2.1 Euler angles

We consider cells placed in a shear flow with flow direction $Oz^{\prime }$ , shear gradient direction $Oy^{\prime }$ and vorticity axis $Ox^{\prime }$ . To characterize the orientation and motion of cells, we shall use the Euler angles. Let us consider, as a first approximation of RBC shape, an ellipsoid of the equation

(2.1) $$\begin{eqnarray}r^{2}x^{2}+y^{2}+z^{2}=1.\end{eqnarray}$$

Here $r$ is the aspect ratio, which will be larger than 1 here (oblate ellipsoid). This ellipsoid may rotate and we use the Euler angles as defined in Jeffery (Reference Jeffery1922) and in figure 1 to describe this rotation in the fixed coordinate system $Ox^{\prime }y^{\prime }z^{\prime }$ that coincides initially with the system $Oxyz$ associated with the ellipsoid (supplemental material is available online at https://doi.org/10.1017/jfm.2019.42 and presents a comparison with the other convention used in the literature).

In the original paper by Jeffery (Reference Jeffery1922), $\unicode[STIX]{x1D703}$ is obtained by rotation around the $Oz^{\prime }=Oz$ axis, then $\unicode[STIX]{x1D719}$ is obtained by rotation around the $Ox^{\prime }$ axis, such that it is defined as the angle between the planes $Ox^{\prime }y^{\prime }$ and $Ox^{\prime }x$ (see figure 1). With this convention, when $\unicode[STIX]{x1D719}=0$ and $\unicode[STIX]{x1D703}=90^{\circ }$ , the cell face is in the $Ox^{\prime }z^{\prime }$ plane.

If the cells are viewed from the velocity gradient axis $Oy^{\prime }$ , as in our experiments, the angle $\unicode[STIX]{x1D6F9}$ defined as the angle between $Ox^{\prime }$ and the projection of the cell axis of revolution $Ox$ onto the plane $Ox^{\prime }z^{\prime }$ , can be easily determined (see figure 1). This angle is also the angle between the projection of the cell and the flow direction. Here $\unicode[STIX]{x1D6F9}$ is related to $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D719}$ through $\tan \unicode[STIX]{x1D6F9}=\tan \unicode[STIX]{x1D703}\sin \unicode[STIX]{x1D719}$ .

Figure 1.

The Euler angles used to describe the flipping regime. Convention used in Jeffery (Reference Jeffery1922) and in this paper. The large black arrow shows the viewing direction in the experiments, where projection onto the $x^{\prime }z^{\prime }$ plane is seen (black ellipse).

Figure 2.

Selected snapshots along half a flipping period of an ellipsoid of aspect ratio $r=3$ following a Jeffery orbit of orbit angle $\unicode[STIX]{x1D703}_{0}=45^{\circ }$ . The black shape is the projection of the ellipsoid parallel in the shear gradient direction; it corresponds to what is seen in our experiment. Here $\unicode[STIX]{x1D6F9}$ is the angle of the main axis of this projected shape with the flow direction. As stated by (2.6), it oscillates between $-\unicode[STIX]{x1D703}_{0}$ and $\unicode[STIX]{x1D703}_{0}$ . The apparent aspect ratio $r_{a}$ oscillates between a minimal value strictly larger than 1 and $r$ , when the cell is seen edge on and $\unicode[STIX]{x1D6F9}=\pm \unicode[STIX]{x1D703}_{0}$ .

2.2 Jeffery orbits

For a shear flow in the $z^{\prime }$ direction with $y^{\prime }$ the shear gradient direction and $x^{\prime }$ the vorticity direction, the motion of a rigid ellipsoid in creeping flow is given by Jeffery (Reference Jeffery1922)

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{\unicode[STIX]{x1D703}}=\dot{\unicode[STIX]{x1D6FE}}\frac{r^{2}-1}{r^{2}+1}\sin \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D703}\sin \unicode[STIX]{x1D719}\cos \unicode[STIX]{x1D719}, & \displaystyle\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{\unicode[STIX]{x1D719}}=\frac{\dot{\unicode[STIX]{x1D6FE}}}{1+r^{-2}}(r^{-2}\cos ^{2}\unicode[STIX]{x1D719}+\sin ^{2}\unicode[STIX]{x1D719}), & \displaystyle\end{eqnarray}$$

with $C\equiv r\tan \unicode[STIX]{x1D703}_{0}$ the orbit parameter. These equations can be solved and give

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle \tan \unicode[STIX]{x1D703}=\frac{C}{r(r^{-2}\cos ^{2}\unicode[STIX]{x1D719}+\sin ^{2}\unicode[STIX]{x1D719})^{1/2}}, & \displaystyle\end{eqnarray}$$

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle \tan \unicode[STIX]{x1D719}=r^{-1}\tan \frac{\dot{\unicode[STIX]{x1D6FE}}t}{r+r^{-1}}. & \displaystyle\end{eqnarray}$$

Here $\unicode[STIX]{x1D703}$ oscillates between $\unicode[STIX]{x1D703}_{0}$ and $\arctan C$ (spinning motion).

We shall refer to these possible motions as flipping motions. Among them, $\unicode[STIX]{x1D703}_{0}=0^{\circ }$ corresponds to what is called rolling motion ( $\unicode[STIX]{x1D703}$ always equal to 0), while $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ corresponds to tumbling ( $\unicode[STIX]{x1D703}$ always equal to $90^{\circ }$ ). Note that when $r>1$ , $\dot{\unicode[STIX]{x1D719}}$ is minimal when $\unicode[STIX]{x1D719}=0$ , which corresponds to the cell aligned with the flow direction.

Simple trigonometry then yields the equation for $\unicode[STIX]{x1D6F9}(t)$ according to Jeffery theory:

(2.6) $$\begin{eqnarray}\tan \unicode[STIX]{x1D6F9}=\tan \unicode[STIX]{x1D703}_{0}\times \sin \frac{\dot{\unicode[STIX]{x1D6FE}}t}{r+r^{-1}}.\end{eqnarray}$$

Therefore, $\unicode[STIX]{x1D6F9}$ oscillates between two extreme positions $-\unicode[STIX]{x1D703}_{0}$ and $\unicode[STIX]{x1D703}_{0}$ , where it stays for more time (see figure 2 for an example).

In this paper, we shall consider the possible flipping motions for a cell, that will be hypothesized to closely follow Jeffery orbits, but also the tank-treading motion. In that case, the cell small axis remains in the shear plane ( $\unicode[STIX]{x1D703}=90^{\circ }$ ) while the cell adopts a constant angle relatively to the flow direction. This definition is unambiguous as long as the cell does not deform. If it does, as pointed out in Dupont et al. (Reference Dupont, Delahaye, Barthès-Biesel and Salsac2016) where oblate capsules are considered, one should refer to the membrane material point that was located, at rest, on the small axis. From this unambiguous definition, a cell of fixed shape in flow for which this point is directed toward the vorticity direction would be called a rolling cell, even though its deformation is such that it resembles a tank-treading cell. From this point of view, the tank-treading motion mentioned in Cordasco, Yazdani & Bagchi (Reference Cordasco, Yazdani and Bagchi2014, figure 12) or in Sinha & Graham (Reference Sinha and Graham2015, figure 15) should be called rolling motion.

