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Low-Mach-number asymptotics for two-phase flows of granular materials

Published online by Cambridge University Press:  12 January 2011

C. VARSAKELIS  and
M. V. PAPALEXANDRIS
C. VARSAKELIS
Affiliation:
Institute of Mechanics, Materials and Civil Engineering, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium
M. V. PAPALEXANDRIS*
Affiliation:
Institute of Mechanics, Materials and Civil Engineering, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium
*
†Email address for correspondence: miltos@uclouvain.be
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Abstract

In this paper, we generalize the concept of low-Mach-number approximation to multi-phase flows and apply it to the two-phase flow model of Papalexandris (J. Fluid Mech., vol. 517, 2004, p. 103) for granular materials. In our approach, the governing system of equations is first non-dimensionalized with values that correspond to a reference thermodynamic state of the phase with the smaller speed of sound. By doing so, the Mach number based on this reference state emerges as a perturbation parameter of the equations in hand. Subsequently, we expand each variable in power series of this parameter and apply singular perturbation techniques to derive the low-Mach-number equations. As expected, the resulting equations are considerably simpler than the unperturbed compressible equations. Our methodology is quite general and can be directly applied for the systematic reduction of continuum models for granular materials and for many different types of multi-phase flows. The structure of the low-Mach-number equations for two special cases of particular interest, namely, constant-density flows and the equilibrium limit is also discussed and analysed. The paper concludes with some proposals for experimental validation of the equations.

JFM classification

Complex Fluids: Complex Fluids Granular media: Granular media Multiphase and Particle-laden Flows: Particle/fluid flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 669 , 25 February 2011 , pp. 472 - 497
Copyright
Copyright © Cambridge University Press 2011

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