Phase-field-based lattice Boltzmann finite-difference model for simulating thermocapillary flows.

A phase-field-based hybrid model that combines the lattice Boltzmann method with the finite difference method is proposed for simulating immiscible thermocapillary flows with variable fluid-property ratios. Using a phase field methodology, an interfacial force formula is analytically derived to model the interfacial tension force and the Marangoni stress. We present an improved lattice Boltzmann equation (LBE) method to capture the interface between different phases and solve the pressure and velocity fields, which can recover the correct Cahn-Hilliard equation (CHE) and Navier-Stokes equations. The LBE method allows not only use of variable mobility in the CHE, but also simulation of multiphase flows with high density ratio because a stable discretization scheme is used for calculating the derivative terms in forcing terms. An additional convection-diffusion equation is solved by the finite difference method for spatial discretization and the Runge-Kutta method for time marching to obtain the temperature field, which is coupled to the interfacial tension through an equation of state. The model is first validated against analytical solutions for the thermocapillary driven convection in two superimposed fluids at negligibly small Reynolds and Marangoni numbers. It is then used to simulate thermocapillary migration of a three-dimensional deformable droplet and bubble at various Marangoni numbers and density ratios, and satisfactory agreement is obtained between numerical results and theoretical predictions.

[1]  Hsan-Yin Hsu,et al.  Optically actuated thermocapillary movement of gas bubbles on an absorbing substrate. , 2007, Applied physics letters.

[2]  R. D. Schroll,et al.  Laser microfluidics: fluid actuation by light , 2009, 0903.1739.

[3]  Asghar Esmaeeli,et al.  An analytical solution for thermocapillary-driven convection of superimposed fluids at zero Reynolds and Marangoni numbers , 2010 .

[4]  L. Duan,et al.  On-board Experimental Study of Bubble Thermocapillary Migration in a Recoverable Satellite , 2008 .

[5]  Junseok Kim.  Phase-Field Models for Multi-Component Fluid Flows , 2012 .

[6]  R. Subramanian,et al.  The stokes motion of a gas bubble due to interfacial tension gradients at low to moderate Marangoni numbers , 1988 .

[7]  D. Juric,et al.  A Front-Tracking Method for Dendritic Solidification , 1996 .

[8]  Shan,et al.  Lattice Boltzmann model for simulating flows with multiple phases and components. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  B. Blanpain,et al.  Lattice Boltzmann method for double-diffusive natural convection. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Chang Shu,et al.  Diffuse interface model for incompressible two-phase flows with large density ratios , 2007, J. Comput. Phys..

[11]  P. Gao,et al.  Thermocapillary migration of nondeformable drops , 2008 .

[12]  Jean-Pierre Delville,et al.  An optical toolbox for total control of droplet microfluidics. , 2007, Lab on a chip.

[13]  Kenneth J. Witt,et al.  Transient Motion of a Gas Bubble in a Thermal Gradient in Low Gravity , 1994 .

[14]  B. Shi,et al.  Effects of force discretization on mass conservation in lattice Boltzmann equation for two-phase flows , 2012 .

[15]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[16]  Sigurd Wagner,et al.  Microfluidic actuation by modulation of surface stresses , 2003 .

[17]  Micaiah John Muller Hill,et al.  IV. On a spherical vortex , 1894, Proceedings of the Royal Society of London.

[18]  François Gallaire,et al.  Thermocapillary valve for droplet production and sorting. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Matthieu Robert de Saint Vincent,et al.  Thermocapillary migration in small-scale temperature gradients: application to optofluidic drop dispensing. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Taehun Lee,et al.  Single bubble rising dynamics for moderate Reynolds number using Lattice Boltzmann Method , 2010 .

[21]  Tianyu Zhang,et al.  Cahn-Hilliard Vs Singular Cahn-Hilliard Equations in Phase Field Modeling , 2009 .

[22]  L. Luo,et al.  A priori derivation of the lattice Boltzmann equation , 1997 .

[23]  P. Gascoyne,et al.  Droplet-based chemistry on a programmable micro-chip. , 2004, Lab on a chip.

