Drop and bubble migration at large Reynolds and Marangoni numbers

Abstract

The migration of a drop or bubble in a fluid, onto which a temperature gradient is being applied, is treated analytically. The original calculation by Young, Goldstein and Block /1/ neglects the convective terms in the momentum equation and in the energy equation. Accounting for these terms by perturbation theory fails due to convergence problems. Numerical methods have been applied instead. In this paper convergence is achieved by splitting the convective contributions into terms proportional to the velocity u 0 of migration and terms proportional to the local flow velocity u 1( r ). Inclusion of the overall convective heat transport u 0· V T( r ) into the homogeneous differential operator of the energy equation entails representing the temperature field by modified spherical Bessel functions and convergence at large Marangoni numbers. Likewise, inclusion of the overall convective momentum transport u 0· V u ( r ) into the homogeneous differential operator of the momentum equation causes a representation of the flow field by modified spherical Bessel functions and convergence at large Reynolds numbers. — For the case of equal physical parameters of the migrating particle and the outer fluid, an exact analytical representation of the velocity of migration in terms of an exponential integral valid for arbitrary Marangoni numbers is derived. The velocity u 0 of migration decreases inversely proportional to the Marangoni number at large Marangoni numbers. - The local contributions u 1· V T( r ) and u 1· V u ( r ) to the convective heat transport and momentum transport are treated by a Green's function formalism. The resulting set of equations for the flow field and the temperature field is solved self-consistently.