On the weak viscous effect of the reflection and transmission over an arbitrary topography

Abstract

In this article, monochromatic viscous waves propagating over an arbitrary topography are studied, specifically the effect of bottom sliding coefficient and molecular viscosity. In the theoretical formulation, the perturbation approximation is directly applied to the Navier-Stokes equation and boundary conditions which are specified to correspond to realistic situations. Furthermore, the arbitrary topography is approximated using successive shelves separated by abrupt steps. On each shelf, the solution is represented in terms of reflection and transmission of monochromatic waves. Matching kinematic and dynamic conditions are implemented to interpret the solutions. The formulation of our modified plane-wave approximation can be analytically reduced to the traditional formulation by Lamb [Hydrodynamics, 6th ed. (Cambridge University Press, Cambridge, 1937)] when the bottom sliding coefficient and molecular viscosity are neglected. Next, the wave transformations over three different trenches are simulated. The results indicate that wave reflection and transmission are certainly affected by molecular viscosity especially in shallow or intermediate fluid depths. In addition, numerical results are in good agreement with existing theoretical results and laboratory experiments without considering bottom sliding coefficient and molecular viscosity.