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Diffuse Interface Methods for Multiple Phase Materials: An Energetic Variational Approach

Published online by Cambridge University Press:  28 May 2015

J. Brannick ,
C. Liu ,
T. Qian  and
H. Sun
J. Brannick*
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
C. Liu
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
T. Qian
Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
H. Sun
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
*
*Email addresses: [email protected] (J. Brannick), [email protected] (C. Liu), [email protected] (T. Qian), [email protected] (H. Sun)
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Abstract

In this paper, we introduce a diffuse interface model for describing the dynamics of mixtures involving multiple (two or more) phases. The coupled hydrodynamical system is derived through an energetic variational approach. The total energy of the system includes the kinetic energy and the mixing (interfacial) energies. The least action principle (or the principle of virtual work) is applied to derive the conservative part of the dynamics, with a focus on the reversible part of the stress tensor arising from the mixing energies. The dissipative part of the dynamics is then introduced through a dissipation function in the energy law, in line with Onsager's principle of maximum dissipation. The final system, formed by a set of coupled time-dependent partial differential equations, reflects a balance among various conservative and dissipative forces and governs the evolution of velocity and phase fields. To demonstrate the applicability of the proposed model, a few two-dimensional simulations have been carried out, including (1) the force balance at the three-phase contact line in equilibrium, (2) a rising bubble penetrating a fluid-fluid interface, and (3) a solid particle falling in a binary fluid. The effects of slip at solid surface have been examined in connection with contact line motion and a pinch-off phenomenon.

Keywords

65F1065N2265N55multiphase flowenergetic variational approach
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 2 , May 2015 , pp. 220 - 236
Copyright
Copyright © Global-Science Press 2015 

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