Tilted isotherms, not Rayleigh number, drive hidden convection in drying droplets

A tiny evaporating droplet sitting on a glass slide should, according to a century-old rule of thumb, remain perfectly still inside. The temperature difference across it is far too small, and the droplet far too shallow, to push its Rayleigh number anywhere near the classical threshold that separates a quiescent fluid from one stirred by buoyancy. Yet a new combined numerical and theoretical study reports that such droplets do circulate, with internal speeds of a few micrometers per second, and that the familiar criterion used to predict the onset of this natural convection simply does not apply at these scales. The culprit, the researchers find, is geometry: the curved, tilted isothermal contours that inevitably form inside a small evaporating droplet destabilize the fluid in a way that the classical Rayleigh criterion, built for flat horizontal layers, cannot capture.

Natural convection is the buoyancy-driven flow that arises when a fluid’s density varies with temperature in a gravitational field. It governs heat and mass transport in systems ranging from thermal energy storage units to microfluidic chips. At the microscale, where evaporating sessile droplets serve as a canonical fluidic unit, weak buoyancy-induced flows can dominate transport behavior and determine drying quality and solute distribution. Understanding exactly when and why such convection starts is therefore central to controlling microscale heat and mass transfer, from printing and coating to the drying patterns left behind by biological fluids.

Internal flows in an evaporating aqueous droplet on a smooth horizontal substrate have long been attributed to two mechanisms. Because evaporation is strongest near the droplet’s pinned edge, an outward capillary flow replenishes the water lost there and preserves the sessile configuration. Meanwhile, the non-uniform evaporation flux cools the surface unevenly, creating surface tension gradients that drive Marangoni flow along the air-droplet interface. The same surface cooling also establishes a temperature gradient through the bulk liquid, which in principle should trigger a third flow: natural convection. Experiments have indeed revealed such convection in plasma droplets, but not in pure water droplets. The distinction arises because adsorbed plasma proteins at the air-droplet interface remove the inhibitory effect that Marangoni flow otherwise exerts on buoyancy-driven motion. All droplets discussed in the new work are therefore plasma droplets, and the authors note that natural convection in an evaporating sessile droplet had not been reported before their earlier publications.

The puzzle is quantitative. Lord Rayleigh argued in 1916 that a liquid can retain stable equilibrium even when denser fluid overlays lighter fluid, provided its viscosity is high enough; a critical temperature difference is needed to overcome viscous drag. In 1926, Jeffries derived critical Rayleigh numbers for fluid layers between two infinite parallel surfaces: 1708 for two rigid, no-slip boundaries and 670 for a rigid lower boundary with a free upper surface. For a water layer with a characteristic length of one millimeter, these thresholds correspond to critical temperature differences of about 11 degrees Celsius and 4.3 degrees Celsius, respectively. A sessile droplet differs fundamentally from both configurations, since its free air-liquid surface is curved rather than horizontal. In the new simulations of an evaporating droplet with a volume of roughly 5 microliters, a contact angle of 36 degrees and a contact radius of 2 millimeters, the largest temperature difference between the substrate and the air-droplet interface is only about 0.38 to 0.46 degrees Celsius, giving a Rayleigh number below 1, orders of magnitude beneath any classical threshold.

Despite this, the simulations show active circulation. For a plasma droplet of 5.5 microliters with a 36-degree contact angle evaporating on a glass slide, the maximum temperature of 295 kelvin occurs near the droplet edge and the minimum of 294.62 kelvin near the apex, and the resulting flow reaches a maximum velocity of 7.25 micrometers per second. When the contact angle is reduced to 20 degrees, shrinking the volume to 2.2 microliters and cutting the characteristic length by roughly a factor that pushes the Rayleigh number essentially to zero, convection persists at 1.35 micrometers per second. These results are consistent with the team’s earlier experimental observations, which recorded natural convection throughout the whole evaporation period of plasma droplets. On that basis, the authors conclude that a critical Rayleigh number is not a feasible criterion for the onset of natural convection in droplet evaporation.

To isolate the effect, the researchers also simulated a cylindrical enclosure, 2 millimeters by 1 millimeter, sized to the droplet’s characteristic length. With its upper and lower walls held at the droplet’s minimum and maximum temperatures, 294.62 kelvin and 295 kelvin, and its side walls adiabatic, the enclosure developed flat, horizontal isothermal contours, and the liquid between the walls remained quiescent, exactly as Jeffries’s analysis predicts for this configuration, whose critical temperature difference is 11 degrees Celsius. But when the two side walls were instead held at the constant temperature of 294.62 kelvin, the isothermal contours became non-horizontal, and active convection appeared, with a maximum flow velocity of 11.3 micrometers per second. Crucially, both cylinder configurations share the same Rayleigh number as conventionally defined, yet exhibit opposite flow states. The droplet, with its inherently non-horizontal isotherms, behaves like the second case. The authors therefore propose that the critical Rayleigh number criterion is fundamentally inapplicable to systems featuring non-horizontal isothermal contours.

