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and the capillary number 

Here η$_{int,ext}$ are the viscosities of fluids inside and outside the membrane respectively, A is the surface area of the vesicle, r$_{0}$ = ( 3 V 4 π ) 1 / 3 is the radius of a sphere containing the same volume as the vesicle, and ˙ γ is the shear rate. It is convenient to choose r$_{0}$ as the unit of distance so that the volume of the vesicle is 

We also use η$_{ext}$ as the unit of viscosity and ˙ γ − 1 as the unit of time. 

Several studies have been recently dedicated to the determination of the phase diagram regarding different types of motion of an almost spherical vesicle in a shear flow. These studies were theoretical[8–11], experimental [12–15], and numerical [16, 17]. Those theoretical works treat the membrane deviation from a spherical shape as a perturbation ( ϵ = √$_{Δ}$ is the small expansion parame- ter), represented as a series of spherical harmonics. Then it is suggested that one or two first orders in the expan- sion of the shape evolution equations must be retained. Generally, the resulting equations differ only because the authors propose distinct rules of neglection. The Leading Order Theory[8] (LOT) keeps only the terms of order of the imposed flow and the effects of the vorticity in the final equations. This theory is precise up to O ( ϵ ) in the expansion scheme. Later studies have included higher or- der terms in the expansion (up to O ( ϵ $^{2}$)) . Lebedev et al. [11] included the next order terms in the membrane Hel- frich force (as compared to LOT), while another study (hereafter called Higher Order Theory[10] (HOT)) ac- counted for the next order terms not only in the Helfrich force but also in the hydrodynamic field as well. 

In the theories of Ref. [10] and Ref. [11], as well as in numerical studies based on dissipative particles dynamics [17], a saturation of the VB/TB phase border for large enough C$_{a}$ was suggested– i.e. the value of λ at which the transition from VB to tumbling occurs becomes almost independent on the capillary number (or equivalently on shear rate). 

Recent advances in three dimensional numerical com- putations based on the boundary integral formulation [16] have made it possible to study vesicle dynamics quantitatively. The results of this study show, that the loss of stability of the TT solution occurs at values of λ significantly higher than those predicted by analytical calculations[10, 11]. Furthermore, no saturation of the VB/TB phase border upon increasing shear rate (or C$_{a}$ ) was found. These results provide a new input that is worth understanding from the analytical point view. 

The main objective of this work is to investigate the reasons for the discrepancies between the results provided by the recent numerical simulations [16] and analytical 

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theories. It was pointed out in Ref.[11] that the critical values of λ at which the steady-state solution loses its stability, scales as O (Δ − 1 / $^{2}$) for a fixed shear rate. How- ever, this behaviour, which also agrees with the result of Keller and Skalak [18], does not provide the correct information about the next order terms, as will be seen here. The next correction turns out to cause a rather big shift of the phase borders. 

This peculiar behavior (i.e. that the critical λ diverges at vanishing Δ ) confers to the vesicle problem a special status: in the small Δ limit, one further order and only one in the expansion scheme in powers of Δ is needed as compared to previous studies. As a consequence, it will be shown, in particular, that the next order term, previously unaccounted for, survives whatever small the deviation from a sphere is. This is the source of devia- tion between the recent numerical work and the existing analytical theories. As will be seen inclusion of one more order in the expansion will allow us to extract a phase diagram that is in quantitative agreement with the re- sults of numerical simulations[16]. It will also be shown that fourth order spherical harmonics (in previous theo- ries only second order harmonics were included) can not be neglected, and especially beyond a certain shear rate. Indeed, it will be revealed that these harmonics give rise to a new important feature recently revealed in numerical simulations [16]. More precisely, the VB/TB phase bor- der does not saturate for fixed Δ when shear rate exceeds a certain value, rather a significant widening is revealed, in contrast to previous theoretical [10, 11], numerical [17] and experimental investigations [14]. 

## II. SHAPE EVOLUTION EQUATIONS 

## A. Basic definitions and the expansion scheme 

Here we perform the expansion of the shape evolution equations one order higher as compared to the most ad- vanced of previous papers[10]. Since a slightly deflated vesicle in the absence of flow takes a centro-symmetric shape, and the shear flow has a symmetry center in ev- ery point, we consider only symmetric shapes. 

Naturally, we take the origin of coordinates at the cen- ter of the vesicle moving with the velocity of the undis- turbed shear flow. That the vesicle moves with the same velocity as the applied shear flow is a consequence of the symmetry of the problem. We use the conventional parametrization of the vesicle membrane with reference points belonging to a sphere of radius unity 

The shape function f ( x ) is then expanded in spherical harmonics of x . Note that the shape depends also on time, but the temporal variable will be specified only when need be. Following Ref.[9, 10] we introduce a for- mal expansion parameter ε = √$_{Δ}$ which helps classifying 

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