File:A boundary-layer model of thermocapillary flow in a cold corner (IA aboundarylayermo109459275).pdf

A pool of liquid with a horizontal free surface is bounded on one side by a vertical solid wall, which is maintained at a cold temperature relative to the core flow region. Strong temperature gradients along the surface give rise to surface tension variations (thermocapillary stress), which drives flow. Thin viscous boundary layers form along the surface and wall. A boundary- layer model is designed which captures the dynamics of the cold corner, applicable for any Marangoni number M and Prandtl number P in the convective inertial regime. Analytical expressions for the velocity and boundary-layer thicknesses are developed, which allow accurate prediction of the flow field. The core flow region (outside the viscous boundary layers) is treated as irrotational flow and Laplace's equation is solved using both a Green's function approach and a complex variables approach in the quarter-plane. The flow along the wall is treated as a plane wall jet. The two dimensional unsteady heat equation is solved using an alternating direction implicit method. Results show that the flow into the corner is strong enough to contain the thermal field, compressing the isotherms along the wall after steady-state is reached. Additionally, a uniform stream function prediction is developed, by matching the inner and outer flows giving a relatively accurate depiction of the flow.