Since the numerical studies we will refer to in the following have used the term tank-treading, and since in most experimental studies the deformation seems to remain weak, we shall however go on using the (potentially improper) term of tank-treading for cells whose small axis of symmetry lies in the shear plane.

3.1 Experimental papers

One of the first major contributions to the study of RBC dynamics under shear flow can be found in Goldsmith & Marlow (Reference Goldsmith and Marlow1972) where statistics on the angle distribution is obtained. They observed dilute RBCs in a large tube where, at the cell scale, the flow can be assimilated to simple shear flow, with the same outer viscosity $\unicode[STIX]{x1D702}_{0}$ as plasma. They also performed experiments in a plate geometry at high $\unicode[STIX]{x1D702}_{0}$ (520 mPa s) and low $\dot{\unicode[STIX]{x1D6FE}}$ (around $1~\text{s}^{-1}$ ). Both experiments correspond to an increasing $\dot{\unicode[STIX]{x1D6FE}}$ : in the Poiseuille flow experiment the observation tube is narrower than the upstream tubes, and in the shear chamber experiment, RBCs are initially at rest.

The main outcomes of Goldsmith and Marlow’s study are as follows.

1. (i) In plasma and at low shear rate ( $\dot{\unicode[STIX]{x1D6FE}}<10~\text{s}^{-1},\unicode[STIX]{x1D70F}<0.01~\text{Pa}$ ), the time evolution $\unicode[STIX]{x1D719}(t)$ can be well described by Jeffery’s equation (2.5) if an effective aspect ratio is introduced, which is 25 % lower than the aspect ratio of the convex envelope of the cell. This result was previously found for rigid circular disks (Anczurowski & Mason Reference Anczurowski and Mason1967), and the discrepancy between the two ratios is expected to decrease as the aspect ratio gets closer to 1.

2. (ii) At higher $\dot{\unicode[STIX]{x1D6FE}}$ , departure from Jeffery’s equation is observed, with longer duration of alignment with flow and the initial $\unicode[STIX]{x1D719}\rightarrow -\unicode[STIX]{x1D719}$ symmetry for $\dot{\unicode[STIX]{x1D719}}$ is lost. This is interpreted as the consequence of cell deformability, which induces different shapes depending on the cell orientation relatively to the extensional direction of the flow. At the same time, more cells drift into rolling motion ( $\unicode[STIX]{x1D703}=0$ ). Those feature are lost for hardened cells.

3. (iii) Contrary to what the Jeffery theory states, $\unicode[STIX]{x1D703}$ was found not to be periodic as a function of $\unicode[STIX]{x1D719}$ , and the motion appears to be less regular. This is interpreted as a consequence of rotary Brownian motion of cells.

4. (iv) In the highly viscous suspending medium, cells are seen to align strongly with the flow. However, no clear discrimination is made between tank-treading and rolling cells.

The drift to rolling is studied in Bitbol (Reference Bitbol1986). The cells were observed in a cone-plate geometry allowing for $\dot{\unicode[STIX]{x1D6FE}}$ between 1 and $200~\text{s}^{-1}$ . Viscosity $\unicode[STIX]{x1D702}_{0}$ of the suspending medium was between 1 and 10 mPa s. The main results are as follows.

1. (i) If one considers the single parameter $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D702}_{0}\dot{\unicode[STIX]{x1D6FE}}$ , which measures the flow stress on the cell, a rough phase diagram is obtained from direct observations as follows: if $\unicode[STIX]{x1D70F}<0.02~\text{Pa}$ , cells rotate in the flow (that is, flipping motion is observed). In the intermediate range $0.02~\text{Pa}<\unicode[STIX]{x1D70F}<1~\text{Pa}$ , cells drift to rolling. For higher stresses, the number of rolling cells plateaus at viscosity $\unicode[STIX]{x1D702}_{0}=1~\text{mPa}~\text{s}$ while it decreases when $\unicode[STIX]{x1D702}_{0}>5~\text{mPa}~\text{s}$ : cells start to tank-tread.

2. (ii) In the rolling regime, the cell diameter increases with $\unicode[STIX]{x1D702}_{0}$ and $\dot{\unicode[STIX]{x1D6FE}}$ , as already found in Goldsmith & Marlow (Reference Goldsmith and Marlow1972).

3. (iii) In the considered viscosity range, the typical time for cell drift towards rolling is of the order of $100\times \dot{\unicode[STIX]{x1D6FE}}^{-1}$ .

Several new features were highlighted in Dupire et al. (Reference Dupire, Socol and Viallat2012), which followed a first paper by the same group (see Abkarian et al. Reference Abkarian, Faivre and Viallat2007). In these papers, the case of decreasing $\dot{\unicode[STIX]{x1D6FE}}$ is considered for the first time, and off-shear plane motion is also described for the first time. The carrying fluid viscosities $\unicode[STIX]{x1D702}_{0}$ are 7 and 29 mPa s, and the $\dot{\unicode[STIX]{x1D6FE}}$ goes from 0 to $15~\text{s}^{-1}$ in the first case and to $2.7~\text{s}^{-1}$ in the second case. The cells flow in a large parallelepiped flow chamber where around 20 cells were tracked individually and visualized along both the shear and vorticity axes. The main results of this paper are as follows.

1. (i) At low $\dot{\unicode[STIX]{x1D6FE}}$ , the orbit angle $\unicode[STIX]{x1D703}_{0}$ of a flipping cell is not constant but varies between $50^{\circ }$ and $90^{\circ }$ .

2. (ii) If the shear rate exceeds a threshold $\dot{\unicode[STIX]{x1D6FE}}_{t}$ lying between 0.01 and 0.05 Pa depending on the observed cell, this orbit angle stabilizes to a constant value (that decreases with $\dot{\unicode[STIX]{x1D6FE}}$ , until the rolling regime is reached).

3. (iii) In the flipping regime, the time evolution of $\unicode[STIX]{x1D719}$ is well described by the Jeffery equation.

4. (iv) At some critical shear rate $\dot{\unicode[STIX]{x1D6FE}}_{c}^{+}$ , transition towards tank-treading is observed. Tank-treading is associated to swinging, that is, periodic oscillation of the inclination angle.

5. (v) If $\dot{\unicode[STIX]{x1D6FE}}$ is then decreased, the tank-treading motion remains stable until some other critical shear rate $\dot{\unicode[STIX]{x1D6FE}}_{c}^{-}<\dot{\unicode[STIX]{x1D6FE}}_{c}^{+}$ at which a transient intermittent regime is observed. It shows an alternance (in time) between tank-treading and flipping motions with high $\unicode[STIX]{x1D703}_{0}$ . When an orbit angle lower than $50^{\circ }$ is reached, flipping motion with a fixed orbit angle becomes stable, with the same orbit angle equal to that observed for the same shear rate in the increasing $\dot{\unicode[STIX]{x1D6FE}}$ case. From this observation we deduce that $\dot{\unicode[STIX]{x1D6FE}}_{t}<\dot{\unicode[STIX]{x1D6FE}}_{c}^{-}$ . Although not explicitly stated in Dupire et al. (Reference Dupire, Socol and Viallat2012), it seems that upon a further decrease of $\dot{\unicode[STIX]{x1D6FE}}$ the same branch as in the increasing $\dot{\unicode[STIX]{x1D6FE}}$ case is followed.

