Interfacial thermocapillary migration of a liquid droplet near a solid planar wall in a Brinkman medium

Abstract

Abstract The quasi-steady thermocapillary motion of a spherical liquid droplet embedded within a Brinkman porous medium adjacent to an impermeable plane wall is investigated. A uniform temperature gradient is imposed normal to the wall, leading to thermocapillary-driven migration of the droplet. The governing energy and momentum equations, incorporating the Brinkman model for the porous medium, are solved involving superposition of fundamental solutions and a collocation technique. This analysis is conducted under the assumptions of small Péclet and Reynolds numbers, ensuring linearity, and a small capillary number, guaranteeing the droplet remains spherical. Thermal equilibrium is assumed between the fluid and solid matrix, and a slip condition governs the flow at the planar wall. The results reveal that the migration velocity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> notably decreases as the droplet approaches the wall, with an approximate 30% reduction in velocity as the spacing parameter increases from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0.1</mml:mn> </mml:mrow> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0.99</mml:mn> </mml:mrow> </mml:math> . Increasing the Brinkman parameter (α) significantly suppresses motion, reducing the velocity by nearly 70%, indicating the strong resistance of the porous medium. Increasing the thermal conductivity ratio ( k ) or viscosity ratio (σ) further lower the migration speed by about 20%–40%. The study offers new insights into droplet transport under thermal gradients in confined porous domains. In limiting cases, the model recovers established solutions for either clear fluid domains or wall-free porous environments. This research holds substantial relevance for applications in microfluidics and targeted drug delivery systems, where precise control over droplet transport through complex porous biological environments is crucial.