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TECHNICAL PAPERS

# Annular Extrudate Swell of Newtonian Fluids: Effects of Compressibility and Slip at the Wall

[+] Author and Article Information
Evan Mitsoulis

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

J. Fluids Eng 129(11), 1384-1393 (Jun 05, 2007) (10 pages) doi:10.1115/1.2786491 History: Received December 14, 2006; Revised June 05, 2007

## Abstract

Numerical simulations have been undertaken for the benchmark problem of annular extrudate swell present in pipe extrusion and parison formation in blow molding. The effects of weak compressibility and slip at the wall are studied through simple linear laws. The finite element method is used to provide numerical results for different inner/outer diameter ratios $κ$ under steady-state conditions for Newtonian fluids. The present results provide the shape of the extrudate, and, in particular, the thickness and diameter swells, as a function of the dimensionless compressibility and slip coefficients, $B$ and $Bsl$, respectively. The pressures from the simulations have been used to compute the excess pressure losses in the flow field (exit correction). Weak compressibility slightly affects the thickness swell (about 1% in the range of simulations $0⩽B⩽0.02$) mainly by a swell reduction, while slip drastically reduces the swelling to 1–2% for obvious slip $(Bsl≈1)$ and to 0 for perfect slip $(Bsl>10)$. The exit correction increases with increasing compressibility levels and is highest for the tube $(κ=0)$ and lowest for the slit $(κ=1)$. It decreases monotonically to 0 as the dimensionless slip coefficient reaches its asymptotic limit of perfect slip. All results are ordered with the diameter ratio $κ$, between the limits of tube $(κ=0)$ and slit $(κ=1)$.

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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 diagram 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 (see also Table 1 for details).

Figure 4

Poiseuille flow of weakly compressible Newtonian fluids. Axial pressure and velocity distributions along the centerline: (a) pressure (planar flow), (b) velocity (planar flow), (c) pressure (axisymmetric flow), and (d) velocity (axisymmetric flow).

Figure 5

Extrudate swell of Newtonian fluids for κ=0 (tube): (a) incompressible fluid (B=0); (b) weakly compressible fluid (B=0.02)

Figure 6

Thickness swell B2 as a function of the compressibility coefficient B for Newtonian fluids obeying a linear equation of state (Eq. 4)

Figure 7

Diameter swell B1 as a function of the compressibility coefficient B for Newtonian fluids obeying a linear equation of state (Eq. 4)

Figure 8

Inner diameter swell B3 as a function of the compressibility coefficient B for Newtonian fluids obeying a linear equation of state (Eq. 4)

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 (Eq. 4)

Figure 10

Exit correction nen as a function of the compressibility coefficient B for Newtonian fluids obeying a linear equation of state (Eq. 4)

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 (Eq. 4)

Figure 12

Annular extrudate swell of Newtonian fluids for κ=0.25: (a) Bsl=0 (no slip); (b) Bsl=1 (obvious slip)

Figure 13

Thickness swell B2 as a function of the slip coefficient Bsl for Newtonian fluids obeying a linear slip law (Eq. 6)

Figure 14

Diameter swell B1 as a function of the slip coefficient Bsl for Newtonian fluids obeying a linear slip law (Eq. 6)

Figure 15

Inner diameter swell B3 as a function of the slip coefficient Bsl for Newtonian fluids obeying a linear slip law (Eq. 6)

Figure 16

Thickness swell B2 as a function of the diameter ratio κ for Newtonian fluids obeying a linear slip law (Eq. 6)

Figure 17

Exit correction nen as a function of the slip coefficient Bsl for Newtonian fluids obeying a linear slip law (Eq. 6)

Figure 18

Exit correction nex as a function of the diameter ratio κ for Newtonian fluids obeying a linear slip law (Eq. 6)

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