STEADY FLOW OF AN OLDROYD-B FLUID THROUGH A FOUR-TO-ONE ABRUPT CONTRACTION

Authors:

Khalifa Mohammad Helal,

DOI NO:

https://doi.org/10.26782/jmcms.2020.03.00001

Keywords:

Viscoelastic fluid,Oldroyd-B fluid,Navier-Stokes equations,tensorial transport equations,finite element method,abrupt contraction,

Abstract

This study looks the steady problem which models the behavior of incompressible non-Newtonian viscoelastic Oldroyd-B fluid through a four-to-one abrupt contraction in a bidimensional domain The constitutive equations for the Oldroyd-B fluids consist of highly non-linear system of partial differential equations (PDE) of combined elliptic-hyperbolic type. The numerical results are obtained by a technique of decoupling the system into the Navier-Stokes like problems for the velocity and pressure (elliptic part of the system) and the steady tensorial transport equation for the extra stress tensor (hyperbolic part of the system). To approximate the velocity and pressure,  (Hood-Taylor) finite elements method is used whilethe discontinuous Galerkin finite element method is used to solve the tensorial transport part to approximate the extra stress tensor. Through the flow over four-to-one abrupt contraction domain, the effects of varying the parameters, i.e., i.e., Reynolds number, Weissenberg number, relaxation and retardation time parameter, on the contours of the velocity profile, stream line, pressure and extra stress tensor are presented, analyzed and discussed graphically.   

Refference:

I. A. Ern and J. Guermond, “Discontinuous Galerkin Methods for Friedrichs’ Systems, I. General Theory”, SIAM J. Numer. Anal., Vol. 44, Issue: 2,pp. 753-778, 2006.
II. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer-Verlag 1994.
III. B. Q. Li, Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer. Springer-Verlag, 2006.
IV. C. Fetecau and K. Kannan, “A note on an unsteady fow of an Oldroyd-B fluid”, International Journal of Mathematics and Mathematical Sciences, Vol., 19, pp. 3185–3194, 2015.
V. F. Hecht, “New development in FreeFem++”, Journal of numerical mathematics, Vol. 20, Isssue: 3-4pp. 251-266, 2012.
VI. G. F. Carey and J. T. Oden, Finite elements. Vol.VI. Fluid mechanics. The Texas Finite Element Series, VI. Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1986.
VII. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.
VIII. J. Hron, Numerical Simulation of Visco-Elastic Fluids, In: WDS’ 97, Freiburg, 1997.
IX. K. M. Helal, “Numerical Solutions of Steady Tensorial Transport Equations Using Discontinuous Galerkin Method Implemented in FreeFem++”, Journal of Scientific Research, Vol. 8, Issue: 1, pp.29-39, 2016.
X. K. M. Helal, “Numerical Study and CFD Simulations of Incompressible Newtonian Flow by Solving Steady Navier-Stokes Equations Using Newton’s Method”, Journal of Mechanics of Continua and Mathematical Sciences,Vol. 9, Issue: 2, pp. 1403-1417, 2015.
XI. K. Najib and D. Sandri, On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow, Numer. Math.,Vol. 72, pp. 223-238, 1993.
XII. K. R. Rajagopal, On boundary conditions for fluids of differential type, A. Sequeira (ed.) Navier-Stokes Equations and Related Non-Linear Problems, Plenum Press, 273-278, 1995.
XIII. M. Jamil, C. Fetecau, and M. Imran, “Unsteady helical flows of Oldroyd-B fluids”, Commun. Nonlinear. Sci. Numer. Simulat.,Vol. 16, pp.1378–1386, 2011.
XIV. M. M. Rhaman and K. M. Helal, “Numerical Simulations of Unsteady Navier-Stokes Equations for incompressible Newtonian Fluids using FreeFem++ based on Finite Element Method”, Annals of Pure and Applied Mathematics, Vol.6, Issue: 1, pp. 70-84, 2014.
XV. M. Sulaiman, A. Ali and S. Islam, “Heat and Mass Transfer in Three-Dimensional Flow of an Oldroyd-B Nanofluid with Gyrotactic Micro-Organisms”, Mathematical Problems in Engineering, Vol. 2018, ID 6790420.
XVI. M. Jamil, C. Fetecau, and M. Imran, “Unsteady helical flows of Oldroyd-B fluids”, Commun. Nonlinear. Sci. Numer. Simulat.,Vol. 16, pp.1378–1386, 2011.
XVII. M. Pires, A. Sequeira, “Flows of Generalized Oldroyd-B Fluids in Curved Pipes”, In: Escher J. et al. (eds) Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 80. Springer, Basel, 2011.
XVIII. M. M. Rhaman and K. M. Helal, “Numerical Simulations of Unsteady Navier-Stokes Equations for incompressible Newtonian Fluids using Free Fem++ based on Finite Element Method”, Annals of Pure and Applied Mathematics, Vol.6, Issue: 1, pp. 70-84, 2014.
XIX. M. Sulaiman, A. Ali and S. Islam, “Heat and Mass Transfer in Three-Dimensional Flow of an Oldroyd-B Nanofluid with Gyrotactic Micro-Organisms”, Mathematical Problems in Engineering, Vol. 2018, ID 6790420.
XX. Oldroyd, James, “On the Formulation of Rheological Equations of State”, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 200, Issue: 1063,pp. 523–541, 1950
XXI. P. Lesaint and P. A. Raviart, On a finite element method for solving the neutron transport equation, C. Boor (editor), Mathematical Aspects of Finite Elements in Partial Differential Equations, 89-123, New York, Academic press, 1974.
XXII. P. Saramito, Simulation numeerique decoulements de fluids visco-elastiquespar elements finis incompressible setune methode de directions alternes Applications, These de l’Institut National Polytechnique de Grenoble, 1990.
XXIII. S. A. Shehzad, A. Alsaedi, T. Hayat, and M. S. Alhuthali, “Three-Dimensional Flow of an Oldroyd-B Fluid with Variable Thermal Conductivity and Heat Generation/Absorption”, PLoSONE,Vol. 8, 2013.
XXIV. T. Hayat and A. Alsaedi, “On thermal radiation and Joule heating effects on MHD flow of an Oldroyd-B fluid with thermophoresis”, Arb. J. Sci. Eng.,Vol. 36, pp.1113–1124, 2011.
XXV. T. Hayat, S. A. Shehzad, M. Mustafa, and A. A. Hendi, “MHD flow of an Oldroyd-B fluid through a porous channel”, Int. J. Chem. Reactor Eng., Vol. 10, Article ID A8, 2012.
XXVI. V. Girault and P. A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Computational Mathematics. Springer-Verlag, Berlin, 1986.

View Download