Diffuse-interface modelling of thermocapillary flow instabilities in a Hele-Shaw cell

In this paper we present the results of a diffuse-interface model for thermocapillary or Marangoni flow in a Hele-Shaw cell. We use a Galerkin-type spectral element discretization, based on Gauss–Lobatto quadrature, for numerical implementation of the governing equations resulting from the diffuse-interface model. The results are compared to classical results for a linear and circular fixed interface. It is found that the diffuse-interface solution converges to the classical solution in the sharp-interface limit. The results are sufficiently accurate if the interfacial thickness is only small compared to the size of the thermocapillary boundary layer, even if the interfacial thickness used is much larger than the real interfacial thickness. We also consider freely movable interfaces with a temperature gradient perpendicular to the interface. It will be shown that this situation can lead to a destabilizing Marangoni convection.