ca$ but decreases with ${\it\lambda}$ , as expected. It should be noted that the deformation in the rolling regime is almost the same as the one which is found for a viscous spherical capsule (Foessel et al. Reference Foessel, Walter, Salsac and Barthès-Biesel2011). This seems to indicate that the shear plane deformed profile of an elastic capsule depends only on $Ca$ and the viscosity ratio and is independent of the surface to volume ratio. Of course, this observation would need to be verified, but this is outside the scope of the present paper.

3.3 Influence of $a/b$ on the equilibrium configurations

We finally consider how the capsule dynamics is influenced by the aspect ratio. Figure 10 shows how the stable configurations evolve when the aspect ratio of the oblate capsule is increased from $a/b=1/2$ to $a/b=2/3$ . An aspect ratio of $a/b=2/3$ corresponds to $a/\ell =0.76$ and $b/\ell =1.44$ . Aspect ratios lower than 1/2 are difficult to simulate without taking into account bending effects, the capsule shape being characterised by large intrinsic curvatures: no such cases are thus considered with the present model as it is devoid of bending resistance. Increasing $a/b$ from $1/2$ to $2/3$ in the case ${\it\lambda}=1$ results in a lowering of the capillary number value at which the swinging-to-wobbling transition occurs (figure 10 a). The aspect ratio, however, has a limited influence on the capillary number above which rolling takes place. The swinging regime is thus stable for a smaller range of capillary numbers when $a/b$ increases. At ${\it\lambda}=5$ , the main effect of an increase of $a/b$ is to shift the transition to rolling to higher values of capillary number and to reduce the capillary number range for the wobbling motion.

In conclusion, the capsule qualitatively behaves similarly at $a/b=1/2$ and $2/3$ . The aspect ratio has a small effect on some of the capillary numbers at which transitions between regimes occur: it increases the $Ca$ -range of stability of the wobbling regime, but this phenomenon is mainly visible for viscosity ratios larger than 1. This justifies having presently studied in detail the particular case of $a/b=1/2$ .

Figure 10.

Stable equilibrium configurations of an oblate capsule as a function of the capillary number $Ca$ for two values of the aspect ratio: $a/b=1/2$ and $2/3$ . (a) ${\it\lambda}=1$ ; (b)  ${\it\lambda}=5$ .

4 Time to reach the equilibrium configuration

All the previous results indicate that an oblate capsule placed off the shear plane requires a very long time ${\it\tau}$ , of the order of $\dot{{\it\gamma}}{\it\tau}=10^{2}{-}10^{3}$ , to reach its mechanical equilibrium configuration. We have seen that the time ${\it\tau}$ depends on the initial orientation ${\it\zeta}(0)$ , the capillary number $Ca$ and the viscosity ratio ${\it\lambda}$ . In this section, we study the convergence time of an oblate capsule ( $a/b=1/2$ ). Such information can be useful to experimentalists for designing experimental protocols and to numericians for choosing adequately the computational time.

As is clearly apparent from figures 4 and 6, the convergence time is difficult to determine with precision. We define an estimate of ${\it\tau}$ as the time it takes ${\it\xi}(t)$ or ${\it\zeta}(t)$ to be equal to 10 % of the difference between its initial value and its value at equilibrium state. For example, based upon the time evolution of ${\it\zeta}$ , the convergence time ${\it\tau}_{{\it\zeta}}$ would be such that:

(4.1) $$\begin{eqnarray}|{\it\zeta}({\it\tau}_{{\it\zeta}})-{\it\zeta}_{\infty }|=0.1|{\it\zeta}(0)-{\it\zeta}_{\infty }|,\end{eqnarray}$$

where ${\it\zeta}_{\infty }$ is the equilibrium value of ${\it\zeta}$ . The same definition applies to the convergence time ${\it\tau}_{{\it\xi}}$ based on ${\it\xi}$ . The corresponding convergence areas are shown as grey zones in figures 4 and 6. Note that their width depends on the difference between the initial and final orientations.

In the tumbling regime, the initial orientation influences significantly the convergence time as is clearly apparent in figure 4(a). However, in the swinging regime, we have checked that ${\it\zeta}(0)$ has little influence on either ${\it\tau}_{{\it\zeta}}$ or ${\it\tau}_{{\it\xi}}$ . The same conclusion holds true for the rolling regime for $|{\it\zeta}(0)-{\it\zeta}_{\infty }|\leqslant 60^{\circ }$ . We thus study an average off-plane inclination $|{\it\zeta}(0)-{\it\zeta}_{\infty }|=45^{\circ }$ and show the combined effects of $Ca$ and ${\it\lambda}$ . In the wobbling regime, the time to reach equilibrium is fast for ${\it\zeta}$ , but very slow for ${\it\xi}$ and depends significantly on the initial orientation of the capsule (figure 5). We therefore do not try to determine it.