6. (vi) There is no variation in $\dot{\unicode[STIX]{x1D6FE}}_{c}^{-}$ and $\dot{\unicode[STIX]{x1D6FE}}_{c}^{+}$ if many shear rate cycles are applied to the same cell. The corresponding critical shear stress values are 0.023 and 0.086 Pa for $\unicode[STIX]{x1D702}_{0}=29~\text{mPa}~\text{s}$ . These are mean values, because all the critical stresses, as well as the orbit angles associated with a given stress, depend on the considered cell.

7. (vii) All these transitions occur in a shape preserving manner, with diameter variations that remains lower than 10 %.

The transition between flipping and tank-treading for several viscosities and many cells is studied in Fischer & Korzeniewski (Reference Fischer and Korzeniewski2013). They consider a Poiseuille flow, so it is likely that only the increasing $\dot{\unicode[STIX]{x1D6FE}}$ case is considered, though we do not know what the upstream conditions are. The critical $\dot{\unicode[STIX]{x1D6FE}}$ for $\unicode[STIX]{x1D702}_{0}=11,24$ and 104 mPa s are 75, 10 and $0.6~\text{s}^{-1}$ , respectively. The transition shear rate is determined as the value at which half of the RBCs are in tank-treading regime. The authors also establish that the transition is not sharp: the number of cells in tank-treading increases smoothly with the $\dot{\unicode[STIX]{x1D6FE}}$ . The transition becomes more abrupt at high carrying fluid viscosity $\unicode[STIX]{x1D702}_{0}$ . Those thresholds together with that measured in Morris & Williams (Reference Morris and Williams1979) for plasma viscosity, allow a separation line to be drawn between flipping-like motion and tank-treading-like motion in the ( $\unicode[STIX]{x1D702}_{0},\unicode[STIX]{x1D70F}$ ) space, as shown in figure 3.

In Lanotte et al. (Reference Lanotte, Mauer, Mendez, Fedosov, Fromental, Claveria, Nicoud, Gompper and Abkarian2016), experiments similar to those in Fischer & Korzeniewski (Reference Fischer and Korzeniewski2013) are performed in physiological condition, but on a limited amount of cells. Transitions from flipping to rolling are observed from $\dot{\unicode[STIX]{x1D6FE}}=10~\text{s}^{-1}$ until $\dot{\unicode[STIX]{x1D6FE}}=40~\text{s}^{-1}$ ( $\unicode[STIX]{x1D70F}=0.04~\text{Pa}~\text{s}$ ). In the meantime and also for larger shear rates, an increasing proportion of cup-shaped stomatocytes in rolling motion is detected. At even higher shear rates (until the maximum considered, $\dot{\unicode[STIX]{x1D6FE}}=2000~\text{s}^{-1}$ ), highly deformed polylobed shapes are observed. No notion of increasing or decreasing $\dot{\unicode[STIX]{x1D6FE}}$ is introduced in these experiments. They confirmed this picture in a second recent paper with shear rates higher than $60~\text{s}^{-1}$ (see Mauer et al. Reference Mauer, Mendez, Lanotte, Nicoud, Abkarian, Gompper and Fedosov2018).

Finally, in Levant & Steinberg (Reference Levant and Steinberg2016), 17 cells are studied in a four roll mill apparatus which allows to consider a more general flow defined by the vorticity $\unicode[STIX]{x1D714}$ and the strain rate $s$ . Simple shear flow corresponds to the case $2\unicode[STIX]{x1D714}=2s=\dot{\unicode[STIX]{x1D6FE}}$ . It corresponds to a stability threshold for the apparatus, since $\unicode[STIX]{x1D714}/s>1$ is required for the cell to remain trapped. Long time observation is then possible, while varying the ratio $\unicode[STIX]{x1D714}/s$ . More precisely, $\unicode[STIX]{x1D714}$ is fixed and $s$ is varied. The explored range of external viscosities is 20–87 mPa s. The possibility to increase the contribution of the vorticity allows to switch to flipping regimes even in that viscosity range. With the choice to characterize the flow stress by $2s\unicode[STIX]{x1D702}_{0}$ and to consider the extended viscosity $\unicode[STIX]{x1D702}_{0}(w/s)^{-1}$ , three regions are identified, corresponding to tumbling, swinging, and an intermittent regime, which coincide with those identified in Dupire et al. (Reference Dupire, Socol and Viallat2012). Contrary to what was observed in Dupire et al. (Reference Dupire, Socol and Viallat2012), large deformations are associated with the intermittent regime. Although the stress seems to be increased and decreased within the same experiment, no hysteretic behaviour is reported. In figure 3, the explored area in the $(\unicode[STIX]{x1D702}_{0}(w/s)^{-1},2s\unicode[STIX]{x1D702}_{0})$ parameter space is shown on the phase diagram for simple shear rate with parameter space $(\unicode[STIX]{x1D702}_{0},\unicode[STIX]{x1D702}_{0}\dot{\unicode[STIX]{x1D6FE}})$ . We shall note that, in agreement with numerical simulations in Cordasco & Bagchi (Reference Cordasco and Bagchi2013), no off-plane motion is observed in the flipping region, while it is seen in simple shear flow in Dupire et al. (Reference Dupire, Socol and Viallat2012). This partial mapping shows that an increased rotational component in the flow, as in Levant & Steinberg (Reference Levant and Steinberg2016), favours in-plane motion.

The experimental studies summarized above have all brought new features to the problem. They neither contradict each other nor provide cross-validation since the explored parameter ranges, or the experimental method, often differ. Studies on individual cells that are followed in time have allowed more detailed dynamics to be exhibited (see Abkarian et al. Reference Abkarian, Faivre and Viallat2007; Dupire et al. Reference Dupire, Socol and Viallat2012; Levant & Steinberg Reference Levant and Steinberg2016), but full characterization of the transition dynamics could not be addressed, in particular concerning the consequence of dispersion in size and mechanical properties of cells. The decreasing $\dot{\unicode[STIX]{x1D6FE}}$ case has only been clearly addressed in Dupire et al. (Reference Dupire, Socol and Viallat2012). On the other hand studies on statistically relevant populations are often limited to the study of a single feature, e.g. the population in tank-treading regime (see Goldsmith & Marlow Reference Goldsmith and Marlow1972; Fischer & Korzeniewski Reference Fischer and Korzeniewski2013) or in deformed rolling regime (see Mauer et al. Reference Mauer, Mendez, Lanotte, Nicoud, Abkarian, Gompper and Fedosov2018), but they allowed the width of transition zones due to cell dispersity to be estimated, in the increasing $\dot{\unicode[STIX]{x1D6FE}}$ case only.

