A three-field formulation for incompressible viscoelastic fluids

This paper presents a new stabilized finite element method for incompressible viscoelastic fluids. A three-field formulation is developed wherein Oldroyd-B model is coupled with the mass and momentum conservation equations for an incompressible viscous fluid. The variational multiscale (VMS) framework is employed to develop a stabilized formulation for the coupled momentum, continuity and stress equations. Based on the new stabilized method a family of linear and higher-order triangle and quadrilateral elements with equal-order velocity–pressure–stress fields is developed. Stability and convergence property of the various elements is studied and optimal rates are attained in the norms considered. The method is applied to some benchmark problems and accuracy and computational economy of the formulation is investigated for various flow conditions.

[1]  Christopher P. Cheng,et al.  In Vivo Quantification of Blood Flow and Wall Shear Stress in the Human Abdominal Aorta During Lower Limb Exercise , 2002, Annals of Biomedical Engineering.

[2]  M. Fortin,et al.  A new approach for the FEM simulation of viscoelastic flows , 1989 .

[3]  M. Fortin,et al.  A new mixed finite element method for computing viscoelastic flows , 1995 .

[4]  U. Ghia,et al.  High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method , 1982 .

[5]  T. Hughes,et al.  Isogeometric Fluid–structure Interaction Analysis with Applications to Arterial Blood Flow , 2006 .

[6]  M. Crochet,et al.  A new mixed finite element for calculating viscoelastic flow , 1987 .

[7]  G. Thurston,et al.  Rheological parameters for the viscosity viscoelasticity and thixotropy of blood. , 1979, Biorheology.

[8]  T. Tezduyar,et al.  Arterial fluid mechanics modeling with the stabilized space–time fluid–structure interaction technique , 2008 .

[9]  M. Anand,et al.  A SHEAR-THINNING VISCOELASTIC FLUID MODEL FOR DESCRIBING THE FLOW OF BLOOD , 2004 .

[10]  A. Masud,et al.  A stabilized mixed finite element method for the incompressible shear-rate dependent non-Newtonian fluids: Variational Multiscale framework and consistent linearization , 2011 .

[11]  Arif Masud,et al.  A variational multiscale stabilized formulation for the incompressible Navier–Stokes equations , 2009 .

[12]  K. Rajagopal,et al.  A thermodynamic frame work for rate type fluid models , 2000 .

[13]  J. Reddy,et al.  The Finite Element Method in Heat Transfer and Fluid Dynamics , 1994 .

[14]  Raanan Fattal,et al.  Flow of viscoelastic fluids past a cylinder at high Weissenberg number : stabilized simulations using matrix logarithms , 2005 .

[15]  T. Mexia,et al.  Author ' s personal copy , 2009 .

[16]  L. Franca,et al.  A hierarchical multiscale framework for problems with multiscale source terms , 2008 .

[17]  R. Armstrong,et al.  Finite element methdos for calculation of steady, viscoelastic flow using constitutive equations with a Newtonian viscosity , 1990 .

[18]  Franco Brezzi,et al.  $b=\int g$ , 1997 .

[19]  Karl Perktold,et al.  Computational Models of Arterial Flow and Mass Transport , 2003 .

[20]  Thomas J. R. Hughes,et al.  Finite element modeling of blood flow in arteries , 1998 .

[21]  T. Hughes.  Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods , 1995 .

[22]  Kumbakonam R. Rajagopal,et al.  A Model Incorporating Some of the Mechanical and Biochemical Factors Underlying Clot Formation and Dissolution in Flowing Blood , 2003 .

[23]  Charles A. Taylor,et al.  In Vivo Validation of Numerical Prediction of Blood Flow in Arterial Bypass Grafts , 2002, Annals of Biomedical Engineering.

[24]  N. Phan-Thien,et al.  An adaptive viscoelastic stress splitting scheme and its applications: AVSS/SI and AVSS/SUPG , 1996 .

[25]  Robert C. Armstrong,et al.  Finite element method for viscoelastic flows based on the discrete adaptive viscoelastic stress splitting and the discontinuous Galerkin method : DAVSS-G/DG , 1999 .

[26]  T. Hughes,et al.  The variational multiscale method—a paradigm for computational mechanics , 1998 .

[27]  Marek Behr,et al.  Four-field Galerkin/least-squares formulation for viscoelastic fluids , 2006 .

[28]  K. Perktold,et al.  Computer simulation of local blood flow and vessel mechanics in a compliant carotid artery bifurcation model. , 1995, Journal of biomechanics.

[29]  Arif Masud,et al.  A multiscale finite element method for the incompressible Navier-Stokes equations , 2006 .

[30]  K. Rajagopal,et al.  The flow of blood in tubes: theory and experiment , 1998 .

[31]  F. Baaijens.  Mixed finite element methods for viscoelastic flow analysis : a review , 1998 .

[32]  Gianni Pedrizzetti,et al.  Cardiovascular fluid mechanics , 2003 .

[33]  Thomas J. R. Hughes,et al.  Patient-Specific Vascular NURBS Modeling for Isogeometric Analysis of Blood Flow , 2007, IMR.

[34]  Calculation of steady viscoelastic flow using a multimode Maxwell model : application of the explicitly elliptic momentum equation (EEME) formulation , 1990 .

[35]  Arif Masud,et al.  Revisiting stabilized finite element methods for the advective–diffusive equation , 2006 .

[36]  Guglielmo Scovazzi,et al.  A heterogeneous multiscale modeling framework for hierarchical systems of partial differential equations , 2011 .