Thermocapillary motion of a droplet on an inclined plate

We examine the dynamics of a droplet spreading on an inclined solid surface in the presence of constant wall thermal gradients. We use lubrication theory in combination with the Karman-Polhausen integral method to simplify the governing equations for the droplet motion and energy conservation leading to coupled evolution equations for the drop thickness and average temperature. An important feature of the spreading model developed here is the behaviour of the drop at the contact line; this is modeled using a constitutive relation, which is dependent on the local temperature of the wall. We use a finite-element formulation to obtain numerical solutions of the evolution equations and carry out a full parametric study. We investigate the various types of behaviour encountered due to the interplay of Marangoni stresses, gravity and the dynamics of the contact line.