Double Emulsions Get an Exact Formula for Thermocapillary Motion

A theoretical study published in Discover Fluid Mechanics has derived, for the first time, exact closed-form solutions for the thermocapillary migration of a concentric double emulsion—a compound droplet consisting of an inner core wrapped in a liquid shell and suspended in a third, immiscible fluid. The work, by Kareem E. Ragab, M. S. Faltas and Hager A. Hamam, appeared on 20 September 2026 and addresses a long-standing gap between the well-understood migration of simple droplets and the far more complicated behavior of core–shell fluid systems.

Thermocapillary motion, sometimes called thermocapillary migration, arises when a temperature gradient acts along a fluid interface. Because surface tension generally decreases as temperature rises, a gradient in temperature produces a gradient in surface tension, and the resulting imbalance generates tangential Marangoni stresses. These stresses pull fluid along the interface from warm, low-tension regions toward cooler, high-tension regions, setting up internal circulation and, remarkably, propelling the entire droplet through an otherwise quiescent fluid without any external body force. The foundational framework for this phenomenon was established decades ago by Young, Goldstein and Block, who analyzed a single spherical droplet in a uniform temperature gradient under creeping-flow conditions.

Double emulsions—such as water-in-oil-in-water or oil-in-water-in-oil systems—are far more demanding objects of study. They contain two coupled interfaces, multiple viscosity ratios, and recirculating flows in both the core and the shell. When a uniform temperature gradient is imposed, thermocapillary stresses arise on both the inner and outer interfaces simultaneously, generating coupled flow structures across all three fluid phases. The new analysis shows that the encapsulated core is anything but a passive passenger: it modifies the conductive temperature field, redistributes the Marangoni stresses acting at both interfaces, and alters the migration response and internal circulation of the whole system through the intervening liquid shell.

The mathematical formulation considers a compound droplet with an inner core of radius εa enclosed by an outer interface of radius a, where ε is the core-to-shell size ratio ranging from zero to one. Three fluid phases are distinguished: the inner core (phase 1), the encapsulating shell (phase 2), and the surrounding ambient medium (phase 3), each characterized by its own dynamic viscosity and thermal conductivity. Far from the droplet, a uniform temperature gradient of magnitude G is imposed along the z-axis. The analysis is carried out in the creeping-flow limit, where the Reynolds number is negligibly small and inertia is negligible, and in the conduction-dominated limit, where the thermal Péclet number is negligibly small so that heat transport is governed entirely by diffusion. A negligibly small capillary number ensures that both interfaces remain spherical and undeformed at the instant considered.

Within these limits, the steady energy equation in each phase reduces to the Laplace equation, and the fluid dynamics are described by the Stokes momentum equations together with incompressibility, which reduce via a stream-function representation to the classical fourth-order biharmonic equation. The authors imposed thermal continuity of temperature and normal heat flux at both interfaces, kinematic conditions requiring continuity of normal and tangential velocity, and dynamic conditions in which the jump in tangential viscous stress across each interface is balanced by the surface gradient of interfacial tension. Solving this coupled three-region, two-interface boundary-value problem yields exact expressions for the temperature and velocity fields in all three regions, with coefficients that combine the thermal conductivity ratios, the viscosity ratios, the size ratio ε, and a dimensionless parameter Û defined as the ratio of the characteristic thermocapillary velocity scales at the outer and inner interfaces.

The migration velocities themselves follow from force-free conditions. Because the droplet is freely suspended with no gravity or other external forces, the axial hydrodynamic force on the inner core must vanish, and so must the resultant force on the complete core–shell system. Integrating the stress tractions over the inner and outer spherical interfaces shows that the force-carrying coefficients in the shell and external stream functions must both be zero. Solving the resulting system produces closed-form laws for the normalized velocities of the inner core and the outer interface, each explicitly revealing the combined dependence on the viscosity ratios, thermal conductivity ratios, core-to-shell size ratio, and relative thermocapillary forcing. Notably, the common factor appearing in both velocity expressions contains coupled thermal, viscous, and geometric contributions from all three phases, so a change in the size or properties of any phase simultaneously modifies the migration response at both interfaces.

