Numerical Simulations on Thermocapillary Flow on Heated Sinusoidal Topography

The interaction between thermocapillary flow and substrate geometry is analyzed numerically. Taking surface tension into account, the momentum equation is derived and solved using a commercial FEM solver, COMSOL Multiphysics where the effects of surface tension and surface deflection can be easily incorporated into the momentum equation. In the case that the Marangoni number is close to its critical value, i.e., Ma approximate to Mac\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\text{Ma}}\approx {\text{Ma}}}_{c}$$\end{document}, the strong symmetric thermocapillary flow is observed when the wavelength of topography, lambda T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda }_{T}$$\end{document}, and the wavelength of instability motion, lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}, are nearly the same. This interesting phenomenon has been called flow-structure resonance. Through the numerical simulations, various flow modes, such as symmetric two-cell and four-cell modes, asymmetric two-cell mode, and oscillatory asymmetric two-cell mode are identified by changing the Marangoni number and wavelength of topography. It is clearly shown that for a certain lambda T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda }_{T}$$\end{document}-system, the transition from oscillatory mode to steady one is possible by relaxing the previous non-deformable surface condition due to high surface tension, i.e., Ca -> 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{Ca}}\to 0$$\end{document}, here Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{Ca}}$$\end{document} is the capillary number. The present study reveals that the preferred flow mode is the complex function of the various parameters such as the Marangoni number, the Biot number, the wavelength of topography, and the capillary number.