Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type

Non-uniqueness and linear stability of the one-dimensional flow of multiple viscoelastic fluids

“…The existence of the one-fluid Poiseuille/Couette flow for an interpolated Johnson-Segalman fluid has been extensively described by Guillopé and Saut [1]. Similarly to [2] they prove existence and uniqueness, for any pressure gradient (Poiseuille flow) or upper plate velocity (Couette flow), if the dimensionless polymer viscosity c = rj j)ot /( rj sal + rj jx)l ) is less than 8/9.…”

“…Remark 2.1 : In [8], we used local scales in order to have the same volumic équations as [1]. The major drawback of this method is that the jump conditions take the ratios of local scales into account.…”

“…We must notice that, as in the Johnson-Segalman model, the extrastress is unbounded with respect to y if a = ±l, and bounded if a ^ ± 1 (50) This seems to be an asset to these models in addition to their non-zero second normal stress différence This property remains m the axisymmetnc geometry Remark 4 2 Let us stress that the cases a < 0 and a > 0 give opposite conditions on e' This is due to the f act that the trace of T IS of the sign of a If a is négative, the 0 £ (T) term of the PTT model carf be very small and destabilize the differentiaT équation This disadvantage is more acute for the MPTT model, whose # £ (T) term can even get négative To solve these problems, we propose the Tollowing constitutive équation depending on the second invariant of T In Section 4, we have given sufficient conditions to ensure the uniqueness of some stationary solutions of (47), that do always exist As is explained m [1] and [10], in the case of the Navier-Stokes équations in a bounded domam, we know that the nonlmear stability is given by the hnear stabihty, which occurs if and only if the spectrum of the lmeanzed stationary operator is on the nght side of the ïmagmary axis Recent results of M Renardy [14] have brought a new ïnsight on Couette flows of viscoelastic fluids and proved under weak assumptions that the pnnciple of hnear stability holds…”

“…In the same article [1], they study the one dimensional Lyapunov stability of Couette flow and prove under some assumptions thc unconditional L 2 stability of Couette flow and the conditional //" stability. These results are based on projected équations and one dimensional perturbations, using standard a priori estimâtes methods.…”

“…1 is a strictly increasing function a 1 = a" from R to IR if f < 8/9 Thus, in a Poiseuille/Couette flow under the hypothesis e < 8/9, a unique choice of a leads to the fulfillmg of the upper boundary condition (13) If r > 8/9 and hes m the range of multiplicity (see [1]), any choice of 0, solution to (16) leads to multiple solutions in a 1 …”