An approximate solution to flow through a contraction for high Trouton ratio fluids

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Abstract

An approximate analytical solution is presented to predict the flow, vortex and pressure drop behaviour in abrupt axisymmetric contractions. This solution is developed to identify the effect of fluids which exhibit large Trouton ratios at large extensional rates despite having a constant shear viscosity. This behaviour approximates that shown in dilute polymeric solutions, although any transient effects are neglected by removing any strain dependence of the extensional viscosity and by assuming that the steady-state extensional viscosity undergoes a step change from a Trouton ratio of 3 to a large Trouton ratio at a critical rate. The predictions compare well to available experimental results for dilute to semi-dilute solutions of a flexible polymer, polystyrene, prior to the onset of elastic instabilities and for dilute solutions of a semi-rigid polymer, xanthan. The results show that vortex behaviour is determined by the fluid’s characteristic time (through the magnitude of the Weissenberg number of the flow), while the steady-state extensional viscosity is of far more importance than transient effects in determining the steady-state pressure drop.

Introduction

Contraction flows have received much attention within both academic and industrial communities over the last century due to their utilisation in many industrial processes where significant extensional rates are required to effect desired outcomes. One example is the enhancement of drop break-up in homogenisation processes. Many researchers have previously worked on understanding contraction flows either experimentally, numerically or analytically. Both experimental and numerical works have focussed on investigating the relationships between fluid rheology, and vortex behaviour and/or pressure drop [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], whilst most theoretical works have focussed on the use of contraction flows or converging flows to measure an apparent extensional viscosity [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. Several researchers have attempted to validate these theoretical works with experiments aimed at measuring the extensional viscosity of a fluid using a converging flow field.

Two methods in particular have enjoyed popularity. The first of these methods utilises an abrupt contraction where, by measuring the pressure drop and comparing it to an analytical model, the extensional viscosity can be approximated [12], [13], [23]. The works of Binding [12], [13] have received particular attention over the past decade. Binding made the assumption that the fluid adopts a vortex configuration which minimises the energy consumption through the converging section bounded by a radius R, with a zero velocity at r=R. Using variational calculus, R can be evaluated by the requirement that the energy consumption is a minimum. This method has the advantage that there is a significant check on the results because the pressure drop and vortex reattachment length serve to provide independent predictions of the extensional viscosity. It has the disadvantage, however, that the extension rate can vary significantly through the converging section, so that a fluid with a strongly rate dependent shear and extensional response can provide extremely misleading results. Another disadvantage is that it assumes that the vortex will meet the entrance of the contraction at the inner wall, although this may not be the case for all fluids of interest. Mongruel and Cloitre [23] adopted a slightly different approach to the same problem, where they used a spherical solution close to the orifice to predict the extensional viscosity for fluids flowing through an abrupt contraction. Assuming that the vortex makes a small angle to the axial direction when it hits the orifice, they were able to approximate the flow-field and pressure drop for dilute suspensions of rods. They found that close to the orifice the pressure drop is dominated by the extensional response. Conversely, the size and shape of the vortex was determined by the fluid rheology across the entire converging region, even in regions before the likelihood of any fibre alignment.

The second method relies on an a priori assumption for the flow field of interest, and a converging section is thus fabricated to elicit the desired behaviour in the fluid [14], [18], [20]. When using a specially fabricated converging section, two extreme cases exist: zero velocity at the wall and full slip at the wall. In the first case, the apparent extensional viscosity can be determined theoretically, although there will be regions with significant shear, which can increase the measurement errors dramatically. There is also the disadvantage that, if the wall is actually within the region which would otherwise have been a vortex in an abrupt contraction, the results may again be somewhat misleading. The applied strain rate then would not equal the prescribed strain rate. Alternatively, when the wall is lubricated so there is slip, both a constant extension rate and shear-free system are attainable. This has the advantage that a homogeneous, shear-free extensional flow can be developed, but it has certain practical disadvantages relating to the adherence to the slip condition or to the effect of using a lubricant. Kim et al. [18] found for polymer melts that acceptable lubrication was attained if the lubricant had a shear viscosity 1/100th that of the test materials.

In this present work, we develop a new approach based on the initial works of Binding to predict the flow-field in an abrupt axisymmetric contraction. Like Binding [12], we assume that the behaviour is such that the velocity profile in the axial direction at any point in the contraction will have the same velocity distribution as that for fully developed laminar flow in a pipe of that diameter, and that the radius of the converging section will be determined by the fluid’s response to minimise the energy consumption. Unlike Binding [12], who assumed a power-law response to both shear and extensional conditions, we have assumed that the fluid will have a response whereby the shear viscosity is constant, but the extensional viscosity will be such that the Trouton ratio, Tr, is 3 when the Deborah number, De, is less than 0.5 and will suddenly increase to a much higher value when De is greater than 0.5.

The fluid behaviour is chosen to represent the behaviour of dilute polymer or fibre systems, where the shear viscosity will be constant but the extensional response will rapidly increase, with Tr going from 3 to a large value at De0.5, where De is based on the relaxation time for the alignment (in the case of fibres or rod-like polymers) or the coil-stretch transition (in the case of flexible molecules). This model has certain disadvantages in capturing the fluid behaviour compared with rigorous constitutive equations such as the FENE models. Deviations from real fluid behaviour will be particularly noticeable because describing the fluid in such a simplistic manner ignores the role of strain in achieving large extensional viscosities. Mongruel and Cloitre’s [23] constitutive model (consisting of a solvent contribution and an anisotropic contribution due to the fibre orientation) is also more rigorous than our simplified model, however, it is slightly less advantageous, though, as it does not make use of an apparent characteristic time. Consequently, while Mongruel and Cloitre can obtain an apparent extensional viscosity from their experiments, they are less well equipped to measure the dynamics of the fibre alignment, and hence the transition from low to high Trouton ratios.

