0 2 4 6 8 10 -1/2 0 10 20 30 40 50 Higher Order Theory Keller-Skalak Theory Present Theory 0 2 4 6 8 10 -1/2 0 10 20 30 40 50 Present Theory Higher Order Theory Theory λ ( − 1) c 0 λ (0) c 0 LOT[8] 8 $_{23}$√$_{30}$$_{π}$$_{(3}$$_{.}$$_{38)}$ − 32 $^{$_{23}$(}$− 1 . 39) HOT[10] 3 . 91 − 1 . 22 Lebedev et al. [11] 16 $_{23}$√$_{10}$$_{π}$$_{(3}$$_{.}$$_{90)}$ − 32 $^{$_{23}$(}$− 1 . 39) Keller and Skalak [18] 2 $_{3}$√$_{10}$$_{π}$$_{(3}$$_{.}$$_{74)}$ 73 $_{63}$(1 . 17) Present 3.89 1.54 6 Figure 2: λ$_{0}$ and λ ∞ as a function of Δ − 1 / $^{2}$. Table I: λ ( − 1) c 0 and λ (0) c 0 extracted from different theories. It should be noted that Lebedev et al.[11] theory provided only values for λ ( − 1) c 0 and we have formally extracted from their theory the value of λ (0) c 0 for comparison purpose. Theory λ ( − 1) c ∞ λ (0) c ∞ LOT[8] 8 $_{23}$√$_{30}$$_{π}$$_{(3}$$_{.}$$_{38)}$ − 32 $^{$_{23}$(}$− 1 . 39) HOT[10] 4 . 79 − 1 . 38 Lebedev et al. [11] 16 $_{23}$√$_{15}$$_{π}$$_{(4}$$_{.}$$_{78)}$ − 32 $^{$_{23}$(}$− 1 . 39) Present 4.76 1.13 Table II: λ ( − 1) c ∞ and λ (0) c ∞ extracted from different theories. It should be noted that Lebedev et al.[11] theory provided only values for λ ( − 1) c ∞ and we have formally extracted from their theory the value of λ (0) c ∞ for comparison purpose. gibly small, i.e. the shift would vanish in the small excess area limit. In other words, the present theory shows that there is a convergence of the small deformation scheme at order O ( ϵ $^{2}$) . This implies that there is no need to continue the expansion of the shape evolution equation beyond the order O ( ε $^{2}$) since this would neither notably modify the results for Δ < 0 . 43 nor significantly extend the applica- bility to higher values of Δ . Comparison of the present theory with numerical results (see comparison provided in Ref. [16]) shows a good agreement for Δ = 0 . 43 . (the only value explored to date by the numerical scheme). large excess area limit. As a consequence, the truncation (17) becomes illegitimate. Generally, we may assert that Δ < 1 is a good estimate for the applicability limits of the small deformation approximation. Some theories[8, 11, 18] allow for analytical extraction of the coefficients λ ( − 1) c and λ (0) c , and for HOT and the present theory the extraction is numerical by using the slope and offset of the lines on the Fig.2. The combined data are presented in the tables 1 and 2. As can be seen the dependencies shown on Fig.2 are fairly linear for Δ < 1 and are almost indistinguishable from straight lines for Δ < 0 . 5 . The region of 1 $_{/}$√$_{Δ}$ where the curves noticeably deviate from straight lines is wider for present theory than for HOT. This is most likely caused by the introduction of fourth order spherical harmonics, their amplitude is no more small compared to that of the second order harmonics for Δ > 1 . The ex- pression (9) becomes inapplicable, and as a result f$_{4}$$_{,l}$ ( t ) grow uncontrollably with time in solutions of the differ- ential equations. This problem can be fixed by expanding the excess area to the order O ( ε $^{4}$) , but the results of such patching are not trustworthy because they go beyond the precision of the differential equations. HOT and Lebedev et al. theories show a good agree- ment for the coefficients λ ( − 1) c with the present calcula- tion. This means, that expansion of the shape evolution equations to the order O ( ε ) was indeed sufficient in or- der to capture the correct value of λ ( − 1) c even for large values of C$_{a}$, where Lebedev et al. theory loses its ap- plicability. The present theory shows that the correct value of λ (0) c is captured thanks to the fact that the ex- pansion scheme is pushed one step further in the excess area. Keller-Skalak theory gives a slightly different result for λ ( − 1) c , the difference is caused by the fact they used a velocity field having a surface divergence of order O ( ε ) , whereas the membrane local inextensibility requires zero surface divergence instead. On the other hand Keller- Skalak theory provides a better estimate for the value of λ (0) c than the one proposed by HOT and this explains why its results are relatively close to the results of numerical simulations for Δ ∼ 1 . Note also that the second term in (17) tends to align the vesicle to the elongating direction of the straining part of the shear flow which makes the angle π/ 4 with the direction of the flow. In the TT phase close to the loss of the stability of the steady-state solution the vesi- cle is almost parallel to the flow. The effective vorticity (17) turns out to be less than Ω in the TT phase, and it leads to the increased stability of the steady-state so- lution. The correction to the vorticity is proportional to ε and thus increases with ϵ and dominates over Ω in the C. Discussion of the tank-treading phase borders