Unsteady heat conduction in a quarter plane, with an application to bubble growth models

Abstract

A solution is obtained to the problem of transient heat conduction in the quarter plane x, y ⩾ 0. initially at zero temperature, for the case of a suddenly-applied radiation condition on the boundary x = 0, the boundary y = 0 being maintained at zero temperature. The solution is obtained in closed form by a double transform technique as an integral of the known solution for one-dimensional heat transfer in the half space x > 0 subject to the radiation condition on x = 0. The temperature and heat flux on the boundaries have been found by quadrature. The results are used to estimate the evaporation rate at the perimeter of a bubble growing (without a microlayer) on a plane wall of infinite conductivity in a uniformly superheated liquid. It is shown that evaporation at the wall may make a significant contribution to bubble growth at low Jacob number for fluids with large contact angle, or after the evaporation of a microlayer to dryness.