The non-isothermal spreading of a thin drop on a heated or cooled horizontal substrate

We revisit the spreading of a thin drop of incompressible Newtonian fluid on a uniformly heated or cooled smooth planar surface. The dynamics of the moving contact line are modelled by a Tanner Law relating the contact angle to the speed of the contact line. The present work builds on an earlier theoretical investigation by Ehrhard and Davis (JFM, 229,365{388 (1991)) who derived the non-linear partial differential equation governing the evolution of the drop. The (implicit) exact solution to the two-dimensional version of this equation in the limit of quasi-steady motion is obtained. Numerically calculated and asymptotic solutions are presented and compared. In particular, multiple solutions are found for a drop hanging beneath a suffciently cooled substrate. If time permits, some basic models for evaporative spreading will be considered.