# Analysis of viscoelastic fluid flow in different geometries abrupt contraction

### Abstract

In this work, two-dimension incompressible laminar viscoelastic fluid flow in an expansion-contraction duct has been investigated numerically. The effect of expansion-contraction shape, Deborah number, Reynolds number, expansion ratio and contraction length on the flow field variables has been investigated. Four cases for the expansion-contraction duct configurations are tested, the first case is the expansion duct, the second case is the contraction-expansion duct, the third case is the triangular expansion-contraction duct and the fourth case is the two-side triangular expansion-contraction duct. The study investigated the effect of the parameters: Reynolds number (Re), Deborah number (De), contraction length and expansion ratio (ER) on the flow field variables (pressure, stresses and velocity).

The FENE-P (Finitely Extensible Nonlinear Elastic in the Peterlin approximation) viscoelastic fluid model is chosen for the modeling of viscoelastic fluid flow. The code is written in Simply Fortran 2 computer language. Numerically the momentum equations and the FENE-P model equation are solved by using the finite volume method passed on the collocated grid arrangement by using the Rhie-Chow interpolation. The power low scheme is used for the convective terms in the momentum equations, the up-wind scheme is used in the convective terms in the FENE-P model equation and the TDMA algorithm is used to solve the algebraic equations.

The results show that the expansion ratio when decreasing causes an increase in the stresses and the pressure for all the studied cases. Further, the contraction length increment playing a major effect on the increasing of the pressure. It is discovered that increasing contraction length by 50% causes an increase in the pressure by 50%. For Deborah number, the results show that Deborah number effects all variables in the flow field, where the stresses show increasing with the increasing of Deborah number at the expansion side of all the cases. Moreover, the pressure is also increased as Deborah number increase for all the cases. The expansion duct it shows that increasing Deborah number from 0 to 1 causes an increase of 1.2% of the pressure, while for the contraction-expansion duct this percentage is increased. It is shown that increasing Deborah from 0 to 0.5 causes an increasing by 5% of the pressure. **Furthermore,** the effect of Deborah number on the recirculation zones is to shrink their size for all the cases.

A comparison between the studied cases are presented and shows that the contraction-expansion duct has the maximum stresses and maximum required pressure, while the triangle contraction-expansion has the smallest required pressure. The limit of Deborah number gets to 1.2 in the expansion duct, while it is limited to 0.5 for the other cases.