The case of physiological values for the external fluid viscosity has seldom been addressed, probably because of sedimentation issues, and is therefore much less documented than the case of more viscous fluids.

3.2 On the hysteresis and the intermittent regimes

A striking feature is the existence, at least for high enough viscosities of the external fluid, of an hysteresis in the $(\unicode[STIX]{x1D702}_{0},\dot{\unicode[STIX]{x1D6FE}})$ space upon variations of $\dot{\unicode[STIX]{x1D6FE}}$ , which has been described differently in different papers. We aim at clarifying this in the following.

In the first experimental paper to account for such a feature (see Abkarian et al. Reference Abkarian, Faivre and Viallat2007), the observed low shear rate motion is surprisingly always tumbling. Transition to tank-treading (or, rather, swinging) is characterized by the existence of an intermediate regime where cells alternatively tumble and swing, which is observed for both decreasing and increasing $\dot{\unicode[STIX]{x1D6FE}}$ . The experimental constraints did not allow a conclusion on whether this intermittent regime is transient or not. Transition shear rates denoted $\dot{\unicode[STIX]{x1D6FE}}_{c}^{{>}}$ and $\dot{\unicode[STIX]{x1D6FE}}_{c}^{{<}}$ are identified, with $\dot{\unicode[STIX]{x1D6FE}}_{c}^{{>}}>\dot{\unicode[STIX]{x1D6FE}}_{c}^{{<}}$ , a signature of hysteretic behaviour. They correspond respectively to the first observed transition from tumbling to swinging when increasing $\dot{\unicode[STIX]{x1D6FE}}$ and to the first observed transition from swinging to tumbling when decreasing $\dot{\unicode[STIX]{x1D6FE}}$ , for an observation over a time scale of order 20s.

In the same paper, an improvement of the analytical model by Keller & Skalak (Reference Keller and Skalak1982) (a droplet enclosed by an ellipsoidal fluid membrane) is proposed for RBC dynamics. Membrane shear elasticity is taken into account (as, simultaneously, in Skotheim & Secomb (Reference Skotheim and Secomb2007)), and in Dupire et al. (Reference Dupire, Abkarian and Viallat2015) the possibility for a stress-free configuration different from the equilibrium shape is explored. In that model, considering more inflated stress-free configurations amounts to multiplying the shear modulus by some constant smaller than 1. Considering the transition threshold and other characteristics such as cell oscillation periods, it was deduced in Dupire et al. (Reference Dupire, Abkarian and Viallat2015) that the stress-free configuration is likely to be a spheroid, in agreement with other modelling approaches (see Lim, Wortis & Mukhopadhyay Reference Lim, Wortis and Mukhopadhyay2002; Cordasco & Bagchi Reference Cordasco and Bagchi2014; Peng et al. Reference Peng, Mashayekh and Zhu2014). Returning to our initial question, the analytical model results in differential equations for the angles characterizing cell dynamics, for the case where the cell axis of symmetry remains in the shear plane. In other words, only transitions between tumbling-like and tank-treading-like motions can be explored (a recent model proposed by Mendez & Abkarian (Reference Mendez and Abkarian2018) should solve this issue in the future). These equations can be solved numerically for a given shear rate and a given set of parameters for the cell mechanical properties, and an intermittent regime is observed in an interval $[\dot{\unicode[STIX]{x1D6FE}}_{c}^{-};\dot{\unicode[STIX]{x1D6FE}}_{c}^{+}]$ . In Abkarian et al. (Reference Abkarian, Faivre and Viallat2007) or in Dupire et al. (Reference Dupire, Abkarian and Viallat2015) it is however not specified whether the threshold values are obtained by considering increasing or decreasing $\dot{\unicode[STIX]{x1D6FE}}$ . Knowing the dependency of the solutions of the cell angle evolution equations on the history of $\dot{\unicode[STIX]{x1D6FE}}$ values probably requires a precise stability analysis such as that done in Kessler et al. (Reference Kessler, Finken and Seifert2009) in the case of quasi-spherical cells.

Though they are clearly related to each other, how the different thresholds $\dot{\unicode[STIX]{x1D6FE}}_{c}^{\langle ,\rangle }$ and $\dot{\unicode[STIX]{x1D6FE}}_{c}^{-,+}$ compare with each other is not clear as they were determined differently (limited observation time in the experiments and more importantly no direct study of potential hysteresis in the numerical analysis).

The intermittent behaviour has been observed and thoroughly studied using numerical simulations (starting from a rest configuration) in Cordasco & Bagchi (Reference Cordasco and Bagchi2014) and in Peng et al. (Reference Peng, Mashayekh and Zhu2014), with the axis of revolution also restricted to shear plane, though in some cases this configuration is said to be metastable according to Cordasco & Bagchi (Reference Cordasco and Bagchi2014). In Peng et al. (Reference Peng, Mashayekh and Zhu2014), the minor bound $\dot{\unicode[STIX]{x1D6FE}}_{c}^{-}$ of the intermittent regime is determined and shown to be smaller than the transition shear rate between rolling and tank-treading. Finally, in the experimental paper by Levant & Steinberg (Reference Levant and Steinberg2016) a stationary intermittent regime is also observed for, apparently, increasing and decreasing flow stress although it is only explicitly highlighted for a decrease of the flow stress $2s\unicode[STIX]{x1D702}_{0}$ . Note that in this paper the flow geometry is not the same and no off-shear plane motion is observed.

In Dupire et al. (Reference Dupire, Socol and Viallat2012), the picture turns out to be more complex: the decreasing $\dot{\unicode[STIX]{x1D6FE}}$ path would be characterized by the apparition of a transient intermittent (tank-treading/flipping with angle larger than $50^{\circ }$ ) phase when tank-treading regime is left, whose duration in time is not clear. After this intermittent phase, flipping motion with orbit angle strictly smaller than $50^{\circ }$ would follow. The shear rate at which this transition takes place is denoted by $\dot{\unicode[STIX]{x1D6FE}}_{c}^{-}$ , and no notion of interval over which this intermittent regime would be possible is introduced. The analogy with the notation $\dot{\unicode[STIX]{x1D6FE}}_{c}^{-}$ introduced in Abkarian et al. (Reference Abkarian, Faivre and Viallat2007) suggests that the transient intermittent regime would only take place in the low part of the whole expected range for the intermittent regime $[\dot{\unicode[STIX]{x1D6FE}}_{c}^{-};\dot{\unicode[STIX]{x1D6FE}}_{c}^{+}]$ , while stable tank-treading would be preferred in the top range.

On the other hand, upon an increase of shear rate, no intermittent regime is observed but rather a continuous drift from tumbling to rolling, then a transition to swinging. In Dupire et al. (Reference Dupire, Socol and Viallat2012), the threshold for this transition is denoted by $\dot{\unicode[STIX]{x1D6FE}}_{c}^{+}$ , suggesting that when increasing $\dot{\unicode[STIX]{x1D6FE}}$ the route to swinging through the intermittent regime is replaced by the orbital drift. In Levant & Steinberg (Reference Levant and Steinberg2016), the intermittent regime is associated with large cell deformations: we hypothesize that the orbital drift is another route that prevents such large elastic distortions. This would explain why the intermittent regime is not seen in the increasing shear rate case, unless forced by restricting the motion to the shear plane, artificially in the simulations or because of a specific flow geometry as in Levant & Steinberg (Reference Levant and Steinberg2016).

