Linear stability of shear-thinning fluids in deformable channels: Effect of inertial terms

In the present work, the effect of inertial terms is numerically investigated on the hydroelastic stability of shear-thinning fluids in pressure-driven flow between two parallel plates lined with a Mooney-Rivlin hyperelastic polymeric gel. Having obtained the basic flow for the fluid and the basic deformation for the solid, the vulnerability of the basic solutions are examined when subjected to infinitesimally-small, normal-mode perturbations. By dropping all nonlinear perturbation terms, an eigenvalue problem is obtained, in the form of two coupled fourth-order ODEs, which are solved using the shooting or spectral methods. Based on the numerical results obtained in this work, it is concluded that the Mooney-Rivlin gels having a negative first-normal-stress-difference are more stable than the neo-Hookean gels as far as hydroelastic instability in plane Poiseuille flow is concerned. While under creeping-flow conditions the effect of shear-thinning is predicted to be stabilizing, in inertial flows its effect is predicted to be stabilizing or destabilizing depending on the Reynolds number.