Classical studies on surfactant-laden droplets in a tubular Poiseuille flow by Pak et al. (J Fluid Mech 753:535–552, 2014) and Dandekar and Ardekani (J Fluid Mech 902:A2, 2020) have elucidated remarkable cross-stream migration phenomena as a second-order effect in low surface Péclet number \(Pe_\textrm{s}\) \((=V_\textrm{c}a/D_\textrm{s})\) limit, where \(V_\textrm{c}\) is the characteristic flow velocity, a denotes the droplet radius and \(D_\textrm{s}\) is the diffusivity of the surfactant. However, in many biomedical lab-on-a-chip devices, droplet motion occurs between two parallel plates due to combined shear and pressure-driven flow. We unravelled that such a combined flow can lead to better steering of the droplet with enhanced migration velocity. Our model incorporates heat generation by living cells via a thermal dipole within the droplet, surrounded by a non-isothermal medium. We found that the cross-stream migration velocity can be at the leading order in \(Pe_\textrm{s}\) , provided the thermocapillary effects combined with the strength of the thermal dipole are strong enough. A novel aspect of this work is the definition of flow control parameters based on a constant volumetric flow rate condition, which governs the droplet migration and flow patterns. We illustrate that a careful transition between flow profiles, specifically from Poiseuille to Couette, provides optimal control in droplet migration, and employing a rectangular geometry instead of a tubular gives better regulation on steering a droplet for accurate microfluidic investigations.
