content_image/399716.jpg content_image/399718.jpg content_image/399721.jpg content_image/399722.jpg content_image/399715.jpg content_image/399717.jpg content_image/399719.jpg content_image/399720.jpg content_image/399724.jpg content_image/399723.jpg
The left hand side term of the equation is a special derivative of f$_{k,l}$ that naturally arises in a non-rotating coordinate system, and its definition is 

<img src='content_image/399716.jpg'>

To the order at which our expansion is performed, the rotational velocity of the vesicle is not equal to the vor- ticity of the imposed flow (unlike lower order calculations [8–11]), but rather it has a new contribution originating from the shape function 

Note that if only the first term Ω$_{i}$ is retained [8–11], then (16) coincides with the Jaumann derivative. The functions F$_{k}$ can be written as 

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The coefficients a$_{i}$, b$_{i}$, c$_{i}$, and d$_{i}$ are rational functions of λ, ¯ κ = ε/C$_{a}$, and Z$_{0}$. The exact expressions are listed in the appendix. 

## A. The analytical phase diagram and comparison with previous works 

The phase diagram for Δ = 0 . 43 is presented on the Fig.1. The value Δ = 0 . 43 is chosen for the sake of com- parison with available numerical data [16]. Also shown on that Figure are the results obtained in two previous theories[10, 11]. As can be seen the basic structure of the phase diagram bears similarity with previous ones. There are however some important distinctions. First, the phase borders are significantly shifted towards the region of higher viscosity contrasts. Second, the VB/TB transition curve does not saturate upon increasing shear rate (or C$_{a}$ ), rather it exhibits a striking widening. We have also compared the results with those obtained re- cently by full three dimensional simulations. A remark- able agreement between the two approaches is found (see comparison in Ref. [16]). 

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## III. THE PHASE DIAGRAM 

<img src='content_image/399724.jpg'>

## B. Basic reasons for necessity of higher order expansion 

Our first concern is to understand why the inclusion of the terms of order O ( ε $^{2}$) provides such a dramatic shift of the phase borders even though the shape is almost spher- ical (the relative excess area is only 0 . 43 / (4 π ) = 0 . 034 ). We have studied the evolution of the phase diagram as a function of Δ in order to gain further insight. We have tracked the viscosity contrast λ at which the loss of sta- bility of TT motion occurs in the two asymptotic limits C$_{a}$ → 0 (the corresponding value is denoted as λ$_{c}$$_{0}$ ) and C$_{a}$ = ∞ (denoted as λ$_{c}$ ∞ ). It was reported in [11] that λ$_{c}$ ∝ ε − $^{1}$. Since we expect the expansions of λ$_{c}$ to be an- alytic in ε, we have thus attempted the following ansatz: 

To check the validity of this expansion we plot these crit- ical values as a function of Δ − 1 / 2 = 1 /ε. We expect then a linear behavior. This is presented on Fig.2 where we also compare our results with those of previous theories. A key point to be discussed further below is the fact that previous theories provide correct values for the dominant term λ ( − 1) c in the expansions (18), but not for λ (0) c which does not vanish even in the limit of almost spherical vesi- cles (note that the next order terms in the expansion (18) tend to 0 with the excess area). Because λ (0) c remains fi- nite whatever small is the deviation from a sphere, the discrepancy between the TT phase borders obtained by different theories (with one lower order below the present one, namely of the order of O ( ε ) ) persists for any excess area. In contrast the present theory valid to order O ( ϵ $^{2}$) , has the property that even if one wishes to make an ex- pansion to the next order (i.e. order O ( ϵ $^{3}$) ), then the shift in the borders in the phase diagram would be negli- 

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