Implications of inertia for hydroelastic instability of Herschel-Bulkley fluids in plane Poiseuille flow

This study investigates the effects of inertia on the hydroelastic instability of a pressure-
-driven Herschel-Bulkley fluid passing through a two-dimensional channel lined with a po-
lymeric coating. The no-viscous hyperelastic polymeric coating is assumed to follow the
two-constant Mooney-Rivlin model. In this work, analytical basic solutions are determined
for both the polymeric gel and the fluid at very low Reynolds numbers. Next, the basic so-
lutions are subjected to infinitesimally-small, normal-mode perturbations. After eliminating
the nonlinear terms, two 4-th order differential equations are obtained. The equations with
appropriate boundary conditions are then numerically solved using the shooting method.
The results of the solution show that the inertia terms in the perturbed equations destabili-
ze the pressure-driven Herschel-Bulkley fluid flow. The investigation reveals that the elastic
parameter has a stabilizing effect on the flow. Also, based on the obtained results, the yield
stress, depending on the power-law index, has a stabilizing or destabilizing effect on the
flow. Since in this work the inertia terms are included in the pertinent governing equations,
therefore, the results of this study are much more realistic and reliable than previous works
in which inertia terms were absent. In addition, unlike the previous works, the present study
considers both the shear-thinning and shear-thickening types of fluids. Hence, the results of
this work embrace all the fluids which obey the Herschel-Bulkley model.