Thermocapillary and Photocapillary Effects on a Spherical Droplet Embedded in a Brinkman Medium

Abstract

This study develops an analytical formulation to investigate the thermocapillary and photocapillary migration dynamics of a spherical fluid droplet suspended within a saturated, homogeneous, and isotropic porous matrix, assuming local thermal equilibrium. A uniform external thermal gradient, parallel to an adiabatic boundary, drives the migration. The analysis operates under low Péclet ($Pe{\scriptstyle \ll}1$)and Reynolds $Re{\scriptstyle \ll}1$) numbers, implying negligible advective heat transport and inertial effects, respectively. Consequently, the thermal field is governed by Laplace's equation, internal droplet fluid motion by Stokes equations, and interstitial flow by the Brinkman equation. Analytical expressions for the droplet's translational velocities and associated hydrodynamic forces are derived, examining the influence of the Brinkman parameter $(\alpha)$, thermal conductivity ratio $(k)$, and viscosity ratio $(\sigma)$ on droplet kinematics. Furthermore, the limiting Stokes (high permeability) and Darcy (low permeability) flow regimes are analyzed.