Finally, the fact that, to the best of the authors’ knowledge, no analytical or numerical analysis has considered the decreasing shear rate case so far limits the discussion on the relationship between hysteretic and (possibly transient) intermittent regimes. A secondary goal of the present study is to reinforce the experimental input on this question.

5.1 Premise

We assume that, for a given shear rate, the suspension is composed at each time, on average, of a proportion $p_{T}$ of cells in tank-treading-like (or swinging) motion and a proportion $1-p_{T}$ of cells in flipping motion following a Jeffery orbit, where the orbit angle $\unicode[STIX]{x1D703}_{0}$ can be between 0 (rolling) and $90^{\circ }$ (tumbling). In Dupire et al. (Reference Dupire, Socol and Viallat2012), it is shown that above a threshold value, orbits are unstable and the cells switch from one orbit to another, with angles between this threshold and $90^{\circ }$ . In contrast, below this value orbits are stable. In order to explore this idea, we look for two subpopulations of flipping cells: (i) with proportion $p_{s}$ , those with stable orbits that we suppose to lie between two extremal values $\unicode[STIX]{x1D703}_{0}^{-}$ and $\unicode[STIX]{x1D703}_{0}^{+}$ , with equal probability; while this may appear as a strong statement, this is the only reasonable choice amongst the possible distributions that would lead to a reasonable number of fitting parameters; (ii) with proportion $(1-p_{s})$ , those with unstable orbits with angles between $\unicode[STIX]{x1D703}_{0}^{+}$ and $90^{\circ }$ , with equal probability. In addition, shear stress may have an influence on the cells’ aspect ratio, which should also be determined.

In the following, we derive the expression of the theoretical densities that include the geometrical and dynamical characteristics of the tank-treading and flipping regimes, but also the dispersion in size and shape within a sample and the uncertainties associated with the shape characterizations.

Figure 5.

Probability density for the length of the projected shapes of RBCs, in the vorticity and flow direction, in the $\unicode[STIX]{x1D702}=25~\text{mPa}~\text{s}$ solution, for $\dot{\unicode[STIX]{x1D6FE}}=200~\text{s}^{-1}$ .

5.2 Tank-treading

Viewed from the shear axis direction, depending on the elongation of the cell (increasing with shear rate), but also on its angle $\unicode[STIX]{x03C0}/2-\unicode[STIX]{x1D719}$ compared with the flow direction (decreasing with shear rate), a tank-treading (or swinging) cell will appear as an ellipse whose long axis may be in the flow or in the vorticity direction. We therefore look for two populations of cells, in proportion $\unicode[STIX]{x1D6FD}$ and $1-\unicode[STIX]{x1D6FD}$ , with orientations 0 and $90^{\circ }$ . The theoretical distribution $d_{\unicode[STIX]{x1D6F9},TT}$ on $[0^{\circ };90^{\circ }]$ is then, in principle,

(5.1) $$\begin{eqnarray}d_{\unicode[STIX]{x1D6F9},TT}[\unicode[STIX]{x1D6FD}](\unicode[STIX]{x1D6F9})=\unicode[STIX]{x1D6FD}2\unicode[STIX]{x1D6FF}_{0}(\unicode[STIX]{x1D6F9})+(1-\unicode[STIX]{x1D6FD})2\unicode[STIX]{x1D6FF}_{90}(\unicode[STIX]{x1D6F9}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6F9}_{0}}$ is the Dirac distribution centred on $\unicode[STIX]{x1D6F9}_{0}$ .

Here and in the following, we place the parameters of the distribution inside square brackets $[-]$ while the studied variable is placed inside parentheses $(-)$ . For the aspect ratio, we note that the size distribution is large, as at rest: typical range for the long axis of a cell at rest is 7– $9~\unicode[STIX]{x03BC}\text{m}$  (see Canham & Burton Reference Canham and Burton1968). In addition, the distribution of apparent cell length in the flow direction is not symmetric but has a shorter tail on the large values side (see figure 5). The distribution of width in the vorticity direction is not symmetric either, with a shorter tail on the low value side. This may be due to many factors: uneven distribution of the deformability among the cells (see Dobbe et al. Reference Dobbe, Streekstra, Hardeman, Ince and Grimbergen2002b ), saturation of deformation of a given cell under shear (it would not be possible to elongate a cell and shrink it in the vorticity direction beyond a given threshold), a result of swinging dynamics that favours some projected lengths (not valid for width). For the aspect ratio $r_{zx}=$ (flow direction axis/vorticity direction axis) we shall therefore expect an asymmetric distribution (see Dobbe et al. Reference Dobbe, Hardeman, Streekstra, Strackee, Ince and Grimbergen2002a ). It appeared that the skew-normal distribution, defined by

(5.2) $$\begin{eqnarray}p_{sG}[r_{T},w_{T},\unicode[STIX]{x1D6FC}](r_{zx})\propto \left(1+\text{erf}\left(\frac{\unicode[STIX]{x1D6FC}(r_{zx}-r_{T})}{\sqrt{2}w_{T}}\right)\right)\text{e}^{-(r_{zx}-r_{T})^{2}/(2w_{T}^{2})},\end{eqnarray}$$

works well. Here $\unicode[STIX]{x1D6FC}$ is an asymmetry parameter that will be negative in our case. In our general analysis, we chose to consider the aspect ratio $r_{a}$ between the long axis and the short axis of the cells, a parameter that is defined even for cells not parallel or perpendicular to flow. Here, we shall therefore consider the inverse aspect ratio for the cells with aspect ratio $r_{zx}$ lower than 1, hence the following theoretical distribution $d_{r_{a},TT}$ for $r_{a}$ :

(5.3) $$\begin{eqnarray}d_{r_{a},TT}[r_{T},w_{T},\unicode[STIX]{x1D6FC}](r_{a})=\unicode[STIX]{x1D6E9}(r_{a}-1)\left[p_{sG}[r_{T},w_{T},\unicode[STIX]{x1D6FC}](r_{a})+\frac{1}{r_{a}^{2}}p_{sG}[r_{T},w_{T},\unicode[STIX]{x1D6FC}]\left(\frac{1}{r_{a}}\right)\right],\end{eqnarray}$$

with $\unicode[STIX]{x1D6E9}$ the Heaviside function. The proportion $\unicode[STIX]{x1D6FD}$ of cells whose projection is aligned with the flow (5.1), is equal to $\int _{1}^{\infty }p_{sG}[r_{T},w_{T},\unicode[STIX]{x1D6FC}](r_{a})\,\text{d}r_{a}$ .