The simplicity afforded by choosing a step change in extensional behaviour enables the general procedure adopted by Binding [12] to be used. This has the advantage of providing a relatively simple technique to identify features in contraction flow which can be caused by a high steady state Tr in strong flow and a Trouton ratio of 3 in weak flow with a critical relaxation time between the two regions. Based on the resultant behaviour from this model, the critical time for the onset of a high Trouton ratio can be approximated by the shape and length of the vortex, while the extensional viscosity enhancement can be predicted by measuring the pressure drop if the shear viscosity is known.

Section snippets

Definitions and assumptions

The following dimensionless quantities are used:De=λϵ˙,Re=Qρ2πηsR0,Wi0=Qλ2πR03,X=L2R1,β=R1R0,P=PeπR038Qηs.where ηs is the shear viscosity, λ the characteristic relaxation time for the onset of strong flow, ϵ˙ the local extension rate, R the radius at a position z, Q the volumetric flow rate, ρ the fluid density, R1 the upstream pipe radius, R0 the downstream pipe radius and L is the vortex reattachment length (i.e. the distance between the contraction and where the vortex first detaches from

Flow kinematics

At z<0, R=R1 and, using the assumption of fully developed flow, it is apparent that vr=0 and thus the Navier–Stokes equations can be solved to provide:vz=2QπR121rR12.Assuming the same velocity profile throughout the converging section, R1 can be replaced with R(z) and, using the continuity equation, the velocity and deformation profiles can be given as:vz=2QπR(z)21rR(z)2,vr=2QR(z)πR(z)2rR(z)1rR(z)2,Drr=2QR(z)πR(z)313rR(z)2,Dθθ=2QR(z)πR(z)31rR(z)2,Dzz=4QR(z)πR(z)32rR(z)21,andDrz=QπR(z)3

Weak flow solution

The first step to solving the problem is to determine the solution for regions where the rate is less than that required for the onset of strong flow. These regions are not known in advance, and their determination will be shown in Sections 5 Onset of strong flow, 6 Envelope of strong flow. In weak flow, E˙es=0, so the rate of energy dissipation at any z can be given by:E˙=ηsQ23πR4(24+9R4+RR(RR8)2R2(4+RR))+16Q2ηsR2πR4.Using variational calculus, an equation in R(z) can be set-up to

Onset of strong flow

The next step in the problem solution is to find the onset of strong flow. This is where this work differs fundamentally from the works of Binding [12], [13]. By examining Eq. (12), it can be seen that ϵ˙ will be at a maximum at r=0 (and will have a value of equal magnitude but opposite sign at r=R). Consequently, the onset of strong flow will occur first at r=0, and can be determined by substituting Eq. (19), the weak flow solution, into (12) and solving for De=1/2. The limitations and

Envelope of strong flow

For the range of Wi0 where there will be regions of strong flow, the next step is to determine the envelope where strong flow will occur. Finding this will enable E˙es to be determined.

Noting that ϵ˙, and hence De, can be both positive and negative, Eq. (12) gives:4QλR(z)πR(z)32rR(z)21=±12,orr=R12±(R/R0)332Wi0R.Noting that R is negative, the two envelopes are 0rR1/2+(R/R0)3/(32Wi0R) and R1/2(R/R0)3/(32Wi0R)rR. Therefore:E˙es=0R1/2+(R/R0)3/(32Wi0R)2πTr+ϵ˙2rdrδz+R1/2(R/R0)3/(32Wi0R

Pressure drop

Using the same rationale as Binding [12], the significant contributions to the pressure drop through the contraction zone are the same as those accounted for when determining R(z). Consequently, the pressure drop through the contraction can be expressed as (assuming that the pressure drop which would otherwise exist due to fully developed laminar flow is negligible):Pe=E˙Q.

Dividing E˙ into two parts, before and after the onset of strong flow we obtain:Pe=0zcritE˙dz+zcritLE˙dzQ,=0zcritE˙s+E˙ew

Effect of non-zero velocity at the vortex boundary

If, rather than the velocity at r=R(z) being zero, the velocity is some small value, then the qualitative behaviour – such as the transition from biaxial extension at the edge of the vortex, through to zero extension and through to uniaxial extension at the centre – will remain unchanged. This can be shown as follows:

Suppose that the velocity at r=R is given a value, V(z). V(z) will have the following properties:

  • (1)

    V(z) will be tangential to the streamline for r=R(z). Consequently, vz at r=R(z)

Results and discussion

Fig. 2 compares the predicted R(z) from Eq. (19)(dotted line) with a streamline near the vortex boundary from a numerical solution performed using a commercial package, FEMLAB 3.0 (dashed line). The numerical solution is for a Newtonian fluid in a 4-1 contraction, neglecting the inertial terms. The vortex size and shape is independent of fluid viscosity or pressure drop, as is expected for Newtonian fluids in the limit of zero Reynold’s number. From Fig. 2, it can be seen that, while the

Conclusion

An approximate solution based on Binding’s analysis [12] has been presented for flow through an abrupt axisymmetric contraction for a fluid representing a suspension of elongated fibres or particles. This solution examines the effect of a critical relaxation time and a high Trouton ratio on flow through an axisymmetric contraction. Despite being a very simplistic three parameter analysis, the model is able to capture several features identified in experimental works with flexible to semi-rigid

References (24)

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