9 0 0,5 1 1,5 2 0 0,1 0,2 0,3 0,4 0,5 0,6 φ$^{0}$ Experiment,  =0.15 Experiment,  =0.24 Experiment,  =0.42 Present Theory,  =0.15 Present Theory,  =0.24 Present Theory,  =0.42 Lebedev  et al.  Theory TT inclination angle for C$_{a}$=100 Figure 6: inclination angle of vesicles in TT phase for large C$_{a}$. + b$_{5}$∂$_{i}$f$_{2}$∂$_{ij}$E$_{2}$∂$_{j}$f$_{2}$ + b$_{6}$E$_{2}$f$_{4}$ ) + O ( ε $^{3}$) , F$_{4}$ = ε ( c$_{1}$f$_{4}$ + c$_{2}$f 2 $_{2}$) + ε $^{2}$( c$_{3}$f 3 2 + c$_{4}$f$_{2}$f$_{4}$ )+ + εd$_{1}$E$_{2}$f$_{2}$ + ε $^{2}$( d$_{2}$f$_{4}$E$_{2}$ + d$_{3}$f 2 $_{2}$E$_{2}$ + d$_{5}$f$_{2}$∂$_{i}$E$_{2}$∂$_{i}$f$_{2}$ )+ O ( ε $^{3}$) . The coefficients are then written as a$_{1}$ = − 24 Z$_{0}$ + 6¯ κ 23 λ + 32 a$_{2}$ = 24 (49 λ + 136) Z$_{0}$ + (432 λ + 1008)¯ κ (23 λ + 32) 2 b$_{1}$ = 120 23 λ + 32 b$_{2}$ = 2400 λ − 2 (23 λ + 32) 2 b$_{6}$ = − 40 241 λ + 344 (23 λ + 32) 2 c$_{1}$ = − 40 Z$_{0}$ + 20¯ κ 19 λ + 20 c$_{2}$ = 16 3 $^{(92}$− 3 λ ) Z$_{0}$ + (1822 λ + 3112)¯ κ (19 λ + 20)(23 λ + 32) d$_{1}$ = 20 3 1047 λ + 1072 (23 λ + 32)(19 λ + 20) d$_{2}$ = 10 257 λ − 32 (23 λ + 32)(19 λ + 20) d$_{3}$ = 1 $_{6}$− 125055 λ $^{3}$+ 594716 λ $^{2}$+ 1107168 λ + 392320 (2 λ + 5)(23 λ + 32)(19 λ + 20) 2 a$_{3}$ = − 8 175 (2479595 λ $^{3}$+ 6703156 λ $^{2}$+ 18601472 λ + 16622592 (23 λ + 32) $^{3}$(19 λ + 20) $^{Z$_{0}$}$− − 8 175 78181390 λ $^{3}$+ 390845256 λ $^{2}$+ 713624832 λ + 429094912 (23 λ + 32) $^{3}$(19 λ + 20) ¯ κ F$_{2}$ = a$_{1}$f$_{2}$ + εa$_{2}$f 2 $_{2}$+ ε $^{2}$( a$_{3}$f 3 $_{2}$+ a$_{4}$f$_{2}$f$_{4}$ )+ b$_{1}$E$_{2}$ + εb$_{2}$E$_{2}$f$_{2}$ + + ε $^{2}$( b$_{3}$f$_{2}$∂$_{i}$f$_{2}$∂$_{i}$E$_{2}$ + b$_{4}$∂$_{i}$f$_{2}$∂$_{ij}$f$_{2}$∂$_{j}$E$_{2}$ + phase border in a quite large range of the capillary num- ber. We hope that this work will incite future experimental research. It will also be interesting to investigate ex- perimentally the presence of widening of the VB band as a function of shear rate. Finally, we did not in- clude third order spherical harmonics in the expansion of the shape function, restricting our calculations only to centro-symmetric vesicles. In the full numerical simu- lation [16], no symmetry restriction is imposed and so far no manifestation of the third order harmonic has been observed. This confers to our assumption of centro- symmetry a certain legitimacy. Acknowledgements: We would like to thank G. Danker, S. S. Vergeles, and P. M. Vlahovska for helpful discussions. A.F. and C.M. acknowledge financial sup- port from CNES (Centre National D’études Spatiales) and ANR (Agence Nationale pour la Recherche); "MOSI- COB project". Appendix A: The shape evolution equations and the expression of various coefficients in terms of physical parameters We remind that