Onset of Convection in a Cylindrical Container with a Free Boundary

Abstract

The problem of small perturbations of the equilibriumstate of a viscous, heat-conducting fluid in a cylindrical container with a deformable free upper boundary, on which heat exchange with the ambient medium is preassigned, is studied. The mathematical modeling of convection is based on Oberbeck–Boussinesq equations. The spectral problem thus obtained is solved using the tau method. As a result, the dependence of the imaginary part of the complex decrement on the Marangoni number is obtained. In the case of monotonic perturbations the neutral curves are plotted as functions of a geometrical parameter, namely, the cylinder height-to-radius ratio. The dependence of the Marangoni number on the physical parameters of the fluid is also obtained.