5.3 Flipping motion

Although cells are not exactly ellipsoids, we assume that the dynamics of a flipping cell may still be described by the Jeffery orbit of an ellipsoid having the same aspect ratio $r$ as the convex envelope of the cell, as in Anczurowski & Mason (Reference Anczurowski and Mason1967).

In experiments, the optical axis is $y^{\prime }$ and we visualize its projection on the $x^{\prime }z^{\prime }$ plane (see figure 1 for the notation and figure 2 for an example). The relationship between the coordinates $(x,y,z)$ in the moving $Oxyz$ coordinate system and $(x^{\prime },y^{\prime },z^{\prime })$ in the $Ox^{\prime }y^{\prime }z^{\prime }$ coordinate system is, according to the convention used here,

(5.4) $$\begin{eqnarray}\left(\begin{array}{@{}c@{}}x\\ y\\ z\end{array}\right)=\left[\left(\begin{array}{@{}ccc@{}}1 & 0 & 0\\ 0 & \cos \unicode[STIX]{x1D719} & -\text{sin}\unicode[STIX]{x1D719}\\ 0 & \sin \unicode[STIX]{x1D719} & \cos \unicode[STIX]{x1D719}\\ \end{array}\right)\left(\begin{array}{@{}ccc@{}}\cos \unicode[STIX]{x1D703} & -\text{sin}\unicode[STIX]{x1D703} & 0\\ \sin \unicode[STIX]{x1D703} & \cos \unicode[STIX]{x1D703} & 0\\ 0 & 0 & 1\\ \end{array}\right)\right]^{-1}\left(\begin{array}{@{}c@{}}x^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right).\end{eqnarray}$$

Equations (2.1) and (5.4) yield the equation of the ellipsoid in the $Ox^{\prime }y^{\prime }z^{\prime }$ coordinate system. This is a quadratic equation in $x^{\prime }$ , $y^{\prime }$ and $z^{\prime }$ , with parameters $r$ , $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D719}$ . The contour of the projection of the ellipsoid along the $y^{\prime }$ axis corresponds to the $x^{\prime }z^{\prime }$ points for which the ellipsoid equation has exactly one solution for $y^{\prime }$ . It yields the quadratic equation of an ellipse for $x^{\prime }$ and $z^{\prime }$ , which can be written as $A(r,\unicode[STIX]{x1D703},\unicode[STIX]{x1D719}){x^{\prime }}^{2}+B(r,\unicode[STIX]{x1D703},\unicode[STIX]{x1D719})x^{\prime }z^{\prime }+C(r,\unicode[STIX]{x1D703},\unicode[STIX]{x1D719}){z^{\prime }}^{2}=1$ .

The angle $\unicode[STIX]{x1D6F9}$ of the long axis relatively to flow direction $z^{\prime }$ and the aspect ratio (long axis/short axis) $r_{a}$ of this ellipse are given by

(5.5) $$\begin{eqnarray}\displaystyle & \displaystyle \cot \unicode[STIX]{x1D6F9}=(-A+C-\sqrt{(A-C)^{2}+B^{2}})/B, & \displaystyle\end{eqnarray}$$

(5.6) $$\begin{eqnarray}\displaystyle & \displaystyle r_{a}=\sqrt{(A+C+\sqrt{(A-C)^{2}+B^{2}})/(A+C-\sqrt{(A-C)^{2}+B^{2}})}. & \displaystyle\end{eqnarray}$$

For a given aspect ratio $r$ and orbit parameter $\unicode[STIX]{x1D703}_{0}$ we can calculate the trajectories $\unicode[STIX]{x1D6F9}(r,\unicode[STIX]{x1D703}_{0},t)$ and $r_{a}(r,\unicode[STIX]{x1D703}_{0},t)$ over one period and deduce the associated probability function for $\unicode[STIX]{x1D6F9}$ and $r_{a}$ , as the probability is inversely proportional to the time derivative of the considered parameter. For $r_{a}$ it yields a very complex function, which furthermore has to be integrated to normalize the probability, making the calculation unfeasible. For $\unicode[STIX]{x1D6F9}$ , thanks to the more direct (2.6), this can indeed be more easily done, and one finds, for $\unicode[STIX]{x1D6F9}$ cast onto $[0;90^{\circ }]$ , the probability density

(5.7) $$\begin{eqnarray}p_{\unicode[STIX]{x1D6F9}}[\unicode[STIX]{x1D703}_{0}](\unicode[STIX]{x1D6F9})=\frac{1+\tan ^{2}\unicode[STIX]{x1D6F9}}{90\tan \unicode[STIX]{x1D703}_{0}\sqrt{1-\tan ^{2}\unicode[STIX]{x1D6F9}/\tan ^{2}\unicode[STIX]{x1D703}_{0}}}\quad \text{if }\unicode[STIX]{x1D6F9}<\unicode[STIX]{x1D703}_{0},\text{ and 0 otherwise.}\end{eqnarray}$$

Interestingly, for a given $\unicode[STIX]{x1D703}_{0}$ , the apparent angle probability does not depend on $r$ . This could already be seen from the expression (2.6), where the aspect ratio $r$ only appears in the expression through the period.

To calculate the probability density $p_{r_{a}}$ for $r_{a}$ we chose to sample $r_{a}(r,\unicode[STIX]{x1D703}_{0},t)$ given by (5.6) over one period $T$ with time step $10^{-3}T$ and infer the associated probabilities $p_{r_{a}}[r,\unicode[STIX]{x1D703}_{0}](r_{a})$ with bin size 0.05 as for the experimental data, for all aspect ratios $r$ between $r_{min}=1.025$ and $r_{max}=4.975$ with step $\unicode[STIX]{x1D6FF}r=0.05$ and all orbit angles $\unicode[STIX]{x1D703}_{0}$ between 0 and $90^{\circ }$ with step $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{0}=0.25^{\circ }$ . This set of $80\times 361=28\,880$ discretized probability distributions will be used in the fitting procedure.

Following the premise of this modelling section, the expected distributions are eventually obtained by considering that the orbit angle lies between two extremal values $\unicode[STIX]{x1D703}_{0}^{-}$ and $\unicode[STIX]{x1D703}_{0}^{+}$ , with equal probabilities of sum $p_{s}$ or between $\unicode[STIX]{x1D703}_{0}^{+}$ and $90^{\circ }$ , with equal probabilities of sum $(1-p_{s})$ and that, because the cell population is not fully homogeneous, there can be some variation in their aspect ratio. We consider normal $p_{G}[r_{F},w_{F}](r)$ distribution for the cell aspect ratio (see Rodak, Fritsma & Doig Reference Rodak, Fritsma and Doig2007), with mean $r_{F}$ and standard deviation $w_{F}$ . We would thus seek a $\unicode[STIX]{x1D6F9}$ and $r_{a}$ distribution in the flipping regime