The finding echoes classical theory that has largely been overlooked in droplet studies. Goldstein’s theoretical analysis showed that the stability of fluid systems with non-horizontal isothermal contours depends primarily on the curvature of the density stratification rather than on fluid viscosity, and he termed the resulting weak, conduction-dominated flows oscillatory motions. Sparrow and colleagues similarly demonstrated that the stability of a quiescent steady state is strongly modulated by the geometric morphology of the temperature distribution, with non-horizontal isotherms amplifying perturbations that fluid motion then acts to mitigate. Physically, a tilted isotherm decomposes gravity into a vertical component, along which the usual Rayleigh analysis still applies, and a horizontal component that renders the system non-equipotential. The fluid moves to restore horizontal, equal-density contours and minimize gravitational potential energy, regardless of the global Rayleigh number. The direction of circulation follows the sign of the cosecant of the isotherm inclination angle: counterclockwise when that quantity is positive, clockwise when it is negative.

The team also derived a quantitative scaling law. Under the Boussinesq approximation, the tilted isotherm generates an effective horizontal buoyancy component proportional to the product of density, gravity, the thermal expansion coefficient, the temperature difference and the sine of the inclination angle. Balancing this driving force against viscous dissipation in a steady boundary-layer-type flow yields a characteristic convection velocity that scales with the sine of the angle. Numerical simulations of a rotated two-dimensional rectangular enclosure, swept through inclination angles from 0.1 to 90 degrees, confirmed the prediction: a zero-intercept linear regression gave a maximum velocity of 131.27 times the sine of the angle in micrometers per second, with a coefficient of determination of 0.9989 and a root-mean-square residual of 2.42 micrometers per second. Strikingly, even the smallest tested tilt of 0.1 degree immediately drove the system into an unsteady flow state, and within the explored parameter space the researchers identified no distinct critical angle at which convection switches on.

The velocities involved are tiny, on the order of micrometers per second, and their consequences are scale-dependent. In large systems such as the atmosphere, such motion is negligible, but in millimeter- and sub-millimeter-scale systems, including sessile droplets, microfluidic devices and thin-film processes, it can contribute considerably to heat and mass transfer and to the drying patterns and deposit morphologies that engineers seek to control. The authors situate the work relative to earlier studies of drying colloidal droplets with small initial contact angles, in which natural convection’s influence on solute distribution and deposit morphology was found to be negligible and capillary-flow-driven horizontal motion dominated. The new mechanism matters instead for droplets with larger contact angles or conditions favoring turbulent-like fluctuations, where surface skin or crust layers introduce interfacial constraints that can favor buoyancy-driven flow.

The study carries clear limitations that the authors state explicitly. All simulations use a quasi-steady framework, justified because thermal diffusion within the droplet adjusts in roughly half a second while evaporation takes minutes to hours. The free surface is idealized as a rigid boundary to exclude surface tension and Marangoni interference, which isolates the convection induced purely by non-horizontal isotherms but neglects real-time interface deformation. The predicted velocities are so weak that they are sensitive to numerical discretization and convergence, even after mesh-independence checks showing relative deviations below 1 percent. Direct experimental validation of the angle-dependent convection mechanism is currently lacking, and because the simulations assume plasma droplet conditions in which interfacial protein adsorption eliminates Marangoni suppression, the conclusions cannot be directly extended to pure water droplets. Within those boundaries, the work redefines the stability paradigm for confined thermal flows, shifting attention from a single scalar threshold to a vector analysis of temperature contours, and offers a fundamental explanation for long-observed but poorly understood convective motions inside small-scale systems such as sessile droplets.

Subject of Research: Natural convection driven by non-horizontal isothermal contours in evaporating sessile droplets

Article Title: Natural convection driven by the non-horizontal isothermal contours in an aqueous sessile droplet

Article References: Du, F., Zhang, X., Yang, G., & Li, H. (2026). Natural convection driven by the non-horizontal isothermal contours in an aqueous sessile droplet. Results in Engineering, 32, Article 113658. https://doi.org/10.1016/j.rineng.2026.113658

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113658

Keywords: natural convection, sessile droplet, evaporation, Rayleigh number, isothermal contours, microfluidics, buoyancy, Marangoni flow, plasma droplet, drying patterns, fluid dynamics, thermal engineering