Stability of two-layer Newtonian plane Couette flow past a deformable solid layer

Abstract

The linear stability of two-layer plane Couette flow of Newtonian fluids (designated by labels A and B) of thicknesses (1−β)R and βR, and viscosities μa and μb past a soft, deformable linear viscoelastic solid of thickness HR, shear modulus G, and viscosity ηw is determined using a combination of low wavenumber asymptotic analysis and a numerical method. There are two qualitatively different interfacial modes in this system, viz., the two-fluid interfacial mode due to viscosity stratification [“mode 1;” C. S. Yih “Instability due to viscosity stratification,” J. Fluid Mech. 27, 337 (1967)], and the fluid–solid interfacial mode [“mode 2;” Kumaran, Fredrickson, and Pincus, “Flow induced instability of the interface between a fluid and a gel at low Reynolds number,” J. Phys II 4, 893 (1994)]. The respective effects of solid layer deformability and fluid viscosity stratification on mode 1 and mode 2 are analyzed in detail using both asymptotic and numerical methods. Results of our low wavenumber asymptotic analysis show that the deformability of the solid layer has a dramatic effect on the interfacial instability (mode 1) between the two Newtonian fluids: When the more viscous fluid is of smaller thickness (an unstable configuration for the two-fluid mode 1 instability), the solid layer could completely stabilize the two-fluid interfacial instability, when the nondimensional elasticity parameter Γ=Vμb/(GR) increases beyond a critical value. Here V is the dimensional velocity of the top moving plate. When the more viscous fluid is of larger thickness compared to the less viscous fluid (a stable configuration in rigid channels), it is shown that the solid layer could destabilize or stabilize the two-fluid interfacial mode, depending on the solid layer thickness H. Numerical results at finite values of wavenumber k reveal that the stabilization of the two-fluid interfacial mode predicted by the low wavenumber analysis extends to moderate values of k. For high values of k, the perturbations are localized near the two-fluid interface. Increase in Γ therefore does not have any effect on the high k unstable modes, which are stabilized by the presence of nonzero interfacial tension in the two-fluid interface. When Γ is further increased, the interfacial mode between fluid B and the solid layer becomes unstable. It is demonstrated here that the parameters Γ (representing the shear modulus of the solid), solid layer thickness H, and the solid layer viscosity ηw can be chosen such that both the interfacial modes are stabilized at all wavenumbers, for a fixed top plate velocity.