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.
- Creeping flow
- Deformable solids
- Interfacial instability
- Linear stability analysis
- Viscoelastic fluids