Steady three-dimensional thermocapillary flows and dryout inside a V-shaped wedge

We consider a liquid meniscus inside a wedge of included angle 2 beta that wets the solid walls with a contact angle theta. Under an imposed axial temperature gradient, the Marangoni stress moves fluid toward colder regions whereas the capillary pressure gradient drives a reverse flow, leading to a steady state. The fluxes driven by these two mechanisms are found by numerical integration of the parallel flow equations. Perturbation theory is applied to derive an expression for the capillary pressure, which is typically dominated by the transverse curvature of the circular arc inside the cross section perpendicular to the flow axis, and corrected by a higher order axial curvature resulting from the axial variation of the interface. Lubrication theory is then used to derive a thin film equation for the shape of the interface. Solutions are determined by two primary parameters: D, a geometric parameter giving the relative importance of the two curvatures and M, a modified Marangoni number. Numerical solutions indicate that for sufficiently large M, the Marangoni stress creates a virtual dry region. The value of M at dryout is found to depend linearly on D. A simplified analytical model is developed which agrees very well with the numerical solution for large values of D. It is found that dryout occurs more easily for larger wedge and/or contact angles except for a special case of beta+theta=pi/2. In that case the axial curvature dominates and the dependence of the dryout condition on beta and theta is nonmonotonic, but only weakly so. (C) 2006 American Institute of Physics.