Figure 11.

Convergence time for ${\it\zeta}(0)={\it\xi}(0)=45^{\circ }$ . (a) Effect of the capillary number for ${\it\lambda}=1$ ; (b) effect of the viscosity ratio for moderate $Ca$ on the convergence time based on ${\it\xi}$ (open symbols: swinging regime, full symbols: rolling regime). The grey zone and the vertical line represent the transition from in-plane to rolling motion.

The convergence time depends significantly on whether it is based on ${\it\zeta}(t)$ or on ${\it\xi}(t)$ as is apparent in figure 11(a), where $\dot{{\it\gamma}}{\it\tau}$ is shown as a function of $Ca$ for ${\it\lambda}=1$ . Indeed, in the swinging regime, the convergence time ${\it\tau}_{{\it\xi}}$ is roughly twice ${\it\tau}_{{\it\zeta}}$ . We note that the convergence time increases with $Ca$ in the swinging regime and decreases in the rolling regime. The effect of the viscosity ratio is that it significantly increases ${\it\tau}_{{\it\xi}}$ in the swinging regime as one could expect, since more energy is dissipated when the viscosity ratio increases. In the rolling regime, the response time decreases when the viscosity ratio increases because the capsule is less deformed and tends to behave like a rolling axisymmetric solid body. Note that experimentally, it is ${\it\zeta}(t)$ which is readily measured: to obtain ${\it\xi}(t)$ , one would need to attach a marker to the membrane and follow its motion over time. One must also be careful experimentally when an apparent steady inclination is obtained for ${\it\zeta}$ : it can be misleading, since the membrane will not be at equilibrium with the fluid until ${\it\xi}$ has reached its final position.

5 Discussion and conclusion

The mechanical equilibrium configurations of an oblate capsule subjected to a simple shear flow have been determined numerically by initially positioning the capsule revolution axis off the shear plane and following the capsule dynamics. The study has been conducted for a capsule with a strain-hardening behaviour of the Skalak law type (Skalak et al. Reference Skalak, Tozeren, Zarda and Chien1973) and different values of the viscosity contrast ${\it\lambda}$ . We have mainly considered the aspect ratio $a/b=1/2$ , but have compared the results with $a/b=2/3$ to look at the effect of changing the capsule shape. We have been very careful to make sure that the equilibrium states were time converged. We show that, to characterise the motion of the capsule unambiguously, it is necessary to monitor two angles which measure the general orientation ${\it\zeta}(t)$ of the deformed profile and the position ${\it\xi}(t)$ of the capsule initial apex.

Contrary to previous results in the literature, we find that the equilibrium motion of an oblate capsule is independent of the initial position. For example at ${\it\lambda}=1$ (an extensively studied case), the tumbling and swinging regimes, obtained when the capsule axis is in the shear plane, are mechanically stable equilibrium configurations at low and moderate values of $Ca$ ( $Ca<0.9$ ). When $Ca$ increases, the capsule axis migrates away from the shear plane: a wobbling precession about the vorticity axis occurs for intermediate values of $Ca$ until a stable rolling motion is reached, where the capsule initial apex is located on the vorticity axis ( $Ca\geqslant 1.5$ ). When the aspect ratio is increased, we find that the swinging regime is stable over a smaller range of capillary numbers: the capsule has an out-of-shear-plane motion from lower capillary numbers onwards. It is interesting to note that for a prolate capsule with $a/b=2$ , the stable equilibrium states at ${\it\lambda}=1$ occur in reverse order as $Ca$ increases: rolling, wobbling and swinging (Dupont et al. Reference Dupont, Salsac and Barthès-Biesel2013).

