The stability of plane Couette flow of an upper-convected Maxwell (UCM) fluid of thickness R, viscosity η and relaxation time τR past a deformable wall (modeled here as a linear viscoelastic solid fixed to a rigid plate) of thickness HR, shear modulus G and viscosity ηw is determined using a temporal linear stability analysis in the creeping-flow regime where the inertia of the fluid and the wall is negligible. The effect of wall elasticity on the stable modes of Gorodtsov and Leonov [J. Appl. Math. Mech. 31 (1967) 310] for Couette flow of a UCM fluid past a rigid wall, and the effect of fluid elasticity on the unstable modes of Kumaran et al. [J. Phys. II (Fr.) 4 (1994) 893] for Couette flow of a Newtonian fluid past a deformable wall are analyzed. Results of our analysis show that there is only one unstable mode at finite values of the Weissenberg number, W=τRV/R (where V is the velocity of the top plate) and nondimensional wall elasticity, Γ=Vη/(GR). In the rigid wall limit, Γ≪1 and at finite W this mode becomes stable and reduces to the stable mode of Gorodtsov and Leonov. In the Newtonian fluid limit, W→0 and at finite Γ this mode reduces to the unstable mode of Kumaran et al. The variation of the critical velocity, Γc, required for this instability as a function of W̄=τRG/η (a modified Weissenberg number) shows that the instability exists in a finite region in the Γ c-W̄ plane when Γc>Γ c,Newt and W̄<W̄max, where Γc,Newt is the value of the critical velocity for a Newtonian fluid. The variation of Γc with W̄ for various values of H are shown to collapse onto a single master curve when plotted as ΓcH versus W̄/H, for H≫1. The effect of wall viscosity is analyzed and is shown to have a stabilizing effect.
Bibliographical noteFunding Information:
SK thanks the Shell Oil Company Foundation for support through its Faculty Career Initiation Funds program, and 3M for a Nontenured Faculty Award. Also, acknowledgment is made to the Donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research.
Copyright 2008 Elsevier B.V., All rights reserved.
- Creeping flow
- Deformable solids
- Interfacial instability
- Linear stability analysis
- Viscoelastic fluids