M.Sc Thesis

M.Sc StudentHaj Majda
SubjectThermocapillary Migration Of Long Viscous Droplet in
a Capillary Tube
DepartmentDepartment of Applied Mathematics
Supervisors PROFESSOR EMERITUS Alexander Nepomniashchy
PROF. Alexander Leshansky


We consider the problem of the migration of a long droplet of an incompressible liquid. The droplet is contained in a tube filled with another incompressible liquid. In one case, the tube is a channel, and in the another case it has a cylindrical shape. In both cases, the tube is subjected to a constant axial temperature gradient, which creates a gradient of the interfacial tension and induces a droplet's motion with a constant velocity toward the hotter regions. Assuming small Peclet, Reynolds, Bond, and capillary numbers, allows uncoupling of the temperature field from the flow field, a use of creeping flow and lubrication approximation, the assumption of axisymmetry and negligible gravity, and a use of matched expansions in Ca, respectively. Most important, a small capillary number, Ca, along with the assumption of a long droplet, allows us to define five different regions for the flow around the droplet, a region of constant thickness film bounded by constant curvature cap regions through transition regions. The flow equations with boundary conditions are presented in the transition regions. In one case, where the droplet is between two parallel planes, moving with the droplet, the flux of the fluid in droplet is vanishing. As a result, we obtain an expression for the pressure gradient to obtain a leading order equation for the constant flux across any cross section. In this equation the ratio of viscosities is of the same order as the ratio of layers thicknesses. The solution of the flow equation is obtained numerically and we use the asymptotic behavior to determine the dependence between the ratio of layers thicknesses and the small capillary number. A global mass conservation relation is then used to complete the solution and relate the droplet speed to the constant film thickness. It is found that the speed increases with the temperature gradient, as expected, and with lower droplet's viscosity, and has a nonlinear dependence on  that is ultimately connected with the nonlinearity of the fluid dynamics in the transition regions. The solution analysis of the leading order flow equation in a cylindrical tube is reduced to that of the previous case through the transformation of the equation parameters.