Matched asymptotic solution for flow in a semi-hyperbolic die

doi:10.1016/j.ces.2004.12.043

Chemical Engineering Science

Volume 60, Issue 11, June 2005, Pages 3107-3110

https://doi.org/10.1016/j.ces.2004.12.043 Get rights and content

Abstract

A semi-hyperbolic converging geometry finds application as an inexpensive elongation rheometer under certain flow conditions. We provide a matched asymptotic solution for the flow of a Newtonian fluid under no-slip boundary conditions. The predicted velocity and pressure profiles agree nearly quantitatively with CFD simulated values. Our theoretical approach has certain advantages over the known similarity solution proposed by James (1991. A.I.Ch.E. Journal 37, 59–64).

Introduction

The elongation viscosity of polymeric fluids is an important material property that is relevant for several processing techniques. A converging semi-hyperbolic geometry is known to produce a constant elongation rate under conditions of full slip at the walls (Feigl et al., 2003) or in the limit of infinite Reynolds numbers (inviscid flow) (James, 1991) and thus can be used as a rheometer for measuring the elongation viscosity of polymeric fluids. The solution for the flow of a Newtonian fluid in a semi-hyperbolic die at moderate Reynolds numbers under no-slip boundary conditions was provided by James (1991). His elegant analysis provided a similarity solution for the velocity profile with respect to local reference states. The analysis, however, does not clearly suggest under what flow conditions and where in the geometry the solution is most accurate. Also, the velocity profile is parameterized by a function of axial distance, which itself is not known a priori. We present here an alternative approach for the flow problem based on matched asymptotic solution methodology.

Section snippets

Theory

We consider isothermal steady state flow in an axisymmetric semi-hyperbolic geometry given by $R^{2} (z + B) = C$ , where $C = {LR}_{0}^{2} R_{e}^{2} / (R_{0}^{2} - R_{e}^{2})$ and $B = {LR}_{e}^{2} / (R_{0}^{2} - R_{e}^{2})$ are constants. We assume a plug flow entry velocity $V_{0}$ . The conservation equations in cylindrical coordinates written in a non-dimensional form are given by

Continuity equation: $\frac{1}{r^{*}} \frac{\partial}{\partial r^{*}} (r^{*} V_{r}^{*}) + \frac{\partial V_{z}^{*}}{\partial z^{*}} = 0,$

$z$ -component momentum balance: $\frac{δ^{2} V_{0}}{ν L} [V_{r}^{*} \frac{\partial V_{z}^{*}}{\partial r^{*}} + V_{z}^{*} \frac{\partial V_{z}^{*}}{\partial z^{*}}] = - \frac{δ^{2} P_{0}}{η V_{0} L} \frac{\partial P^{*}}{\partial z^{*}} + \frac{1}{r^{*}} \frac{\partial}{\partial r^{*}} (r^{*} \frac{\partial V_{z}^{*}}{\partial r^{*}}) + \frac{δ^{2}}{L^{2}} \frac{\partial^{2} V_{z}^{*}}{\partial z^{*^{2}}},$

$r$ -component momentum balance: $\frac{δ^{2} V_{0}}{ν L} \frac{δ^{2}}{L^{2}} [V_{r}^{*} \partial]$

Predictions

The form of the velocity profile predicted by Eq. (10) can be easily tested in the limits of large and small values of the spatial variables ( $r, z$ ) as well as for creeping ( $R \to 0$ ) and inertially dominated ( $R \to \infty$ ) flows. Since the importance of the inertial term in Eq. (9) depends on the value of $β$ (or equivalently, $γ (z)$ ), the limits $γ (z) \to 0$ and $γ (z) \to \infty$ correspond, respectively, to $R \to 0$ and $R \to \infty$ . Also, the solution to Eq. (13) shows that $γ (z)$ is a decreasing function of $z$ , so that the limits $γ (z) \to 0$ and $γ ($

Conclusions

We have provided an alternative approach based on matched asymptotic solutions for predicting the velocity profiles of a Newtonian fluid in a semi-hyperbolic die under flow conditions wherein a laminar boundary layer close to the wall co-exists with a plug inviscid core flow. The predicted solutions were found to agree nearly quantitatively compared with CFD simulations. In comparison with the method developed by James (1991), our approach has the advantage of having a closed form solution for

References (3)

K. Feigl et al.
A numerical study of the measurement of elongational viscosity of polymeric fluids in semihyperbolically converging die
Journal of Non-Newtonian Fluid Mechanics
(2003)

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Cited by (2)

Exploring the utility of an axisymmetric semi-hyperbolic die for determining the transient uniaxial elongation viscosity of polymer melts
2007, Journal of Non-Newtonian Fluid Mechanics
The estimation of elongation viscosity of polymeric fluids from pressure drop measurements across converging dies remains an attractive method due to its cost effectiveness. However, the elongation viscosity so measured can be contaminated by large shear effects at the walls unless specific stress boundary conditions can be imposed. Such experiments are in general difficult to perform. In this work we examine an alternative simpler method that relies on a combination of experimental pressure drop measurements in semi-hyperbolic dies and matching viscoelastic CFD simulations to obtain the transient elongation viscosity data. The calculations of the transient uniaxial elongation viscosity from this methodology for an LDPE melt are compared with data obtained using extensional rheometers in order to explore the utility of the strategy.
Measuring the elongation viscosity of lyocell using a semi-hyperbolic die
2006, Rheologica Acta

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Shorter communicationMatched asymptotic solution for flow in a semi-hyperbolic die

Abstract

Introduction

Section snippets

Theory

Predictions

Conclusions

A numerical study of the measurement of elongational viscosity of polymeric fluids in semihyperbolically converging die

Journal of Non-Newtonian Fluid Mechanics

Shorter communication
Matched asymptotic solution for flow in a semi-hyperbolic die