Thermocapillary migration of a drop with internal heat generation in non-isothermal Poiseuille flow

Thermocapillary migration of drops, mediated by temperature-dependent surface tension, fosters diverse applications in numerous small-scale devices. However, most studies related to this topic focus either on axisymmetric scenarios in uniformly heated channels or on even simpler cases of linear temperature gradients, ignoring the impact of thermal convection, and thus cannot address paradigms such as cross-stream migration of drops. Moreover, the impact of any initial temperature difference between the drop and its ambient, and that of any internal heat generation within the drop, both of which are commonplace in modern micro-reactor based applications, remains hitherto unexplored. As such, here we address the thermocapillary motion and cross-stream migration of a neutrally buoyant spherical drop suspended in an unbounded Poiseuille flow. The drop is initially considered to have a uniform but distinct temperature from that of the surrounding phase, whilst also harbouring internal heat generation. We develop a combined semi-analytical-cum-numerical framework, and establish that thermal convection, the surface tension's sensitivity to temperature, and the rate of internal heat generation all play critical roles in dictating the drop’s trajectory. In particular, we show that an initially warm drop tends to migrate towards the flow centerline, while an initially cold one tends to move in the opposite direction. Internal heat generation, however, can significantly influence these trends and may even cause a reversal in the direction of the drop’s migration. We envisage that the above findings will improve our understanding of droplet behaviour in thermally heterogeneous environments, with possible implications in active droplet manipulation.