Influence of Heat Defect on the Characteristics of a Two-layer Flow with the Hiemenz Type Velocity

Abstract

An exact solution is derived in the frame of the creeping flow model to describe thermocapillary convection in a two-layer system with heat defect when the heat is transferred through the interface. The solution is characterized by the Hiemenz-type velocity and temperature distribution which is quadratic in the longitudinal coordinate. The heat defect is connected with changes in the internal energy of the interface caused by the action of thermocapillary forces on the transformation of the area and shape of the surface. A model linear problem is studied to estimate the impact of this effect on the formation of typical flow regimes and stability of these regimes. There is only a nonlinear term in the energy balance condition at the interface corresponding to the heat defect in the model problem. Depending on the values of a parameter defining the character of thermal load on the lower boundary of the system this problem may not have any solution, or it may have one or two exact solutions obtained in an explicit form. In the frame of the linear theory the stability of one of these exact solutions is investigated both taking into account the heat defect and under classical condition of heat balance at the interface setting an equality of heat fluxes on this surface. The interface position and velocity and temperature perturbation fields are calculated. With the decrease of the liquid layer thickness the changes in the internal energy of the interface can result in oscillations of the surface and saw-shaped deformations. Such behavior of the interface does not appear in the system without the heat defect.