Marangoni Flows in a Bilayer Liquid Microfilm Interface on Wave-Contoured Hot Substrates

This study explores the thermal Marangoni hydrodynamics in an immiscible, binary-liquid thin-film system, which is open to the gas phase at the top and rests on a heated substrate with wavy topology. The sinusoidal contour of the heated (constant-temperature) substrate results in temperature gradients along the liquid-liquid and liquid-gas interfaces, causing fluctuations in the interfacial tension, ultimately leading to Marangoni hydrodynamics in the liquid-liquid films. This type of flow is notable in liquid film coatings on patterned surfaces, which are widely used in MEMS/NEMS applications (Weinstein, S. J.; Palmer, H. J. Liquid Film Coating: Scientific Principles and Their Technological Implications; 1997, pp 19-62; Palacio, M.; Bhushan, B. Adv. Mater. 2008, 20, 1194-1198) and biological cell sorting operations (Witek, M. A.; Freed, I. M.; Soper, S. A. Anal. Chem. 2019, 92, 105-131). We solve the coupled Navier-Stokes and energy equations by the perturbation technique to obtain approximate analytical solutions and an understanding of the thermal and hydrodynamic transport in the system domain. Our study explores the parametric influence of the relative thermal conductivity of the liquid layers (k), film thickness ratio (r), and the system's Biot number (Bi) on these transport phenomena. While the strength of the thermal Marangoni effect that is generated reduces with an increase in the relative thermal conductivity (k), the impact of r depends on the k value. We observe that for k > 1 the intensity of Marangoni flow increases with r; however, the opposite holds for k < 1. Furthermore, larger values of Bi induce higher resistance to the vertical conduction from the wavy substrate compared to the convection resistance offered at the top surface, destructively interfering with the ability of the patterned substrate to generate interfacial temperature fluctuations and hence weakening the Marangoni flow.