Fluid Mechanics for graduate students
Non-Newtonian fluid mechanics

Delayed die swell

Luca Brandt & Christophe Duwig

1. Introduction

When a viscoelastic fluid is extruded from a pipe, the diameter of the extruded jet usually increases, even to three or four times the diameter of the pipe. At low rates of extrusion the swell appears to start at the pipe exit. At high rates of extrusion the tendency of the jet to swell at the exit is suppressed and there is a delay in the swell.
The die swell is a general phenomenon occurring in any viscoelastic fluid and it is a very important problem in industry because so many processes involve the extrusion of viscoelastic fluids, for example plastic in their molten state. It was observed that delayed die swell is a critical phenomenon, in the sense that it occurs beyond a critical speed of the fluid; this is not so different than the wave speed of small perturbation in a viscoelastic fluid ( think of it in analogy with the speed of sound in compressible flows).
In fact, it may be shown that, if U, velocity of the fluid in the pipe, is larger than the wave speed c, characteristic of the particular fluid, the vorticity of flows perturbing Poiseuille flows of an upper convected Maxwell model obeys an hyperbolic equation.

a) Die Swell

The phenomenon of die swell may be explained by elastic recovery. The molecules are stretched by the shear forces in the pipe and the average axial stress at the exit is a tension. The first normal stress difference, N1, is so fundamental to determine the actual elastic recovery and increase on the diameter of the jet once the shear stresses are removed outside the pipe. The die swell ratio is defined as the ratio of the maximum diameter of the jet and the diameter of the pipe.

2. Governing parameters

We introduce here the governing adimensional parameters with a short discussion.

Weissenberg number
This may be seen as the ratio of the elastic time scale, the relaxation time, lambda, and the convective time scale of the flow U/d, where d is the diameter of the pipe and U the average velocity on the section.
Reynolds number
The well known adimensional number, ratio between inertia and viscous forces but also between the viscous time scale rho d^2/viscosity and the convective one U/d.

Two other parameters, which can be derived from these two, are considered more relevant in the study of delayed die swell.
The viscoelastic Mach number, U/c, where c is the speed of shear waves into a fluid at rest and the Elasticity number.
This is the ratio between the relaxation time and the viscous time scale; it does not depend on the flow condition (inertia) but only on the type of fluid and the experimental apparatus.

a) Speed of shear waves into a fluid at rest

To introduce the speed of shear waves into a fluid at rest we can assume a viscoelastic fluid, described by the Maxwell model, perturbed by pure shear motions ( u=u(y,t) ). The momentum equation linearized for small amplitudes of the perturbation and with fluid at rest, reduces to the second equation in the figure with i=x and j=y.
Introducing the stress tensor as defined by the first relation, we obtain an equation for the disturbances (see figure).
This is the classical hyperbolic wave equation plus a dumping term ( ut); the solutions may be expressed as u=u exp [k (x-ct)], with c complex constant.
Note that in Newtonian fluids, where the relaxation time is zero, the perturbation is damped and spread by viscosity.

3. Experimental results

We present here shortly some conclusions from the experiments of Joseph et al (1987).
For more pictures and details you can look at the slides of our presentation.

In general the delay starts when a point of inflection in the swell profile suddenly appears at the pipe exit as the extrusion velocity is raised above a critical value. The distance from the pipe to this inflection point is denoted L, while L is the distance where the maximum swell diameter is attained.
It is a general phenomenon in elastic liquids; the delay is less apparent in fluids of small viscosity and time of relaxation (quasi-newtonian fluids). The flow at the critical conditions can be steady or unsteady; the latter type of flows emerge at an angle from the axis of the nozzle and the sidewise deviation from the vertical rotates around the pipe.
Fluids with the longest mean times of relaxation resulted unsteady.
The critical velocity, the swell ratio and the terminal swell distance, L, are decreasing functions of the pipe diameter; the critical Mach number at the exit is always larger than one, but after the swell it is definitely less than one.
More energy is put into the flow as he extrusion velocity is increased. The delay distance, L, increases monotonically with the extrusion velocity, whereas the swell ratio first increases and then decreases.
For higher extrusion velocity the experiments produced unsteady flow of fluids with large mean times of relaxation and smooth steady flow for fluids with small mean times of relaxation.

4. Numerical investigations of the delay occurrence.

To understand the phenomenon, numerical simulations have been performed by Delvaux & Crochet (1990). Due to numerical instabilities and difficulties, the values of E (0.03-0.05) used in their computations are out of the range of available experimental data (E>1.00).
Therefore, a validation by comparison with previous experiments is not possible, but despite this restriction, the conclusions obtained from the numerical simulations are fundamental to understand the physical mechanisms involved in the phenomenon.

a) Inertia effects

The delay effect has been noticed when increasing the extrusion velocity. The influence of inertia is investigated by varying the Reynolds number. The figure below, figure 9 in the paper by Delvaux & Crochet (1990), shows the vorticity contour lines and the shape of the jet from two computations performed using an Oldroyd B model. The flow is assumed axisymmetric and only one half of a longitudinal section is displayed. In figure (a) Re=24.1 and We=0.7; in (b) the Reynolds number has been set to zero to eliminate inertia, while the Weissenberg number is kept constant. It is shown clearly that, without inertial effects, the swell occurs just after the pipe exit and no delay is observed. The vorticity varies almost linearly in the wall normal direction inside the pipe, but it changes immediately to zero outside because of the different boundary conditions (no shear stress). For a large enough value of Re instead, the vorticity in the middle of the jet results unaffected by the change in boundary conditions for a certain distance downstream and the swell is delayed. Therefore inertial effects are needed to understand delayed die swell phenomena.

b) Influence of elasticity

The influence of elasticity has also been investigated by Delvaux & Crochet(1990). Several calculations have been carried out keeping M constant above the critical value and varying E between 0.03 and 0.05. The Giesekus fluid model has been used and the results are reported in figure 13. The influence of the E on the position of the inflection point is noticed, i.e. it appears closer to the pipe exit for larger values. Note also that the swell ratio increases when E increases !


c) Information transmission

To investigate the influence of information transmission, the type of the governing partial differential equations is investigated. An Oldroyd B model has been used  for these calculations. The dark areas in figure 6(c) correspond to the regions of hyperbolicity of the vorticity equation and different viscoelastic Mach numbers have been used to show the evolution of the delay. The finite elements meshes and the the stream function are also displayed in figure 6. Comparison between the streamlines and the filled zones shows that there is a close correspondence between the hyperbolic governed regions and the regions where delay is observed (no swell of the lines).

5. Conclusions

The following points are summarizing the (our) understanding on this phenomenon.


Question

34. Describe the die swell and delayed die swell phenomena, what are the causes and which dimensionless numbers are used to describe them ?
 
 
 
 

Bibliography

Joseph D. D., Matta J. E., Chen K.  'Delayed die swell', J. Non-newtonian Fluid Mech 1987, 24:31,65.

Joseph D. D.,  'Fluid Dynamics of Viscoelastic Liquids', Spinger 1990, Ch 13.

Delvaux V., Crochet M. J.  'Numerical simulations of delayed die swell', Rheologica Acta 1990, 29:1-10.

Joseph D. D., Christodoulou C.  'Independent confirmation that delayed die swell is a hyperbolic transition', J. Non-newtonian Fluid Mech 1993, 79:225,235.

Cloitre M., Hall T., Mata C., Joseph D. D.  'Delayed die swell and sedimentation of elongated particles in wormlike micellar solutions', J. Non-newtonian Fluid Mech 1998, 79:157,171.

 Overheads of the presentation