# Numerical simulation of the hydrodynamically developing flow of a viscoelastic fluid

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## Abstract

Hydrodynamically developing flow of Oldroyd*B* fluid in the planar die entrance region has been investigated numerically using SIMPLER algorithm in a non-uniform staggered grid system. It has been shown that for constant values of the Reynolds number. the entrance length increases as the Weissenberg number increases. For small Reynolds number flows the center-line velocity distributions exhibit overshoot near the inlet, which seems to be related to the occurrence of numerical breakdown at small values of the limiting Weissenberg number than those for large Reynolds number flows. The distributions of the first normal stress difference display the development of the flow characteristics from extensional flow to shear flow.

## Key Words

Hydrodynamically Developing Flow Viscoelastic Fluid Oldroyd*B*Fluid SIMPLER Algorithm Modified Hybrid Scheme

## Nomenclature

*D*Rate of strain tensor

- \(\bar L\)
Slit halfheight

*P*Pressure, indeterminate part of the Cauchy stress tensor

*R*The Reynolds number

*t*Time

*U*Average velocity in the slit

- \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} \)
Velocity vector

*u, v*Velocity components

*W*The Wissenberg number based on the difference between stress relaxation time and retardation time

*W*_{1}The Weissenberg number based on stress relaxation time

*x,y*Tectangular Cartesian coordinates

- ε
Tatio of retardation time to stress relaxation time

- η
Zero-shear-rate viscosity,

*η*_{1}+*η*_{2}*η*_{1}Non-Newtonian contribution to η

*η*_{2}Newtonian contribution to η

*λ*_{1}Stress relaxation time

*λ*_{2}Ratardation time

- π
Density

- (σ, γ, π)
*xx, yy*and*xy*components of\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\tau } _1 \), respectively- \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\tau } \)
Determinate part of the Cauchy stress tensor

- \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\tau } _1 \)
Non-Newonian contribution to\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\tau } \)

- \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\tau } _2 \)
Newtonian contribution to\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\tau } \)

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## References

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