Thermocapillary migration of a concentric double emulsion in creeping flow

The thermocapillary migration of a concentric spherical compound droplet suspended in an unbounded immiscible fluid and subjected to a uniform temperature gradient is investigated theoretically. The analysis is carried out in the creeping-flow and conduction-dominated limits, with negligible inertia and undeformable spherical interfaces. Exact closed-form solutions for the temperature and Stokes-flow fields are obtained in the three fluid regions by imposing the thermal, kinematic, and tangential-stress conditions at the inner and outer interfaces. The migration velocities associated with the inner core and outer interface are subsequently determined from the corresponding force-free conditions. The principal contribution of the present study is the derivation of an exact analytical solution for the coupled three-region, two-interface problem in an unbounded concentric configuration. The resulting migration laws explicitly reveal the combined dependence on the viscosity ratios, thermal conductivity ratios, core-to-shell size ratio, and relative thermocapillary forcing. The results show that the encapsulated core modifies the conductive temperature field and redistributes the Marangoni stresses acting at both interfaces, while the intervening liquid shell transmits the resulting hydrodynamic interaction between them. Consequently, variations in the thermophysical properties and geometric configuration affect both the migration response and the internal circulation of the compound droplet. The closed-form solution provides physical insight into thermocapillary transport in double emulsions and may serve as an analytical benchmark for future numerical, transient, and wall-confined investigations.