The motion of a viscous droplet in unbounded Poiseuille flow under thecombined influence of bulk-insoluble surfactant and linearly varyingtemperature field aligned in the direction of imposed flow is studiedanalytically. Neglecting fluid inertia, thermal convection and shapedeformation, asymptotic analysis is performed to obtain the velocity of aforce-free surfactant-laden droplet. The present study is focused on twolimiting situations of surfactant transport: (i) small surface Peclet number,and (ii) high surface Peclet number. Thermocapillary-induced Marangoni stress,strength of which relative to viscous stress is represented by thermalMarangoni number, has strong influence on the distribution of surfactant on thedroplet surface. Temperature field not only affects the axial velocity of thedroplet but also has significant effect on the cross-stream velocity of thedroplet in spite of the fact that the temperature gradient is aligned with thePoiseuille flow direction. When the imposed temperature increases in thedirection of Poiseuille flow, the droplet migrates towards the flow centerline.The magnitude of both axial and cross-stream velocity components increases withthe thermal Marangoni number. However, when the imposed temperature decreasesin the direction of Poiseuille flow, the magnitude of both axial andcross-stream velocity components may increase or decrease with the thermalMarangoni number. Most interestingly, the droplet moves either towards the flowcenterline or away from it. Present study shows a critical value of the thermalMarangoni number beyond which the droplet moves away from the flow centerlinewhich is in sharp contrast to the motion of a surfactant-laden droplet inisothermal flow for which droplet always moves towards the flow centerline.
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