Determination of the quench velocity and rewetting temperature of hot surfaces. Part I: analytical solution of the micro-scale hydrodynamic model

Abstract

The hydrodynamic micro-scale model, developed previously, is used to solve the non-isothermal interface equation. The complex interface equation is simplified in a coordinate frame that moves with the three-phase contact line. This equation accounts for effects of evaporation, thermo-capillary and intermolecular forces. The new non-isothermal interface equation provides generalization of de Gennes’ equation that applies to the isothermal case. The simplified third-order differential equation is solved numerically, and the effect of numerical parameters and selection of boundary conditions on solution convergence are established for a wide range of properties of solid–liquid pairs. In contrast to the smooth isothermal interfaces, non-isothermal interfaces are characterized by an undulating or wavy geometry. This behavior is a reflection of evaporation and mass transfer occurring across the interface, and unique capillary and thermocapillary effects that arise under non-isothermal conditions. A parametric study of the interface solution shows that increase of the capillary, C, and thermocapillary, Cθ 0 2/ F numbers produces steeper interface profiles, whereas the factor N, evaporation coefficient S, and the Hamaker constant Ā, produce the reverse effect. Larger values of N, S and Ā result in higher undulation frequencies. These effects intensify and become dominant under rewetting conditions. The new interface equation provides an advanced tool for further studies of hydrodynamic mechanisms that govern the motion of thin liquid films on hot solid surfaces, that involve high temperature gradients and intense evaporation. This furnishes a hydrodynamic foundation for analysis of rewetting phenomena, and the definition of rewetting temperature and quench velocity, that are presented in a subsequent paper.