Wall-mode instability in plane shear flow of viscoelastic fluid over a deformable solid.

Abstract

The linear stability analysis of a plane Couette flow of an Oldroyd-B viscoelastic fluid past a flexible solid medium is carried out to investigate the role of polymer addition in the stability behavior. The system consists of a viscoelastic fluid layer of thickness R, density ρ, viscosity η, relaxation time λ, and retardation time βλ flowing past a linear elastic solid medium of thickness HR, density ρ, and shear modulus G. The emphasis is on the high-Reynolds-number wall-mode instability, which has recently been shown in experiments to destabilize the laminar flow of Newtonian fluids in soft-walled tubes and channels at a significantly lower Reynolds number than that for flows in rigid conduits. For Newtonian fluids, the linear stability studies have shown that the wall modes become unstable when flow Reynolds number exceeds a certain critical value Re(c) which scales as Σ(3/4), where Reynolds number Re=ρVR/η,V is the top-plate velocity, and dimensionless parameter Σ=ρGR(2)/η(2) characterizes the fluid-solid system. For high-Reynolds-number flow, the addition of polymer tends to decrease the critical Reynolds number in comparison to that for the Newtonian fluid, indicating a destabilizing role for fluid viscoelasticity. Numerical calculations show that the critical Reynolds number could be decreased by up to a factor of 10 by the addition of small amount of polymer. The critical Reynolds number follows the same scaling Re(c)∼Σ(3/4) as the wall modes for a Newtonian fluid for very high Reynolds number. However, for moderate Reynolds number, there exists a narrow region in β-H parametric space, corresponding to very dilute polymer solution (0.9≲β<1) and thin solids (H≲1.1), in which the addition of polymer tends to increase the critical Reynolds number in comparison to the Newtonian fluid. Thus, Reynolds number and polymer properties can be tailored to either increase or decrease the critical Reynolds number for unstable modes, thus providing an additional degree of control over the laminar-turbulent transition.