On the Heat Generation In a viscous Flow between a Horizontally and Vertically oscillating plane and a Fixed Plane

The heat generation in an unsteady flow between an oscillating plane and a fixed plane is studied theoretically. It is assumed that one of the two planes is oscillating in the horizontal and vertical directions. The flow is two dimensional and the fluid is viscous and incompressible. The equations governing the flow and the heat are analyzed taking the viscous dissipation into account. The coordinate in the direction along the surfaces of the planes is found to be separable in these equations. The governing equations depending on only two independent variables, the coordinate in the direction perpendicular to the plane surfaces and the time, are solved numerically using a finite difference method. It is found that, under the condition that the Prandtl number is very large, the oscillation combined in the horizontal and vertical directions strongly enhances the heat generation in the viscous flow. If the small vertical oscillation is superposed on the horizontal one, the temperature depends on the coordinate along the plane surface in the antisymmetrical form through the convection term in the energy equation. This dependency plays an important role in the heat generation. The mechanism on the heat generation due to the combined oscillation is physically clarified using an order-of-magnitude analysis.