Instability behavior of odd viscosity-induced viscous fluid over a vibrating bed

Abstract

The manuscript focuses on the theoretical stability analysis of the viscous liquid over a vibrating inclined rigid bed when the fluid undergoes an impact of odd viscosity. Such an impact emerges in the classical fluid owing to the broken time-reversal symmetry. The rigid bottom vibrates in streamwise and cross-stream directions. The time-dependent Orr-Sommerfeld eigenvalue problem is obtained using the normal mode approach and resolved based on the Chebyshev-collocation method and the Floquet theory. The effect of the odd viscosity coefficient on the different types of instability gravitational, subharmonic, and harmonic are identified. The gravitational instability arises in the longwave region, whereas the resonated wave instability appears in the finite wavenumber region. The gravitational instability is generated in the fluid flow owing to the gravity driving force, whereas the subharmonic instability appears for lower forcing amplitude and the harmonic instability emerges for comparatively higher forcing amplitude. It is found that the subharmonic and harmonic resonances appear when the forcing amplitude surpasses its critical value. A higher odd viscosity leads to stabilizing the gravitational instability, whereas a larger odd viscosity diminishes the subharmonic resonance along with the harmonic resonance instigated at a high forcing amplitude. Further, a new instability, named shear instability in the finite wavenumber range emerges together with the aforementioned three instabilities when the Reynolds number is sufficiently high with a low angle of inclination and becomes weaker when the time-reversal symmetry breaks.