Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 

 

An existence theorem for the multi-fluid Stokes problem


Authors: A. Nouri, F. Poupaud and Y. Demay
Journal: Quart. Appl. Math. 55 (1997), 421-435
MSC: Primary 35Q30; Secondary 76D05, 76D07
DOI: https://doi.org/10.1090/qam/1466141
MathSciNet review: MR1466141
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Time-dependent flows of viscous incompressible immiscible fluids are studied in the limit of vanishing Reynolds numbers. The velocity fields associated to each fluid solve Stokes equations in a time-dependent domain. Classical immiscibility conditions on the varying fluids interfaces are taken into account by a new formulation of the problem: the viscosity solves a transport equation and the velocity field solves a Stokes problem with this nonconstant viscosity. This formulation, based on the use of a pseudoconcentration function, has already been used for numerical computations (see [9] and [4]). For this nonlinear system of equations, existence of solutions is proved, using the Schauder fixed point theorem and the concept of renormalized solutions introduced recently by R. J. DiPerna and P. L. Lions.


References [Enhancements On Off] (What's this?)

  • [1] G. Astarita and G. Marrucci, Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, New York, 1974
  • [2] Claude Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d’approximation; application à l’équation de transport, Ann. Sci. École Norm. Sup. (4) 3 (1970), 185–233 (French). MR 0274925
  • [3] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), no. 3, 511–547. MR 1022305, https://doi.org/10.1007/BF01393835
  • [4] A. Fortin, Y. Demay, and J. F. Agassant, Computation of stationary interfaces in coextrusion flows, Polymer Engrg. Sci. 34, 1101-1108 (1994)
  • [5] K. Friedrichs, Symmetric positive systems of differential equations, Comm. Pure Appl. Math. 7, 345-392 (1954)
  • [6] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). MR 0259693
  • [7] O. A. Oleinik, Boundary value problems for elasticity theory in unbounded domains, Russian Math. Surveys 43 5, 65-119 (1988)
  • [8] Roger Temam, Navier-Stokes equations, 3rd ed., Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. Theory and numerical analysis; With an appendix by F. Thomasset. MR 769654
  • [9] S. F. Shen, Grapplings with the simulation of non-Newtonian flows in polymer processing, Internat. J. Numer. Methods Engrg. 34, 701-723 (1992)
  • [10] C. W. Hirt and B. D. Nichols, Volume of Fluids (VOF) method for the dynamics of free boundaries, J. Comput. Phys. 39, 201-225 (1981)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 35Q30, 76D05, 76D07

Retrieve articles in all journals with MSC: 35Q30, 76D05, 76D07


Additional Information

DOI: https://doi.org/10.1090/qam/1466141
Article copyright: © Copyright 1997 American Mathematical Society


Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website