Instability of creeping Couette flow past a neo-Hookean solid

Abstract

Fluid flow over a deformable solid can become unstable due to the fact that waves may propagate along the solid–fluid interface. In order to understand the role that nonlinear rheological properties of the solid play in these elastohydrodynamic instabilities, we apply linear stability analysis to investigate creeping Couette flow of a Newtonian fluid past an incompressible and impermeable neo-Hookean solid of finite thickness. As inertial effects are assumed to be negligible, the problem is governed by three dimensionless parameters: an imposed strain, a thickness ratio, and an interfacial tension. In the base state, there is a first normal stress difference in the neo-Hookean solid, and this leads to instability behavior that is significantly different from what is observed with a linear constitutive equation. In the absence of interfacial tension, the first normal stress difference gives rise to a shortwave instability. For sufficiently thin solids, a large range of high-wavenumber modes becomes unstable first as the strain that is imposed on the system increases, while for sufficiently thick solids, a small band of O(1) wavenumbers first becomes unstable. The presence of interfacial tension removes the shortwave instability and leads to larger critical imposed strains and smaller critical wavenumbers. In comparison to the linear elastic constitutive equation, the neo-Hookean model leads to smaller values of the critical imposed strain and larger values of the critical wavenumber, but the difference rapidly diminishes as the solid thickness increases. Analysis of the continuity of velocity boundary condition at the interface reveals that at the critical conditions, the mean flow tends to amplify horizontal interface perturbations, while horizontal velocity perturbations tend to suppress them. The results of this work highlight the importance of properly accounting for large displacement gradients when modeling elastohydrodynamic instabilities.

Manipulation of interfacial instabilities by using a soft, deformable solid layer

“…In this section, we extend the above low-k results to arbitrary wavenumbers to ensure whether the predicted suppression in long wave limit holds for finite and high wavenumber perturbations as well. Furthermore, flow past deformable solid surface (involving only a fluid-solid interface) could become unstable on increasing the deformability in the absence of inertia (Kumaran et al 1994;Gkanis & Kumar 2003), and several additional unstable fluid-solid modes proliferate when inertia is present (Chokshi & Kumaran 2008;Gaurav & Shankar 2009, 2010b. All these fluid-solid unstable modes are not captured by the low-k analysis presented in the previous section.…”

Stability of fluid flow through deformable tubes and channels: An overview

“…A similar instability was predicted in the creeping-flow limit for flow in a deformable tube by Kumaran (1995). Much later, Gkanis & Kumar (2003) pointed out that when the flow is predicted to be unstable, the nondimensional strain in the base state is an O(1) quantity, and hence the use of linearized elastic model for the solid (which assumes that the strain in the solid is 1) becomes inconsistent. They suggested the use of the frame-invariant neo-Hookean model to address this limitation.…”

“…In a similar manner, even for Newtonian fluid flow past a neo-Hookean solid medium, there is a jump in the first normal stress difference across the fluid-solid interface, since the first normal stress difference is absent in the Newtonian fluid. Gkanis & Kumar (2003) first pointed out that this yields a short-wave instability much similar to that in the interface between two viscoelastic fluids.…”

“…It is possible to express the dynamical quantities and governing equations in the solid also in a consistent Eulerian fashion (Chokshi 2007;Choskshi & Kumaran 2008). Equivalently (following Gkanis & Kumar 2003), it is convenient to refer the governing equations for the solid in terms of a reference (Lagrangian) configuration, where the independent variables are the positions X = (X, Y, Z) of material particles in the reference (i.e. unstressed solid) configuration.…”

“…However, this is incorrect, since the linearization is carried out about a deformed base state, and if the base state deformation is not small, then non-trivial terms would arise in the stability analysis coupling the base-state deformation and fluctuations in the deformation field. The lacunae involved in using a linearized elastic model were first pointed out by Gkanis & Kumar (2003) in the context of plane Couette flow past a deformable solid layer. Much of the earlier work in this subject were based on the linearized elastic model (Yeo & Dowling 1987;Yeo 1988;Yeo et al 1994;Lucey & Peake 2003;Kumaran 1995Kumaran , 1998aShankar & Kumaran 1999, and it is important to examine whether the predictions obtained using the linearized elastic model would be modified by using a more accurate nonlinear model.…”