7 D. Discussion of TB/VB transition IV. COMPARISON WITH EXPERIMENTS 0 2 4 6 8 10 12 14 16 18 20 S 1 2 3 =0.16 =0.43 =0.8 Lebedev  et al.   Theory We plan to dedicate a separate research to the proper- ties of different types of vesicle motion, so here we only briefly discuss further implication of the new theory on the TB/VB transition. Neglecting harmonics of orders higher than two, as done in previous theories, provides two major simplifications. First, the vesicle shape has three symmetry planes, which make the definition of the orientation angle obvious, and second the in-plane mo- tion is defined by only two independent variables. As a consequence, the dynamics relaxes with time to a cyclic motion (which degenerates into a point in the TT phase). Thus only a simple limit cycle can exist, and this shows that each of the VB and TB modes has its own region of existence in parameter space. The above assertions can not be made in general, and especially when the fourth order harmonics are included. It also turns out that near the transition region the vesicle tends to finish its TB quasi-cycles by assuming an oblate almost axisymmetric shape in the shear plane rather than an elongated shape perpendicular to the flow. For such shape the error in the definition of the orientation angle is very large. Figure 3: Phase diagrams for various values of Δ in S − Λ coordinates. Let us now discuss how each type of motion is de- termined from our evolution equations. We start with some initial values for f$_{kl}$ ( t ) and wait a ceratin time in- terval until the initial data are irrelevant. Then it was checked whether during one oscillation quasi-cycle f ( x ) has a maximum for x lying in the shear plane and perpen- dicular to the shear velocity. Despite some noise due to the aforementioned complications it is clearly seen that unlike with previous theories the VB/TB phase border does not saturate even for C$_{a}$ = 10 , and the VB region broadens with the increase of the capillary number (Fig. 1). Note that it was suggested recently [20] that higher orders of spherical harmonics can be excited close to the VB/TB phase border and this may cause some widen- ing of the VB phase region (but still saturation at large C$_{a}$ is found, unlike our theory). It can be checked that when Z$_{0}$ + 20 ϵ/C$_{a}$ < 0 , the fourth order harmonics are excited (i.e. the corresponding decay time becomes infi- nite to leading order). Here, we found that close to the VB/TB transition at some times during the oscillation the quantity exceeds 20. This is probably the reason why the VB/TB phase border remains unsaturated for much larger values than in previous analytical studies. It was proposed in [20], however, that excitation of higher harmonics was due to thermal fluctuations. This contrasts in spirit with our theory where the fourth order harmonic is excited as non-linear interaction of second harmonics and no refer- ence to temperature is required. Experiments have been performed recently regarding determination of the phase diagram of vesicle motions under shear flow[14]. They referred to a previous the- oretical work [11] which had suggested that the phase diagram should depend on only two independent dimen- sionless control parameters (and not three), namely It has been reported [14] (with a certain degree of uncer- tainty) that the experimental data were consistent with the fact that only the above two parameters determine the phase diagram. Our results do not comply with this report, as shown on Figure 3. Indeed, besides S and Λ , the excess area Δ plays an important role. Experiments mixed data for different Δ ’s in the plane ( S, Λ) . The band of the VB mode in experiments looks quite wide, and we believe that this reflects the sensitivity of the lo- cation of the VB band to excess area (in other words the experimental data may be viewed as juxtaposition of bands each representing a value of Δ ; see the example of Figure 3). It is hoped that a study representing each value of Δ will be performed in the future with the aim of making comparison with theory clearer. Furthermore, it must be noted that our estimates for the TT-VB phase borders are higher than those observed in experiments. This is not very surprising since tran- sients are found to be very long close to TT-VB transi- tion (as also discussed in our recent full numerical simu- lations [16]), and therefore a firm conclusion on the type of motion can not be made on the basis of the current experimental data, which are limited to only few periods of oscillation (while transient can exhibit up to hundreds of cycles as discussed recently [16]). In other words, this kind of long relaxation would convey the impression that dynamics is of VB type, whereas in reality it is a TT one. − Z$_{0}$C$_{a}$ ε (19) Λ = 4 √$_{30}$$_{π}$ ( 1 + 23 32 λ ) √$_{Δ}$$_{, S}$ = 7 π $_{3}$√$_{3}$ C$_{a}$ Δ (20)