Stability of the plane shear flow of dilute polymeric solutions

Three variants of Oldroyd-B model are analyzed for stability of the base profile in plane Couette flow of dilute polymeric fluid at moderate Reynolds number. The stability to two-dimensional disturbances is analyzed for the linearized problem as well as the weakly nonlinear flow. We begin with the classical Oldroyd-B model with emphasis on the disturbances with axial wavenumber alpha similar to Re-1/2, where Re is the Reynolds number based on maximum velocity and channel width. For linearly stable flow, the finite amplitude stability is analyzed using the equilibrium flow method, wherein the nonlinear flow is assumed to be at the transition point. For the classical Oldroyd-B fluid, the threshold kinetic energy for the equilibrium wall mode disturbances is found to be higher for the viscoelastic fluid than for the Newtonian fluid. In the second variant, the Oldroyd-B model with additional artificial diffusivity is studied. In this model, the diffusion modes, not present in the classical Oldroyd-B model, are introduced. For large wavenumber disturbances, the diffusion modes become the slowest decaying modes in comparison to the wall modes. The threshold energy for the diffusive Oldroyd-B model is smaller than that for the Newtonian fluid. The third variant of the Oldroyd-B model accounts for the nonhomogeneous polymer concentration coupled with the polymeric stress field. While the base profile is linearly stable for the first two models, the nonhomogeneous Oldroyd-B fluid exhibits an instability in the linear analysis. The ``concentration mode'' becomes unstable when the fluid Weissenberg number exceeds a certain transition value. This mode of instability, driven by the stress-induced fluctuations in polymer number density, renders the uniform polymer concentration profile unstable leading to the well-known phenomenon of flow-induced demixing