Flow solution properties in full-zone thermocapillary liquid bridges
- Davis, Dominic
- Smith, Frank
2010 Mathematics Subject Classification
- 76D05 76E30 35B40 53A05 65M60
- liquid bridges, floating-zone, thermocapillarity, finite-element, Navier--Stokes equations, transient, three-dimensional, mode interaction, nonlinearity, midzone
Properties of low Prandtl number flows in slender cylindrical liquid bridges driven by interfacial thermocapillary forces are addressed here in a theoretical and computational light. Both `outward' (positive Marangoni number Ma) and `inward' (negative Ma) flow along the liquid-gas interface are considered. In previous investigations (Davis & Smith 2003), a solution curve for steady, axisymmetric flow was determined from asymptotic theory in the context of slender bridges. It indicated both the non-existence of solutions beyond a positive, cut-off value of the scaled Marangoni number and a double branch in the case of solvability (although with only one `attractor'). In the present study full numerical simulation (using a finite-element iterative solver, described herein) reveals the unsteady, three-dimensional nature of the flow solution beyond the cut-off value. Attention is paid to the case where the radius-to-height aspect ratio is 0.5, from which the (nonlinearly-coupled) azimuthal modes m=1 and m=2 are seen to dominate. The branch behaviour is then examined for Ma<0, and asymptotic analysis reveals that a critical value of the scaled Marangoni number exists, on approach to which the pressure gradient across the midzone becomes large and negative. Full computational solutions on the attractor branch for Ma<0 are subsequently presented, and these show encouraging agreement with asymptotic predictions (as well as slender-flow midzone computations) near the critical Marangoni number. The critical value moreover is shown to correspond to the onset of `lemonheads' (non-convex radial velocity profiles near the midzone), in precisely the same manner as the cut-off value for positive Ma.