On the motion of droplets driven by solutal Marangoni convection in alloy systems with a miscibility gap

Abstract

In the first part of this work, we analytically study the motion of two droplets driven by solutal Marangoni convection in a bipolar coordinate. Particular solutions for the Laplace and Stokes equations are found by applying Robin type boundary conditions for mass transfer and by utilizing continuity of stream function and impenetrability at the surface of droplets. The solutions for the Laplace and Stokes equations are connected by the tangential stress balance between the viscosity stress and the Marangoni stress caused by concentration gradients. In the second part, we numerically investigate the motion of two droplets in an immiscible fluid by solving the combined convective Cahn–Hilliard and Navier–Stokes equations, where the capillary tensor is used to account for the Marangoni force. A significant outcome of the present work is that the attraction or repulsion of droplets is determined by droplet radius and the Marangoni number. In both cases, we obtain the stream lines affected by the spacing between droplets and the ratio of the radius of the droplet.