Communications in Mathematical Sciences

Volume 18 (2020)

Number 5

Thermodynamically consistent hydrodynamic models of multi-component compressible fluid flows

Pages: 1441 – 1468



Xueping Zhao (Max Planck Institute for the Physics of Complex Systems, Dresden, Germany)

Tiezheng Qian (Department of Mathematics, Hong Kong University of Science and Technology, Kowloon, Hong Kong)

Qi Wang (Department of Mathematics, University of South Carolina, Columbia, S.C., U.S.A.)


We present a systematic derivation of thermodynamically consistent hydrodynamic models for multi-component, compressible viscous fluid mixtures under isothermal conditions using the generalized Onsager principle and the one-fluid multi-component formulation. By maintaining momentum conservation while enforcing mass conservation at different levels, we obtain two compressible models. When the fluid components in the mixture are incompressible, we show that one compressible model reduces to the quasi-incompressible model via a Lagrange multiplier approach. Several different approaches to arriving at the quasi-incompressible model are discussed. Finally, we conduct a linear stability analysis on all the binary fluid models derived in the paper to show the differences of the models in near equilibrium dynamics.


compressible fluid, quasi-incompressible fluid, multi-component fluid mixtures, phase field model, linear stability

2010 Mathematics Subject Classification

35Q30, 35Q35, 35Q79

Received 25 January 2019

Accepted 14 March 2020

Published 23 September 2020