Numerical simulations and global existence of solutions of two-dimensional flows of fluids with pressure- and shear-dependent viscosities

  • Authors:

  • J. Hron;J. Málek;J. Nečas;K. R. Rajagopal

  • Affiliations:

  • Institute for Applied Mathematics and Numerics, University of Dortmund, Vogelpothsweg 87, 44227 Dortmund, Germany;Mathematical Institute of Charles University, Sokolovská 83, 18675 Prague 8, Czech Republic;Mathematical Institute of Charles University, Sokolovská 83, 18675 Prague 8, Czech Republic and Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL;Department of Mechanical Engineering, Texas A&M University, College Station, TX

  • Venue:
  • Mathematics and Computers in Simulation - MODELLING 2001 - Second IMACS conference on mathematical modelling and computational methods in mechanics, physics, biomechanics and geodynamics
  • Year:
  • 2003

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Abstract

There is a considerable amount of experimental evidence that unequivocally shows that there are fluids whose viscosity depends on both the mean normal stress (pressure) and the shear rate. Recently, global existence of solutions for the flow of such fluids for the three-dimensional case was established by Málek, Necas and Rajagopal. Here, we present a proof for the global existence of solutions for such fluids for the two-dimensional case. After establishing the global-in-time existence, we discretize the equations via the finite element method, outline the Newton type iterative method to solve the non-linear algebraic equations and provide numerical computations of the steady flow of such fluids in geometries that have technological significance.