[24]  Sauro Succi,et al.  Improved lattice boltzmann without parasitic currents for Rayleigh-Taylor instability , 2009 .

[25]  Xiyun Lu,et al.  Numerical simulation of drop Marangoni migration under microgravity , 2004 .

[26]  Welch.  Transient Thermocapillary Migration of Deformable Bubbles. , 1998, Journal of colloid and interface science.

[27]  D. Rothman,et al.  Diffusion properties of gradient-based lattice Boltzmann models of immiscible fluids. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  J. S. Goldstein,et al.  The motion of bubbles in a vertical temperature gradient , 1959, Journal of Fluid Mechanics.

[29]  P. Lallemand,et al.  Theory of the lattice Boltzmann method: acoustic and thermal properties in two and three dimensions. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  R. Fair,et al.  Electrowetting-based actuation of liquid droplets for microfluidic applications , 2000 .

[31]  Dieter Bothe,et al.  Direct numerical simulation of thermocapillary flow based on the Volume of Fluid method , 2011 .

[32]  Héctor D. Ceniceros,et al.  Computation of multiphase systems with phase field models , 2002 .

[33]  G. Whitesides,et al.  Formation of droplets and bubbles in a microfluidic T-junction-scaling and mechanism of break-up. , 2006, Lab on a chip.

[34]  Chang Shu,et al.  Hybrid lattice Boltzmann finite‐difference simulation of axisymmetric swirling and rotating flows , 2007 .

[35]  Raoyang Zhang,et al.  A Lattice Boltzmann Scheme for Incompressible Multiphase Flow and Its Application in Simulation of Rayleigh-Taylor Instability , 1998 .

[36]  Hans C. Mayer,et al.  Microscale tipstreaming in a microfluidic flow focusing device , 2006 .

[37]  Thomas Thundat,et al.  Microfluidic manipulation via Marangoni forces , 2004 .

[38]  R. Wunenburger,et al.  Laser switching and sorting for high speed digital microfluidics , 2008 .

[39]  Ching-Long Lin,et al.  A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio , 2005 .

[40]  Chuguang Zheng,et al.  Force imbalance in lattice Boltzmann equation for two-phase flows. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[41]  Taehun Lee,et al.  Effects of incompressibility on the elimination of parasitic currents in the lattice Boltzmann equation method for binary fluids , 2009, Comput. Math. Appl..

[42]  S. Quake,et al.  Dynamic pattern formation in a vesicle-generating microfluidic device. , 2001, Physical review letters.

[43]  Markus Gross,et al.  Shear stress in nonideal fluid lattice Boltzmann simulations. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  L. Scriven,et al.  The Marangoni Effects , 1960, Nature.

[45]  Chen,et al.  Interface and contact line motion in a two phase fluid under shear flow , 2000, Physical review letters.

[46]  Haihu Liu,et al.  Modeling and simulation of thermocapillary flows using lattice Boltzmann method , 2012, J. Comput. Phys..

[47]  Y L He,et al.  Additional interfacial force in lattice Boltzmann models for incompressible multiphase flows. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[48]  Chang-JinCJ Kim,et al.  Valveless pumping using traversing vapor bubbles in microchannels , 1998 .

[49]  Jie Shen,et al.  A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method , 2003 .

[50]  A Lamura,et al.  Hybrid lattice Boltzmann model for binary fluid mixtures. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[51]  R. Garrell,et al.  Electromechanical model for actuating liquids in a two-plate droplet microfluidic device. , 2009, Lab on a chip.

[52]  Lin Liu,et al.  Lattice Boltzmann simulations of micron-scale drop impact on dry surfaces , 2010, J. Comput. Phys..

[53]  Yeomans,et al.  Lattice Boltzmann simulations of liquid-gas and binary fluid systems. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[54]  S. Zaleski,et al.  Lattice Boltzmann model of immiscible fluids. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[55]  Huaizhi Li,et al.  Scaling the formation of slug bubbles in microfluidic flow-focusing devices , 2010 .

[56]  Günter Wozniak,et al.  Thermocapillary migration of bubbles and drops at moderate to large Marangoni number and moderate Reynolds number in reduced gravity , 1999 .