Commun. Comput. Phys., 6 (2009), pp. 1095-1117.


Phase Field Model of Thermo-Induced Marangoni Effects in the Mixtures and its Numerical Simulations with Mixed Finite Element Method

Pengtao Sun 1*, Chun Liu 2, Jinchao Xu 2

1 Department of Mathematical Sciences, University of Nevada, Las Vegas, 4505 Maryland Parkway, Las Vegas, NV 89154, USA.
2 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA.

Received 29 October 2008; Accepted (in revised version) 30 March 2009
Available online 14 May 2009

Abstract

In this paper, we study the Marangoni effects in the mixture of two Newtonian fluids due to the thermo-induced surface tension heterogeneity on the interface. We employ an energetic variational phase field model to describe its physical phenomena, and obtain the corresponding governing equations defined by a modified Navier-Stokes equations coupled with phase field and energy transport. A mixed Taylor-Hood finite element discretization together with full Newton's method are applied to this strongly nonlinear phase field model on a specific domain. Under different boundary conditions of temperature, the resulting numerical solutions illustrate that the thermal energy plays a fundamental role in the interfacial dynamics of two-phase flows. In particular, it gives rise to a dynamic interfacial tension that depends on the direction of temperature gradient, determining the movement of the interface along a sine/cosine-like curve.

AMS subject classifications: 65B99, 65K05, 65K10, 65N12, 65N22, 65N30, 65N55, 65Z05
Key words: Phase field, Navier-Stokes equations, Marangoni effects, temperature dependent surface tension, mixed Taylor-Hood finite element, Newton's method.

*Corresponding author.
Email: pengtao.sun@unlv.edu (P. Sun), liu@math.psu.edu (C. Liu), xu@math.psu.edu (J. Xu)
 

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