Abstract
In this paper, we develop a numerical scheme for the solution of the coupled Stokes and Navier-Stokes equations with constitutive equations describing the flow of viscoelastic fluids. The space discretization is based on the so-called Marker-And-Cell (MAC) scheme. The time discretization uses a fractional-step algorithm where the solution of the Navier-Stokes equations is first obtained by a projection method and then the transport-reaction equation for the conformation tensor is solved by a finite-volume scheme. In order to obtain consistency, the space discretization of the divergence of the elastic part of the stress tensor in the momentum balance equation is derived using a weak form of the MAC scheme. For stability and accuracy reasons, the solution of the transport-reaction equation for the conformation tensor is split into pure convection steps, with a change of variable from \({\mathbf{c}}\) to \(\log ({\mathbf{c}})\), and a reaction step, which consists in solving one ODE per cell via an Euler scheme with local sub-cycling. Numerical computations for the Stokes flow of an Oldroyd-B fluid in the lid-driven cavity at We = 1 confirm the scheme efficiency.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bird, R.B., Wiest, J.M.: Constitutive equations for polymeric liquids p. 25
Boyaval, S., Lelièvre, T., Mangoubi, C.: Free-energy-dissipative schemes for the Oldroyd-B model. Math. Model. Numer. Anal. 43, 523–561 (2009)
CALIF\(^3\)S: A software components library for the computation of fluid flows. https://gforge.irsn.fr/gf/project/califs
Fattal, R., Kupferman, R.: Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation (2005)
Gallouët, T., Herbin, R., Latché, J.C., Mallem, K.: Convergence of the MAC scheme for the incompressible Navier-Stokes equations. Found. Comput. Math. 18(1), 249–289 (2018)
Grapsas, D., Herbin, R., Kheriji, W., Latché, J.C.: An unconditionally stable staggered pressure correction scheme for the compressible Navier-Stokes equations. SMAI J. Comput. Math. 2, 51–97 (2016)
Oishi, C., Martins, F., Tom, M., Cuminato, J., McKee, S.: Numerical solution of the extended pom-pom model for viscoelastic free surface flows. J. Non-Newton. Fluid Mech. 166(3), 165–179 (2011)
Pan, T.W., Hao, J., Glowinski, R.: On the simulation of a time-dependent cavity flow of an Oldroyd-B fluid. Int. J. Numer. Methods Fluids 60, 791–808 (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Mokhtari, O., Davit, Y., Latché, JC., de Loubens, R., Quintard, M. (2020). A Marker-and-Cell Scheme for Viscoelastic Flows on Non Uniform Grids. In: Klöfkorn, R., Keilegavlen, E., Radu, F.A., Fuhrmann, J. (eds) Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples. FVCA 2020. Springer Proceedings in Mathematics & Statistics, vol 323. Springer, Cham. https://doi.org/10.1007/978-3-030-43651-3_61
Download citation
DOI: https://doi.org/10.1007/978-3-030-43651-3_61
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-43650-6
Online ISBN: 978-3-030-43651-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)