(5.8) $$\begin{eqnarray}\displaystyle & & \displaystyle d_{r_{a},F}[r_{F},w_{F},\unicode[STIX]{x1D703}_{0}^{-},\unicode[STIX]{x1D703}_{0}^{+},p_{s}](r_{a})=\mathop{\sum }_{r=r_{min}}^{r_{max}}p_{G}[r_{F},w_{F}](r)\unicode[STIX]{x1D6FF}r\nonumber\\ \displaystyle & & \displaystyle \quad \times \left(\frac{p_{s}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{0}}{\unicode[STIX]{x1D703}_{0}^{+}-\unicode[STIX]{x1D703}_{0}^{-}+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{0}}\mathop{\sum }_{\unicode[STIX]{x1D703}_{0}=\unicode[STIX]{x1D703}_{0}^{-}}^{\unicode[STIX]{x1D703}_{0}^{+}}p_{r_{a}}[r,\unicode[STIX]{x1D703}_{0}](r_{a})+\frac{(1-p_{s})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{0}}{90-\unicode[STIX]{x1D703}_{0}^{+}+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{0}}\mathop{\sum }_{\unicode[STIX]{x1D703}_{0}=\unicode[STIX]{x1D703}_{0}^{-}}^{90}p_{r_{a}}[r,\unicode[STIX]{x1D703}_{0}](r_{a})\right),\qquad\end{eqnarray}$$

(5.9) $$\begin{eqnarray}d_{\unicode[STIX]{x1D6F9},F}[\unicode[STIX]{x1D703}_{0}^{-},\unicode[STIX]{x1D703}_{0}^{+},p_{s}](\unicode[STIX]{x1D6F9})=\frac{p_{s}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{0}}{\unicode[STIX]{x1D703}_{0}^{+}-\unicode[STIX]{x1D703}_{0}^{-}+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{0}}\mathop{\sum }_{\unicode[STIX]{x1D703}_{0}=\unicode[STIX]{x1D703}_{0}^{-}}^{\unicode[STIX]{x1D703}_{0}^{+}}p_{\unicode[STIX]{x1D6F9}}[\unicode[STIX]{x1D703}_{0}](\unicode[STIX]{x1D6F9})+\frac{(1-p_{s})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{0}}{90-\unicode[STIX]{x1D703}_{0}^{+}+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{0}}\mathop{\sum }_{\unicode[STIX]{x1D703}_{0}=\unicode[STIX]{x1D703}_{0}^{+}}^{90}p_{\unicode[STIX]{x1D6F9}}[\unicode[STIX]{x1D703}_{0}](\unicode[STIX]{x1D6F9}).\end{eqnarray}$$

Sums on $r$ are made with step $\unicode[STIX]{x1D6FF}r$ and sums on $\unicode[STIX]{x1D703}_{0}$ use step $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{0}$ . Both steps are those used for the generation of the reference distributions.

5.4 Error function on angles

Experimental uncertainty on aspect ratio determination is considered by the use of (skew-)normal distributions.

The determination of the cell angle may lead to huge errors when the cell apparent aspect ratio is close to 1. If one considers the ellipse equation $Ax^{2}+Bxz+Cz^{2}=1$ with $A$ , $B$ and $C$ being normally distributed around $1$ , $0$ and $1+\unicode[STIX]{x1D716}$ , respectively, and variances of the order of 0.1 close to the experimental ones, one numerically finds an orientation angle distribution that is centred on 0 but with a long tail such that it is rather well described by a Cauchy distribution. As the considered angles are in the interval $[0;90]$ we will therefore consider, instead of $\unicode[STIX]{x1D6FF}$ functions, the Cauchy distribution folded onto this interval:

(5.10) $$\begin{eqnarray}\tilde{p}_{C}[\unicode[STIX]{x1D6F9}_{0},w_{\unicode[STIX]{x1D6F9}}](\unicode[STIX]{x1D6F9})=p_{C}[\unicode[STIX]{x1D6F9}_{0},w_{\unicode[STIX]{x1D6F9}}](\unicode[STIX]{x1D6F9})+p_{C}[\unicode[STIX]{x1D6F9}_{0},w_{\unicode[STIX]{x1D6F9}}](-\unicode[STIX]{x1D6F9})+p_{C}[\unicode[STIX]{x1D6F9}_{0},w_{\unicode[STIX]{x1D6F9}}](180-\unicode[STIX]{x1D6F9}),\end{eqnarray}$$

with $p_{C}[\unicode[STIX]{x1D6F9}_{0},w_{\unicode[STIX]{x1D6F9}}](\unicode[STIX]{x1D6F9})=1/(w_{\unicode[STIX]{x1D6F9}}\unicode[STIX]{x03C0}(1+((\unicode[STIX]{x1D6F9}-\unicode[STIX]{x1D6F9}_{0})^{2})/w_{\unicode[STIX]{x1D6F9}}^{2}))$ the Cauchy distribution function around $\unicode[STIX]{x1D6F9}_{0}$ .

For the tank-treading regime, we will thus consider the modified distribution function for the angle:

(5.11) $$\begin{eqnarray}\hat{d}_{\unicode[STIX]{x1D6F9},TT}[\unicode[STIX]{x1D6FD},w_{\unicode[STIX]{x1D6F9}}](\unicode[STIX]{x1D6F9})=\unicode[STIX]{x1D6FD}\tilde{p}_{C}[0,w_{\unicode[STIX]{x1D6F9}}](\unicode[STIX]{x1D6F9})+(1-\unicode[STIX]{x1D6FD})\tilde{p}_{C}[90,w_{\unicode[STIX]{x1D6F9}}](\unicode[STIX]{x1D6F9}).\end{eqnarray}$$

Here $w_{\unicode[STIX]{x1D6F9}}$ is a free parameter that is expected to increase when $\dot{\unicode[STIX]{x1D6FE}}$ decreases, to take into account the fact that the aspect ratio approaches 1.

For the flipping cells, the link between aspect ratio and angle is more complex. The apparent aspect ratio is maximal when $\unicode[STIX]{x1D6F9}=\unicode[STIX]{x1D703}_{0}$ (cell viewed from the edge, $\unicode[STIX]{x1D719}=\pm 90^{\circ }$ ). It is minimal when $\unicode[STIX]{x1D6F9}=0$ , but this minimal value depends on the orbit: it reaches 1 only for tumbling ( $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ ). Therefore, the uncertainty on a given angle depends on the orbit the cell is following. Thus, to continue handling the $r_{a}$ distribution and the $\unicode[STIX]{x1D6F9}$ distribution separately, we make the simplifying assumption that, for a given orbit, the width of the modified Cauchy function is 0 when the angle is equal to its maximum value $\unicode[STIX]{x1D703}_{0}$ and increases linearly with the distance between the angle and the orbit angle. We, thus, consider the modified distribution for one orbit:

(5.12) $$\begin{eqnarray}\hat{p}_{\unicode[STIX]{x1D6F9}}[\unicode[STIX]{x1D703}_{0},w_{\unicode[STIX]{x1D6F9}}](\unicode[STIX]{x1D6F9})=\mathop{\sum }_{\unicode[STIX]{x1D709}=0}^{90}p_{\unicode[STIX]{x1D6F9}}[\unicode[STIX]{x1D703}_{0}](\unicode[STIX]{x1D709})\tilde{p}_{C}[\unicode[STIX]{x1D709},w_{\unicode[STIX]{x1D6F9}}|\unicode[STIX]{x1D709}-\unicode[STIX]{x1D703}_{0}|/90](\unicode[STIX]{x1D6F9}).\end{eqnarray}$$

