The thermocapillary migration of gas bubbles in a viscoelastic fluid

The steady thermocapillary flow of a spherical bubble in a linear temperature field is analyzed by considering that the continuous phase is a weak viscoelastic fluid. Convective heat and momentum transfers are neglected but the action of gravity is taken into account. The problem is formulated for non shear thinning elastic fluids which may be described by the Olroyd-B constitutive equation. The analysis is restricted to weak elastic fluids, an assumption that in dimensionless terms is equivalent to assuming that the Weissenberg number Wi=λ/t_c where λ is the relaxation time of the fluid and t_c the scale time of the flow, is small compared to unity. Thus, the corresponding boundary value problem is solved following a perturbation procedure by regular expansions of the kinematic and stress variables in powers of Wi (retarded motion expansion). Velocity fields as well as the force exerted by the fluid upon the bubble are determined at second order in Wi. It is shown that when the motion is driven by buoyancy in the presence of surface tension forces of a comparable order of magnitude, the velocity fields are strongly affected. Unlike the newtonian case where the recirculation region generated is symmetrical, in a non-newtonian fluid elastic effects produce a breaking of symmetry, so that this region is enhanced and shifted in the downstream direction. The analysis also provides the second order correction to both the terminal velocity and the temperature gradient needed to hold the bubble at rest.