#### 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.*

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