Here, $w_{\unicode[STIX]{x1D6F9}}$ is the width of the error distribution when the aspect ratio is close to 1 ( $\unicode[STIX]{x1D703}_{0}=90^{\circ }$ , $\unicode[STIX]{x1D709}=0$ ). For simplicity, we shall consider the same $w_{\unicode[STIX]{x1D6F9}}$ for both tank-treading and flipping distributions. Both tank-treading and flipping populations will coexist in the highly viscous solution. At high shear rate, tank-treading cells have aspect ratio far from 1 so $w_{\unicode[STIX]{x1D6F9}}$ will be smaller than the expected value for the flipping cells, but in that case, there are few of latter. As $\dot{\unicode[STIX]{x1D6FE}}$ decreases, flipping cells will appear but in the meantime the apparent aspect ratio of tank-treading cells will be close to 1, and the $w_{\unicode[STIX]{x1D6F9}}$ values should coincide reasonably.

Finally, the modified distribution function for the flipping regime is

(5.13) $$\begin{eqnarray}\displaystyle \hat{d}_{\unicode[STIX]{x1D6F9},F}[\unicode[STIX]{x1D703}_{0}^{-},\unicode[STIX]{x1D703}_{0}^{+},p_{s},w_{\unicode[STIX]{x1D6F9}}](\unicode[STIX]{x1D6F9}) & = & \displaystyle \frac{p_{s}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{0}}{\unicode[STIX]{x1D703}_{0}^{+}-\unicode[STIX]{x1D703}_{0}^{-}+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{0}}\mathop{\sum }_{\unicode[STIX]{x1D703}_{0}=\unicode[STIX]{x1D703}_{0}^{-}}^{\unicode[STIX]{x1D703}_{0}^{+}}\hat{p}_{\unicode[STIX]{x1D6F9}}[\unicode[STIX]{x1D703}_{0},w_{\unicode[STIX]{x1D6F9}}](\unicode[STIX]{x1D6F9})\nonumber\\ \displaystyle & & \displaystyle +\,\frac{(1-p_{s})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{0}}{90-\unicode[STIX]{x1D703}_{0}^{+}+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{0}}\mathop{\sum }_{\unicode[STIX]{x1D703}_{0}=\unicode[STIX]{x1D703}_{0}^{+}}^{90}\hat{p}_{\unicode[STIX]{x1D6F9}}[\unicode[STIX]{x1D703}_{0},w_{\unicode[STIX]{x1D6F9}}](\unicode[STIX]{x1D6F9}).\end{eqnarray}$$

5.5 Complete fitting function

Considering that we have two populations of cells, one in tank-treading-like regime with proportion $p_{T}$ and one in flipping regime in proportion $(1-p_{T})$ , we finally have the following fitting functions for the distributions in apparent aspect ratio $r_{a}$ and apparent angle $\unicode[STIX]{x1D6F9}$ , respectively, defined on $[1;+\infty ]$ and $[0;90]$ , with 10 fitting parameters:

(5.14) $$\begin{eqnarray}\displaystyle & & \displaystyle d_{r_{a}}[p_{T},r_{T},w_{T},\unicode[STIX]{x1D6FC},r_{F},w_{F},\unicode[STIX]{x1D703}_{0}^{-},\unicode[STIX]{x1D703}_{0}^{+},p_{s}](r_{a})\nonumber\\ \displaystyle & & \displaystyle \quad =p_{T}d_{r_{a},TT}[r_{T},w_{T},\unicode[STIX]{x1D6FC}](r_{a})+(1-p_{T})d_{r_{a},F}[r_{F},w_{F},\unicode[STIX]{x1D703}_{0}^{-},\unicode[STIX]{x1D703}_{0}^{+},p_{s}](r_{a}),\end{eqnarray}$$

with $d_{r_{a},TT}$ given by (5.3) and $d_{r_{a},F}$ by (5.8), and

(5.15) $$\begin{eqnarray}\displaystyle & & \displaystyle d_{\unicode[STIX]{x1D6F9}}[p_{T},r_{T},w_{T},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}_{0}^{-},\unicode[STIX]{x1D703}_{0}^{+},p_{s},w_{\unicode[STIX]{x1D713}}](\unicode[STIX]{x1D6F9})\nonumber\\ \displaystyle & & \displaystyle \quad =p_{T}\hat{d}_{\unicode[STIX]{x1D6F9},TT}[\unicode[STIX]{x1D6FD},w_{\unicode[STIX]{x1D6F9}}](\unicode[STIX]{x1D6F9})+(1-p_{T})\hat{d}_{\unicode[STIX]{x1D6F9},F}[\unicode[STIX]{x1D703}_{0}^{-},\unicode[STIX]{x1D703}_{0}^{+},p_{s},w_{\unicode[STIX]{x1D6F9}}](\unicode[STIX]{x1D6F9}),\nonumber\\ \displaystyle & & \displaystyle \qquad \quad \text{with }\unicode[STIX]{x1D6FD}=\int _{1}^{\infty }p_{sG}[r_{T},w_{T},\unicode[STIX]{x1D6FC}](r_{a})\,\text{d}r_{a}.\end{eqnarray}$$

Here $\hat{d}_{\unicode[STIX]{x1D6F9},TT}$ is given by (5.11) and $\hat{d}_{\unicode[STIX]{x1D6F9},F}$ by (5.13).

The fitting parameters are recalled in table 1. For each shear rate, we minimize the distance between the theoretical and experimental distribution functions for $r_{a}$ and $\unicode[STIX]{x1D6F9}$ , given by

(5.16) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D716} & = & \displaystyle \mathop{\sum }_{\unicode[STIX]{x1D6F9}}|d_{\unicode[STIX]{x1D6F9},exp}(\unicode[STIX]{x1D6F9})-d_{\unicode[STIX]{x1D6F9}}[p_{T},r_{T},w_{T},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}_{0}^{-},\unicode[STIX]{x1D703}_{0}^{+},p_{s},w_{\unicode[STIX]{x1D713}}](\unicode[STIX]{x1D6F9})|\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{r_{a}}|d_{r_{a},exp}(r_{a})-d_{r_{a}}[p_{T},r_{T},w_{T},\unicode[STIX]{x1D6FC},r_{F},w_{F},\unicode[STIX]{x1D703}_{0}^{-},\unicode[STIX]{x1D703}_{0}^{+},p_{s}](r_{a})|\unicode[STIX]{x1D6FF}r_{a}.\end{eqnarray}$$

Because they appear as bounds in sums, the parameters $\unicode[STIX]{x1D703}_{0}^{-}$ and $\unicode[STIX]{x1D703}_{0}^{+}$ are treated differently from the other parameters. For a given choice of these angles, we minimize $\unicode[STIX]{x1D716}$ using the NMinimize function of Mathematica© software. We then explore systematically the region of interest for these angles so as to find the global minimum.