8 0 5 10 15 20 S 1 1,5 2 2,5 Present Theory,  =0.43 Present Theory,  =1 Lebedev  et al.   Theory ~ ~ Figure 4: Phase diagrams of vesicle motions under general planar linear flows. The viscosity contrast λ is fixed to 1 and the ratio of rotating and straining parts of the flow ω$_{0}$/s$_{0}$ is varied. The theory of Lebedev et al. [11] is plotted in dotted lines for comparison. 0 5 10 15 20 S 1 1,5 2 2,5 3 Present Theory,  $_{0}$=1/2 (simple shear flow) Present Theory,  $_{0}$=3/4 Present Theory,  $_{0}$=5/4 Lebedev  et al.   Theory, arbitrary  0 =0.43, s$_{0}$=1/2 ~ ~ Figure 5: Phase diagrams of vesicles under general planar linear flows. The straining and rotational parts of the flow are denoted as s$_{0}$ and ω$_{0}$ respectively. grams for various values of Δ is indeed not as striking as for the simple shear flow (Fig.3): the TT phase borders fall close to each other and VB bands overlap for a wide range of Δ . At the same time, the VB/TB phase borders still vary significantly with the excess area. However, by fixing Δ and varying λ we can produce series of phases diagrams in the ˜ S and ˜ Λ plane corresponding to different values of ω$_{0}$/s$_{0}$, (see Fig.5). Here we find the same kind of discrepancy as on Fig.3 . Finally, let us compare our results with other set of ex- periments in the TT regime[13]. We represent the vesicle orientation angle φ$_{0}$ under strong shear flow(Fig. 3) (for the sake of comparison with experiments which were per- formed at high shear rates). It can be checked that the exact form of the bending energy becomes insignificant under strong flows, because the dominant contribution to the membrane force comes form the tension part which enforces incompressibility of the membrane. Unlike for the VB/TB phase border, the dependence of φ$_{0}$ on λ quickly saturates with C$_{a}$. We take C$_{a}$ = 100 in our cal- culation. We use the artificial parameter Λ (20) instead of λ in order to show the effect of O ( ϵ $^{2}$) terms taken into account by the present calculation. Note that the results of Lebedev it et al. theory [11] do not depend on the excess area Δ in this representation, unlike our results and those reported by experiments (see Fig. 5). The re- sults of HOT depend also on Δ , but that dependence is weak so that they hardly differ from theoretical results of Ref.[11] (we do not plot them here in order to avoid encumbering of the figure). Experimental results[13] are provided for four different values of Δ : 0 . 15 , 0 . 24 , 0 . 42 , and 1 . 43 . We exclude Δ = 1 . 43 , since we do not expect our theory to be applicable to a such large value of excess area. As can be seen, theoretical results and experimen- tal ones show a rather good agreement provided that one is not close to the TT/VB border (i.e. if the angle is not too close to zero). The discrepancy for small incli- nation angles may be attributed to thermal fluctuations or interactions with walls. However, a systematic theo- retical study of these factors should be performed before drawing conclusive answers. Other experiments[15] were performed under linear flow with applied velocity field u$_{x}$ = ( s$_{0}$ + ω$_{0}$ ) y, u$_{y}$ = ( s$_{0}$ − ω$_{0}$ ) x, u$_{z}$ = 0 , which is different from a simple shear flow. The experiments were performed with no viscos- ity contrast and the ratio of the rotational ( ω$_{0}$ ) and the straining ( s$_{0}$ ) parts of the flow was varied instead. The results of the experiments were plotted in coordinates Here we have presented a small deformation theory of a vesicle in shear flow keeping one more order in the expan- sion of the shape evolution equations than prior studies. We have confirmed the leading term in the asymptotic expansion of critical viscosity ratios determined by previ- ous analytical studies, but we also were able to determine accurately the next term in the expansion. This term is constant and survives no matter how small the excess area is. Moreover, this theory provides a correction to the phase borders that is significant in a wide range of excess areas. Unlike previous analytical works, but in agreement with the results of the numerical simulations [16], we observe an unsaturated growth of the VB/TB representing the generalization of (20). It was then claimed [15] that the phase diagram depends on these two parameters only (at least with no viscosity contrast). The Fig.4 simulates this experiment using the present theory. The discrepancy between resulting phase dia- V. DISCUSSION AND CONCLUSION ˜ Λ = 4 ω$_{0}$ √$_{30}$$_{πs}$$_{0}$ ( 1 + 23 32 λ ) √$_{Δ}$$_{,}$ ˜ S = 14 π $_{3}$√$_{3}$ s$_{0}$ κ Δ (21)