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.