A question that arises at this point is whether those different equilibrium states could be predicted from some simple physical considerations, such as minimisation of the system energy. The results obtained on prolate capsules showed that the stable equilibrium state does not necessarily correspond to the minimum viscous energy dissipation, except for small capillary numbers, when the capsule is subjected to small elastic deformation (Dupont et al. Reference Dupont, Salsac and Barthès-Biesel2013): it then converges towards the rolling regime like a solid ellipsoidal particle. But for $Ca>0.6$ , the membrane deformation plays too large a role: the capsule converges towards configurations that no longer correspond to minima in viscous energy dissipation, the equilibrium configuration being dictated by the fluid–structure interactions. This shows that the minimum energy criterion fails to determine the equilibrium state for a deformable capsule subjected to an external flow, even under Stokes flow conditions. As a consequence, it is not possible to test the equilibrium stability by adding an initial perturbation to the energy and observing its convergence/divergence, as the dynamics depends on nonlinear fluid–structure interactions. The only possibility is thus to simulate different cases and deduce the state diagrams.

The present results agree in essence with those presented in Omori et al. (Reference Omori, Imai, Yamaguchi and Ishikawa2012) for oblate capsules in the case of low viscosity ratios: they had also found for ${\it\lambda}\leqslant 2$ that the capsule revolution axis exhibited a transition from an in-plane to an out-of-plane motion and that the capsule converged towards rolling at higher capillary numbers. A good agreement is found for the value of the capillary number at the wobbling-to-rolling transition considering that the aspect ratios are not exactly the same in both studies: at ${\it\lambda}=1$ , Omori et al. (Reference Omori, Imai, Yamaguchi and Ishikawa2012) obtained $Ca=1.45$ for a capsule following Skalak law (unknown value of the parameter $C$ ) with $a/b=0.4$ , which compares well with our value of $Ca=1.2$ obtained for $a/b=0.5$ . Such is, however, not the case for the swinging-to-wobbling transition (0.1 versus 0.9 in our case). Would the value of the $C$ -parameter in the Skalak law account for such a difference?

Cordasco & Bagchi (Reference Cordasco and Bagchi2013) also found that the capsule tends towards the shear plane for $Ca\leqslant 0.6$ , but they observed a kayaking (i.e. an oscillation about the shear plane) or a precession-to-kayaking motion, which depended on the initial orientation. This conclusion is due to too short computational times ( $\dot{{\it\gamma}}t\,\leqslant \,100$ ). Indeed, we also find such a kayaking motion (see figure 4 b), but it is transient and dies out with time. As for Wang et al. (Reference Wang, Sui, Spelt and Wang2013), they considered only moderate values of $Ca$ and thus missed the rolling motion. They find different regime modes which depend on the initial orientation. This may be due to small inertial effects, as the Reynolds number is small but finite in their study, whereas it is exactly zero in our case.

The new results provide detailed information on the influence of the internal-to-external viscosity ratio on the mechanical equilibrium configurations of oblate artificial capsules. We find that, for moderate value ( ${\it\lambda}<4$ ), the internal viscosity limits the range of $Ca$ for which a swinging motion is possible. This can be easily understood, as high internal viscosity leads to high viscous energy dissipation during membrane rotation. In fact for large enough viscosity ( ${\it\lambda}\geqslant 4$ ), the only in-plane motion is the quasi-solid tumbling motion. The other effect of large internal viscosity is to shift to low $Ca$ values the transition from in-plane to rolling motion. A large internal viscosity leads to a significant tension jump across the interface and thus to membrane buckling, as reported by Foessel et al. (Reference Foessel, Walter, Salsac and Barthès-Biesel2011) and Yazdani & Bagchi (Reference Yazdani and Bagchi2013). The capsule deformability thus has a great influence on the mechanical equilibrium configuration, which depends on the fluid–structure interactions. It was finally found that, at higher viscosity ratio ( ${\it\lambda}\geqslant 4$ ), the main effect of increasing the aspect ratio is to shift the transition to rolling to a higher capillary number. The wobbling motion occurs over a larger range of capillary numbers.

