Stability of a liquid film on inclined flexible substrates: Effect of the spontaneous odd viscosity

We report the stability of a falling incompressible odd viscosity fluid on flexible substrates when the time reversal symmetry is broken. The flexible wall equation incorporates the contribution of odd viscosity, where the stress at an interface is determined by the viscosities of the adjacent fluids. The Orr-Sommerfeld (OS) equation is derived using the modified linear flexible wall equation taking the inertia, flexural rigidity, and spring stiffness effects of the elastic plate into account. Here, we solve the above eigenvalue problem using Chebyshev collocation methods to obtain the neutral curve in the k-Re plane and the temporal growth rate under varying values of odd viscosity. There is an interesting finding that, for moderate Reynolds numbers, the presence of odd viscosity leads to an increase in instability when the stiffness coefficient AK is small. However, as the value of the stiffness coefficient AK rises, the effect of odd viscosity changes to suppress the onset of instability. Additionally, at higher Reynolds numbers and extremely small inclination angles, both shear and wall modes of falling film are observed. The results demonstrate that the unstable domain for the wall mode increases as the odd viscosity coefficient value rises, while an opposite trend occurs in the shear mode.