  The migration characteristics of a suspended deformable droplet in a parallel
plate microchannel is studied, both analytically and numerically, under the
combined influence of a constant temperature gradient in the transverse
direction and an imposed pressure driven flow. Any predefined transverse
position in the micro channel can be attained by the droplet depending on the
applied temperature gradient in the cross-stream direction or how small the
droplet is with respect to the channel width. For the analytical solution, an
asymptotic approach is used, where we neglect any effect of inertia or thermal
convection of the fluid in either of the phases. To obtain a numerical
solution, we use the conservative level set method. Variation of temperature in
the flow field causes a jump in the tangential component of stress at the
droplet interface. This jump in stress component, which is the thermal
Marangoni stress, is an important factor that controls the trajectory of the
droplet. The direction of cross-stream migration of the droplet is decided by
the magnitude of the critical Marangoni stress, corresponding to which the
droplet remains stationary. In order to analyze practical microfluidic setup,
we do numerical simulations where we consider wall effects as well as the
effect of thermal convection and finite shape deformation on the cross-stream
migration of the droplet.
