Marangoni convection of a viscous fluid over a vibrating plate

This research presents a new insight into Marangoni convection through investigating, both numerically and analytically, the surface tension driven instability activated by a coupled effect of a vibrating plate and viscous dissipation. A horizontal, thin fluid layer is bounded from below by an impermeable, adiabatic plate that vibrates in the horizontal direction. The upper boundary is modelled by a free surface subject to a thermal boundary condition of the third kind (Robin). The internal heat generation due to viscous dissipation yields a vertical, potentially unstable temperature gradient. The linear stability analysis of the stationary terms of the basic state is performed. The perturbed flow, in the form of plane waves, is superimposed onto the basic state. The obtained system of ordinary differential equations is solved numerically by means of the Runge-Kutta method coupled with the shooting method. For the two limiting cases, the isothermal upper boundary and adiabatic upper boundary, the analytical solutions of the eigenvalue problem are obtained. The values of the critical parameter, which identifies the threshold for the onset of Marangoni convection, are presented.