The authors validated the solution against established limiting cases. As the core radius tends to zero, the outer-interface velocity reduces exactly to the classical Young–Goldstein–Block migration velocity for a single droplet, and a numerical convergence table confirms that the relative error shrinks systematically as ε decreases. A further gas-bubble limit, obtained by letting the shell viscosity and thermal conductivity become negligible, recovers the thermocapillary contribution to the bubble velocity reported by Hardy when buoyancy is omitted. A formal vanishing-shell limit, ε tending to one, makes the two migration velocities identical and yields a structure resembling the single-droplet formula with an effective core-to-ambient thermocapillary scale—a consistency check the authors report cautiously rather than as a general limit.

The parametric study reveals several robust trends. Both normalized velocities decrease monotonically as the thermal conductivity ratio between core and shell increases, with the reduction most pronounced in the intermediate range and an asymptotic approach at large values, because a more conductive core redistributes heat efficiently and flattens the interfacial temperature differences that drive the Marangoni stresses. Increasing the core-to-shell viscosity ratio suppresses migration dramatically; in one representative case the normalized core velocity falls from roughly 0.235 at zero core viscosity to about 0.02 at a viscosity ratio of 100. Increasing the size ratio ε also weakens both velocities, particularly beyond ε of about 0.5, as the thinning shell confines the thermocapillary-driven circulation. Conversely, increasing the velocity-ratio parameter Û strongly enhances migration, with the core velocity rising from nearly zero at Û = 0 to approximately 0.9 at Û = 10 in the small-core limit.

Accompanying streamline and isotherm visualizations reinforce the physical picture. For small cores, internal circulation spreads broadly through a thick shell; as ε grows to 0.8, the streamlines are squeezed into a narrow annular layer, viscous confinement intensifies, and the maximum flow intensity drops. Low thermal conductivity ratios preserve strong interfacial temperature gradients and vigorous circulation, while highly conducting phases smooth the temperature field and weaken the driving mechanism. Throughout the investigated ranges, the inner-core velocity consistently exceeds the outer-interface velocity, reflecting the stronger hydrodynamic and thermal coupling between the core and the shell and the additional viscous resistance the outer interface faces from the ambient fluid.

The authors emphasize that the formulation describes an instantaneous quasi-steady mobility problem evaluated at the concentric reference configuration, not a permanently concentric steady-migration state; when the two velocities differ, the core begins to drift relative to the outer interface, and describing that subsequent evolution would require a separate time-dependent moving-boundary analysis. The model is most applicable to small compound droplets under weak-to-moderate temperature gradients in highly viscous fluids or microfluidic settings. Beyond its physical insight, the exact solution offers a practical analytical benchmark for future numerical, transient, and wall-confined investigations, and the findings bear on double-emulsion applications ranging from drug delivery and flavor encapsulation to microencapsulation and droplet-based microfluidics, where transport behavior, structural integrity, and long-term stability are paramount.

Subject of Research: Exact analytical solution for the thermocapillary migration of a concentric compound droplet in creeping flow

Article Title: Thermocapillary migration of a concentric double emulsion in creeping flow

Article References: Ragab, K. E., Faltas, M. S., & Hamam, H. A. (2026). Thermocapillary migration of a concentric double emulsion in creeping flow. Discover Fluid Mechanics, 2(1), Article 13. https://doi.org/10.1007/s44369-026-00017-3

Image Credits: AI Generated

DOI: 10.1007/s44369-026-00017-3

Keywords: thermocapillary migration, double emulsion, compound droplet, Marangoni stress, creeping flow, Stokes flow, core-shell droplet, interfacial tension, temperature gradient, microfluidics, analytical solution, Young-Goldstein-Block