Abstract
In this work, we are interested in the modelling of diphasic fluids flows in thin films. The diphasic aspect is described by a diffuse interface model, the Cahn-Hilliard equation. The specific geometry (thin domain) allows to replace heuristically the usual Navier-Stokes equations by an asymptotic approximation, a modified Reynolds equation (in which the pressure and the velocity are uncoupled), where the viscosity depends on the composition of the mixture. An existence result on the limit system is stated. since the boundary conditions are chosen in order to model the injection phenomenon, previous results on the Cahn-Hilliard equation cannot be applied, and new estimates have to be obtained. Moreover, we present numerical simulations for lubrication applications to improve the understanding of the cavitation phenomenon.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Assemien, A., Bayada, G., Chambat, M.: Inertial effects in the asymptotic behavior of a thin film flow. Asymptot. Anal. 9(3), 177–208 (1994)
Bayada, G., Chambat, M.: The transition between the Stokes equations and the Reynolds equation: A mathematical proof. Appl. Math. Optim. 14(1), 73–93 (1986)
Bayada, G., Chambat, M., Ciuperca, I.: Asymptotic Navier-Stokes equations in a thin moving boundary domain. Asymptot. Anal. 21(2), 117–132 (1999)
Bayada, G., Chupin, L., Grec, B.: Some theoretical results concerning diphasic fluids in thin films. Submitted (2009)
Bayada, G., Martin, S., Vázquez, C.: About a generalized Buckley-Leverett equation and lubrication multifluid flow. Eur. J. Appl. Math. 17(5), 491–524 (2006)
Boyer, F.: Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot. Anal. 20(2), 175–212 (1999)
Boyer, F.: Theoretical and numerical study of multi-phase flows through order parameter formulation. In: International Conference on Differential Equations, Vols. 1, 2 (Berlin, 1999), 488–490. World Science Publication, River Edge, NJ (2000)
Boyer, F.: Nonhomogeneous Cahn-Hilliard fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 18(2), 225–259 (2001)
Boyer, F., Chupin, L., Fabrie, P.: Numerical study of viscoelastic mixtures through a Cahn-Hilliard flow model. Eur. J. Mech. B Fluids 23(5), 759–780 (2004)
Chupin, L.: Existence result for a mixture of non Newtonian flows with stress diffusion using the Cahn-Hilliard formulation. Discrete Contin. Dyn. Syst. Ser. B 3(1), 45–68 (2003)
Godlewski, E., Raviart, P.A.: Hyperbolic Systems of Conservation Laws, Mathématiques & Applications (Paris) [Mathematics and Applications], Vol. 3/4. Ellipses, Paris (1991)
Marušić-Paloka, E., Stracević, M.: Rigorous justification of the Reynolds equations for gas lubrication. C. R. Mécanique 33(7), 534–541 (2005)
Paoli, L.: Asymptotic behavior of a two fluid flow in a thin domain: From Stokes equations to Buckley-Leverett equation and Reynolds law. Asymptot. Anal. 34(2), 93–120 (2003)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Bayada, G., Chupin, L., Grec, B. (2010). A New Model of Diphasic Fluids in Thin Films. In: Rannacher, R., Sequeira, A. (eds) Advances in Mathematical Fluid Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04068-9_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-04068-9_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04067-2
Online ISBN: 978-3-642-04068-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)