Numerical stability analysis of thermocapillary flow in droplets on heated or cooled substrates

Abstract

The thermocapillary-buoyant flow in droplets in contact with a heated or a cooled substrate is of interest in numerous industrial fields including additive manufacturing, painting with suspension sprays or ink-jet printing. Non-uniform evaporation along the droplet surface can produce the well known coffee-stain effect, which is finding new applications in medical diagnostics and materials science [1]. While fast solidification or drying may be desirable in these fields, cooling or heating of the substrate as well as evaporation-induced thermal gradients [4] can trigger three dimensional instabilities of the flow inside the droplet. This changes the total heat and mass transfer and may even lead to a clustering of suspended particles [7].

The axisymmetric steady toroidal vortex flow in a one-component fluid is determined by buoyancy and thermocapillary forces [3]. Three-dimensional symmetry breaking is caused by Bénard-Marangoni cells in the center of shallow droplets [5] and by hydrothermal waves traveling along the droplet’s periphery [2, 4, 6]. Furthermore, the competition between evaporation and heat conduction through the substrate affects the thermal gradients near the contact line [4], modifying the azimuthal wave number.

A systematic investigation of the symmetry breaking phenomena is lacking. Therefore, we carry out a linear stability analysis of the thermocapillary-buoyant flow, assuming an indeformable spherical liquid–gas interface (small-capillary-number limit) and a perfectly conducting substrate. Critical Reynolds numbers, energy budgets and the most dangerous perturbations are computed for a range of Prandtl numbers, Grashof numbers and contact angles.

The basic flow is solved using a Finite Element library FEniCS, while the eigensolutions of the linearized problem are computed with the Arnoldi method implemented in ARPACK. The no-penetration boundary condition on the free surface is enforced with Nitsche’s weak formulation for non-boundary-fitted coordinates.

[1] R. Guha, F. Mohajerani, A. Mukhopadhyay, M. D. Collins, A. Sen and D. Velegol, Modulation of spatiotemporal particle patterning in evaporating droplets: applications to diagnostics and materials science, ACS Appl. Mater. Interfaces 9, 43352–43362 (2017). [2] G. Karapetsas, O. K. Matar, P. Valluri and K. Sefiane, Convective Rolls and Hydrothermal Waves in Evaporating Sessile Drops, Langmuir 28, 11433–11439 (2012). [3] S. Masoudi and H. C. Kuhlmann, Axisymmetric buoyant-thermocapillary flow in sessile and hanging droplets, J. Fluid Mech. 826, 1066–1095 (2017). [4] K. Sefiane, A. Steinchen, and R. Moffat, On hydrothermal waves observed during evaporation of sessile droplets, Colloids and Surfaces A: Physicochem. Eng. Aspects 365, 95–108 (2010). [5] W.-Y. Shi, K.-Y. Tang, J.-N. Ma, Y.-W. Jia and H.-M. Li, Marangoni convection instability in a sessile droplet with low volatility on heated substrate, Int. J. Therm. Sci. 117, 274–286 (2017). [6] T.-S. Wang and W.-Y. Shi, Influence of substrate temperature on Marangoni convection instabilities in a sessile droplet evaporating at constant contact line mode, Int. J. Heat Mass Transf. 131, 1270–1278 (2019). [7] T. Watanabe, T. Takakusagi, I. Ueno, H. Kawamura, K. Nishino, M. Ohnishi, M. Sakurai and S. Matsumoto, Terrestrial and microgravity experiments on onset of oscillatory thermocapillary-driven convection in hanging droplets, Int. J. Heat Mass Transf. 123, 945–956 (2018).