Domain growth kinetics in a thermally bistable fluid with heat diffusion is studied. The time evolution of interfaces between the stable phases is calculated numerically in two dimensions and compared to some general results derived analytically. The qualitative behavior is found to be similar to the previously studied cases where fluid dynamics was neglected. There are, however, several important differences such as the value of the dynamical exponent, which determines the power law of the systems correlation length growth. The introduction of fluid motion into the model introduces additional properties, unfamiliar to previously studied systems, like the change of the pressure or the size of the system. This behavior is due to the advection of mass. The present model may have general relevance to any system modeled by a real Ginzburg-Landau-type equation coupled to fluid dynamical conservation equations. In particular, it is a step forward on the way to a faithful modeling of thermally bistable cloudy astrophysical media.