Droplets containing networks of embedded obstacles within a viscous medium have garnered significant interest owing to their potential in drug delivery applications. In this article, a mathematical model is developed to investigate the fluid flow inside a droplet encapsulating resistive medium subjected to an externally imposed temperature gradient of arbitrary orientation. The flow inside the droplet obeys the Brinkman equation, while the flow outside droplet is governed by the Stokes equation. By solving the coupled system analytically for a general Stokesian far field, we derive closed-form expressions for the drag and Migration velocity and examine Poiseuille flow in detail as a special case. A tangential stress jump condition is applied at the interface between the fluid and the resistive droplet which couples the thermal and hydrodynamic field. The primary aim of this study is to elucidate the synergistic influence of the Darcy number (Da) and the thermal Marangoni number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(Ma_T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <msub> <mi>a</mi> <mi>T</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> on droplet migration dynamics. As Da increases from <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$10^{-5}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>5</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$10^{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation>, the droplet behavior transitions from that of a rigid sphere to that of a clean fluid droplet, with the migration speed increasing monotonically with the viscosity ratio <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The moderate <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$Ma_T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <msub> <mi>a</mi> <mi>T</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> enhance droplet migration, whereas an opposing thermal field reverses the motion once <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$Ma_T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <msub> <mi>a</mi> <mi>T</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> exceeds a critical threshold. Several classical results are recovered in the appropriate limiting regimes, validating the generality of our formulation. Overall, the present model highlights the competing roles of internal hydrodynamic resistance and thermocapillary stresses in governing droplet motion, with implications for biomedical, lab-on-chip, and microfluidic applications.

Thermocapillary motion of a droplet with an internal resistive medium

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发表日期：

2026年5月4日

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Rupali Sharma,
Arindam Basak,
Rajaram Lakkaraju,
G. P. Raja Sekhar

摘要

Droplets containing networks of embedded obstacles within a viscous medium have garnered significant interest owing to their potential in drug delivery applications. In this article, a mathematical model is developed to investigate the fluid flow inside a droplet encapsulating resistive medium subjected to an externally imposed temperature gradient of arbitrary orientation. The flow inside the droplet obeys the Brinkman equation, while the flow outside droplet is governed by the Stokes equation. By solving the coupled system analytically for a general Stokesian far field, we derive closed-form expressions for the drag and Migration velocity and examine Poiseuille flow in detail as a special case. A tangential stress jump condition is applied at the interface between the fluid and the resistive droplet which couples the thermal and hydrodynamic field. The primary aim of this study is to elucidate the synergistic influence of the Darcy number (Da) and the thermal Marangoni number $(Ma_T)$ $(M a_{T})$ on droplet migration dynamics. As Da increases from $10^{-5}$ $10^{- 5}$ to $10^{0}$ $10^{0}$ , the droplet behavior transitions from that of a rigid sphere to that of a clean fluid droplet, with the migration speed increasing monotonically with the viscosity ratio $(\mu )$ $(μ)$ . The moderate $Ma_T$ $M a_{T}$ enhance droplet migration, whereas an opposing thermal field reverses the motion once $Ma_T$ $M a_{T}$ exceeds a critical threshold. Several classical results are recovered in the appropriate limiting regimes, validating the generality of our formulation. Overall, the present model highlights the competing roles of internal hydrodynamic resistance and thermocapillary stresses in governing droplet motion, with implications for biomedical, lab-on-chip, and microfluidic applications.