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Research Papers: Fundamental Issues and Canonical Flows

# Annular Extrudate Swell of Newtonian Fluids Revisited: Extended Range of Compressible Simulations

[+] Author and Article Information
Evan Mitsoulis1

School of Mining Engineering and Metallurgy, National Technical University of Athens, Zografou, 15780 Athens, Greecemitsouli@metal.ntua.gr

1

Corresponding author.

J. Fluids Eng 131(7), 071203 (Jun 24, 2009) (10 pages) doi:10.1115/1.3155996 History: Received January 15, 2009; Revised May 07, 2009; Published June 24, 2009

## Abstract

In a recent article (Mitsoulis, 2007, “Annular Extrudate Swell of Newtonian Fluids: Effects of Compressibility and Slip at the Wall,” ASME J. Fluids Eng., 129, pp. 1384–1393), numerical simulations were undertaken for the benchmark problem of annular extrudate swell of Newtonian fluids. The effects of weak compressibility and slip at the wall were studied through simple linear laws. While slip was studied in the full range of parameter values, compressibility was confined within a narrow range of values for weakly compressible fluids, where the results were slightly affected. This range is now markedly extended (threefold), based on a consistent finite element method formulation for the continuity equation. Such results correspond to foam extrusion, where compressibility can be substantial. The new extended numerical results are given for different inner/outer diameter ratios $κ$ under steady-state conditions for Newtonian fluids. They provide the shape of the extrudate, and, in particular, the thickness and diameter swells, as a function of the dimensionless compressibility coefficient $B$. The pressures from the simulations have been used to compute the excess pressure losses in the flow field (exit correction). As before, weak compressibility slightly affects the thickness swell (about 1% in the range of $0≤B≤0.02$) mainly by a swell reduction, after which a substantial and monotonic increase occurs for $B>0.02$. The exit correction increases with increasing compressibility levels in the lower $B$-range and is highest for the tube $(κ=0)$ and lowest for the slit $(κ=1)$. Then it passes through a maximum around $B≈0.02$, after which it decreases slowly. This decrease is attributed to the limited length of the flow channel (here chosen to be eight die gaps).

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## Figures

Figure 1

Schematic representation of extrusion through an annular die and notation for the numerical analysis (2)

Figure 2

Schematic of flow domain and boundary conditions

Figure 3

Finite element meshes used in the computations. The upper half shows mesh M3 containing 2240 quadrilateral elements, while the lower half shows mesh M1 containing 560 elements.

Figure 4

Extrudate swell of compressible Newtonian fluids obeying a linear equation of state. Comparison of results with Ref. 17 for a domain with L0=5: (a) extrudate swell and (b) exit correction based on centerline values.

Figure 5

Extrudate swell of compressible Newtonian fluids obeying a linear equation of state. Contour results at the limiting value Blm: (a) tube flow (κ=0, B=0.11) and (b) slit flow (κ=1, B=0.22). STR=stream function, U=axial velocity, P=pressure, and TXY=shear stress.

Figure 6

Thickness swell B2 as a function of the compressibility coefficient B for Newtonian fluids obeying a linear equation of state. The vertical line corresponds to the limit of previous calculations using the nonconsistent FEM (13).

Figure 7

Diameter swell B1 as a function of the compressibility coefficient B for Newtonian fluids obeying a linear equation of state. The vertical line corresponds to the limit of previous calculations using the nonconsistent FEM (13).

Figure 8

Inner diameter swell B3 as a function of the compressibility coefficient B for Newtonian fluids obeying a linear equation of state. The vertical line corresponds to the limit of previous calculations using the nonconsistent FEM (13).

Figure 9

Thickness swell B2 as a function of the diameter ratio κ for various values of the compressibility coefficient B for Newtonian fluids obeying a linear equation of state

Figure 10

Exit correction nex as a function of the compressibility coefficient B for Newtonian fluids obeying a linear equation of state. The vertical line corresponds to the limit of previous calculations using the nonconsistent FEM (13). Note that the exit correction is based on pressure values calculated at the outer die wall.

Figure 11

Exit correction nex as a function of the diameter ratio κ for various values of the compressibility coefficient B for Newtonian fluids obeying a linear equation of state. Note that the exit correction is based on pressure values calculated at the outer die wall.

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