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Structure-Preserving Numerical Approximations to a Non-isothermal Hydrodynamic Model of Binary Fluid Flows

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Abstract

We present two second order, structure-preserving numerical schemes for a newly derived thermodynamically consistent, non-isothermal hydrodynamical phase field model for incompressible binary viscous fluid flows. The schemes preserve the volume of each fluid phase, the total energy and the positive entropy production rate. The entropy quadratization approach is employed to devise the two semi-discrete numerical schemes in time, preserving both the total energy and the positive entropy production rate. The first scheme is weakly nonlinear, which is solved using iterative methods aided by fast Fourier algorithms. The second scheme is linear, in which a time-dependent supplementary variable is added to preserve the positive entropy production rate. The semi-discrete schemes are discretized in space by a finite difference method on staggered grids subsequently to yield two fully discrete schemes. Mesh refinement is carried out to confirm the order of the schemes and several numerical examples are provided to show hydrodynamic as well as thermal effects in resolving thermocapillary convection near the fluid interface in the incompressible binary viscous fluid flow.

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Acknowledgements

The research is partially supported by National Science Foundation DMS-1816783 award, DMS-1815921, OIA-1655740 and a GEAR award from SC EPSCoR/IDeA Program, NSFC awards #11801269 #11301287 and NSAF-U1930402.

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Authors and Affiliations

  1. Beijing Computational Science Research Center, Beijing, 100193, China

    Shouwen Sun

  2. School of Mathematical Sciences, Tianjin Normal University, Tianjin, 300387, China

    Jun Li

  3. Department of Mathematics & Statistics, Utah State University, Logan, UT, 84322, USA

    Jia Zhao

  4. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Qi Wang

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  1. Shouwen Sun
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  2. Jun Li
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  3. Jia Zhao
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  4. Qi Wang
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Correspondence to Qi Wang.

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Sun, S., Li, J., Zhao, J. et al. Structure-Preserving Numerical Approximations to a Non-isothermal Hydrodynamic Model of Binary Fluid Flows. J Sci Comput 83, 50 (2020). https://doi.org/10.1007/s10915-020-01229-6

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  • DOI: https://doi.org/10.1007/s10915-020-01229-6

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