# Thermocapillary effect on the cross-stream migration of a surfactant-laden droplet in Poiseuille flow

## TL;DR

This paper analytically investigates how temperature gradients and surfactant effects influence the cross-stream migration of droplets in Poiseuille flow, revealing critical conditions where droplets switch from moving towards to away from the flow centerline.

## Contribution

It provides a novel analytical framework for understanding thermocapillary effects on surfactant-laden droplet migration, highlighting the impact of thermal Marangoni number on droplet behavior.

## Key findings

- Droplet migrates towards the flow centerline with increasing temperature in flow direction.
- A critical thermal Marangoni number causes droplet to move away from the centerline.
- Temperature significantly affects both axial and cross-stream velocities of the droplet.

## Abstract

The motion of a viscous droplet in unbounded Poiseuille flow under the combined influence of bulk-insoluble surfactant and linearly varying temperature field aligned in the direction of imposed flow is studied analytically. Neglecting fluid inertia, thermal convection and shape deformation, asymptotic analysis is performed to obtain the velocity of a force-free surfactant-laden droplet. The present study is focused on two limiting 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 thermal Marangoni number, has strong influence on the distribution of surfactant on the droplet surface. Temperature field not only affects the axial velocity of the droplet but also has significant effect on the cross-stream velocity of the droplet in spite of the fact that the temperature gradient is aligned with the Poiseuille flow direction. When the imposed temperature increases in the direction of Poiseuille flow, the droplet migrates towards the flow centerline. The magnitude of both axial and cross-stream velocity components increases with the thermal Marangoni number. However, when the imposed temperature decreases in the direction of Poiseuille flow, the magnitude of both axial and cross-stream velocity components may increase or decrease with the thermal Marangoni number. Most interestingly, the droplet moves either towards the flow centerline or away from it. Present study shows a critical value of the thermal Marangoni number beyond which the droplet moves away from the flow centerline which is in sharp contrast to the motion of a surfactant-laden droplet in isothermal flow for which droplet always moves towards the flow centerline.

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Source: https://tomesphere.com/paper/1701.02938