Thermo-gravitational boundary layer near the free surface of inhomogeneous fluid

Abstract

We study a steady axially-symmetric thermogravitational flow of an inhomogeneous fluid in the horizontal layer caused by an uneven heating of the free boundary. Asymptotic expansions of boundary value problem solutions are constructed for the equations of motion in Oberbeck-Boussinesq approximation at small diffusion coefficients of viscosity and thermal conductivity. We obtained a self-similar solution when the free boundary temperature depends on the radial coordinate by the square law. The principal member of asymptotic expansions describes a non-linear thermogravitational boundary layer near the free surface. Thermocapillary effect is not taken into account. We analyzed two cases. In the first case the boundary layer induces itself by the external flow. In the second case the external flow is set, where the speed order in the boundary layer and outside the layer is equal. We calculated two types of modes near the axis of symmetry - basic and rotational. Basic modes describe fluid flow without rotation. We constructed asymptotic formulae for these modes at small temperature gradient values defined along the free boundary. The rotational modes are due to the bifurcation of the basic modes. The rotational modes have current and countercurrent flotation zones near the free boundary. According to the parameters of the problem the temperature in the boundary layer is either monotonic or has one or two points of the local extremum. When the free boundary is heated, there is only one basic mode near the axis of symmetry. When it is cooled, there are either two basic or two rotational modes. Basic modes exist only if the external flow rate exceeds a critical value. There are only rotating modes when the speed does not exceed this value. At the fluid rotation the heat flow in the boundary layer is directed towards the axis of symmetry. The fluid rotation is absent outside the boundary layer.