This paper deals with the remarkably difficult problem of studying the motion of two immiscible fluids in a variable thermal field under the influence of capillary forces at the interface. The p.d.e.’s to be solved in each phase are Navier-Stokes equation and the heat diffusion-convection equation. At the interface \(\Gamma(t)\) velocity, temperature and thermal diffusive flux are taken continuous, while the stress condition \[ \underline P^{(1)} n- \underline P^{(2)} n= \sigma Hn+ \nabla^{(s)}\sigma \] is imposed, where \(\underline P^{(i)}\) is the normal stress tensor for a viscous fluid (the index refers to the phase), \(n\) is the unit normal vector pointing towards the domain \(\Omega^{(2)}\) occupied by phase 2, \(\sigma\) is the temperature dependent surface tension coefficient, \(H\) is twice the mean curvature and \(\nabla^{(s)}= \nabla\cdot n(n\cdot \nabla)\) is the surface gradient. The problem is formulated in Lagrangian coordinates and is shown to have a unique solution in a suitable function space.