Although the particle geometry is not the same, it is still of interest to compare the present results to experimental ones on RBCs. Goldsmith & Marlow (Reference Goldsmith and Marlow1972) highlighted the influence of the membrane deformability on the stable equilibrium configuration. They observed that when the RBC membrane is stiffened, the cell no longer tends towards the rolling motion but instead towards the tumbling motion. This is consistent with the present results, since stiffening the cell membrane corresponds to increasing $G_{s}$ and thus decreasing $Ca$ . In his experimental study, Bitbol (Reference Bitbol1986) varied the internal viscosity and the external shear rates. To estimate the corresponding values of viscosity ratios and capillary numbers, one can use as characteristic length for an RBC $\ell \sim 2.8~{\rm\mu}\text{m}$ , which corresponds to a cell volume of order $100~{\rm\mu}\text{m}^{3}$ (Klöppel Reference Klöppel2012). The value of shear elastic modulus $G_{s}$ depends on the constitutive law. Hochmuth & Waught (Reference Hochmuth and Waught1987) estimated $G_{s}\simeq 4~{\rm\mu}\text{N}~\text{m}^{-1}$ for a strain-hardening law, such as the Skalak law. Using these values, the results by Bitbol (Reference Bitbol1986) for low values of the viscosity ratio are that the cell exhibits tumbling up to $Ca=0.014$ , before drifting towards the vorticity axis in the range $0.014\leqslant Ca\leqslant 0.7$ . For $Ca>0.7$ , the cell then follows a $C=0$ orbit (i.e. rolling). Those results are also corroborated by Dupire et al. (Reference Dupire, Socol and Viallat2012), who have found that tumbling occurs up to approximately $Ca=0.05{-}0.075$ at ${\it\lambda}\simeq 1$ . One must note that the concept of transition is expressed in slightly different terms experimentally and numerically: concerning the tumbling-to-wobbling transition, it is clearly explained in Bitbol (Reference Bitbol1986) and Yao et al. (Reference Yao, Wen, Yan, Sun, Ka, Xie and Chien2001) that it is not as if all the cells would start precessing around the vorticity axis at a given $Ca$ . From the critical capillary number onwards, it is observed that more and more cells start precessing while the others continue tumbling, the ratio of the precessing cells over tumbling ones increasing when the capillary number is increased. Numerically, we find that tumbling stops for $Ca>0.02$ and that rolling occurs for $Ca>1$ , in agreement with the experimental findings when all the cells have the same motion.

Even though the experimental results on RBCs agree well with the present ones on oblate capsules, one could still wonder about the potential effect of the particle shape. The results of § 3.3 obtained by changing the aspect ratio, however, indicate that this parameter has a limited influence on the dynamics of oblate deformable capsules. An increase in aspect ratio provides the same qualitative capsule behaviour and only modifies slightly some of the capillary number values at the transitions between regimes. In the $Ca$ – ${\it\lambda}$ phase diagram, the main influence of $a/b$ is to increase the $Ca$ -range over which wobbling is stable: this effect is mostly notable when the viscosity ratio is quite larger than 1. It is thus surprising that Cordasco & Bagchi (Reference Cordasco and Bagchi2013) indicated that, at $Ca=0.1$ and ${\it\lambda}=1$ , oblate capsules tended towards an in-plane motion, whereas RBCs exhibited a rolling motion. Since their simulations were run for a finite Reynolds number, it is possible that, although small, inertia affects the dynamics of deformable ellipsoidal particles. The differences in behaviour are more likely to be accounted for by inertial effects than by membrane viscosity. The wheel-like rolling motion was indeed simulated by MacMeccan et al. (Reference MacMeccan, Clausen, Neitzel and Aidun2009) for RBCs with ${\it\lambda}=5$ using the lattice Boltzmann method. The simulations agree well with the experimental results of Yao et al. (Reference Yao, Wen, Yan, Sun, Ka, Xie and Chien2001), the small differences being most likely due to uncertainties in RBC material properties and in the RBC distance from the wall in the experiments. Since the numerical models did not include membrane viscosity, it can be expected that membrane viscosity does not play a central role in the wheel-like rolling motion, but that it rather adds up to the viscosity of the inner fluid (i.e. the RBC haemoglobin solution) giving rise to a global viscosity of the particle.

We have finally shown that a capsule initially placed off the shear plane takes a finite time to reach its stable equilibrium configuration depending on its initial orientation, flow strength and viscosity ratio. For a medium displacement from the shear plane ( ${\it\zeta}(0)=45^{\circ }$ ), the non-dimensional convergence time when ${\it\lambda}=1$ varies between 50 and 400 for $Ca\leqslant 0.5$ and is about 600 for $Ca\geqslant 2$ . Long computational times are thus required to study the equilibrium configurations of oblate microcapsules or cells. For experiments, it is interesting to translate the non-dimensional times into dimensional ones. If one considers the case of an RBC subjected to simple shear flow, one finds that the stable tumbling equilibrium regime is reached after ${\sim}50~\text{s}$ (from figure 4 a), stable swinging after ${\sim}10~\text{s}$ (figure 4 b,c) and stable rolling after ${\sim}2~\text{s}$ (figure 6). Since the typical experimental window time for observation is of the order of 1 minute, one must be careful to check that equilibrium conditions have indeed been reached when the measurements are recorded.