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Contraction and expansion flows of non-Newtonian fluids
Contraction and expansion flows of non-Newtonian fluids T. Abdul-Karem,
D.M. Binding and M. Sindelar
An experimental study is reported concerning the flow of non-Newtonian fluids through contractions and expansions, with emphasis on the latter. Flow visualization is used to illustrate how the flow fields are affected by changing the non-Newtonian character of the fluid. In particular, it will be shown that for the range of conditions covered by the study, elastic fluids exhibit vortex enhancement in contraction flow but vortices are inhibited in expansion flows. The effect of inertia is opposite: vortex motion in the contraction is reduced but in an expansion inertia causes vortex enhancement. A different scenario is obtained by adding a very small concentration of inextensible long fibres to a Newtonian fluid. In this case both contraction and expansion flows are characterized by the occurrence of large vortices (although vortex enhancement is not significant). Thus, dilute fibre suspensions behave similarly to polymeric solutions in contractions but not in expansions. The study also indicates that the fibres fail to flow affinely with the bulk flow. This is particularly apparent in expansion flow where the fibres tend to an orientation which is transverse to the flow direction and also close to boundary walls where substantial fibre-free regions develop. Keywords: contraction flows; expansion vortex enhancement; flow visualization
flows;
long fibre
suspensions;
INTRODUCTION The flow of non-Newtonian fluids through expansions has received scant attention in the literature compared with the vast amount of space that has been devoted to studies of contraction flows. This situation has arisen primarily because of the fact that contraction flows have provided researchers with such a variety of flow characteristics that expansion flows have appeared comparatively uninteresting. One of the most important features of abrupt contraction flows is the appearance of a vortex in the upstream region, attached to the contraction plane. The size of the recirculating zone and its behaviour with flow rate are linked in a complex way to the rheometric properties of the fluid as well as to the dimensions of the geometry ’ ~4 . In general terms, the size of the vortex is associated closely with the Trouton ratio, i.e., the ratio of the extensional to shear viscosities of the fluid. Vortex enhancement (increasing vortex size with increasing flow rate) is generally observed whenever the Trouton ratio is an increasing function of strain rate. A fluid with a large but constant value for the Trouton ratio might be expected to exhibit a large vortex but not vortex enhancement. The influence of elasticity, as measured by the Weissenberg number, for example, is more difficult to quantify, although some numerical results5 suggest that it causes a reduction in vortex size. Fluid inertia invariably has the effect of reducing the vortex size. In expansion flows it is known that inertia will increase the size of vortices in the downstream corners. It is also generally accepted that the vortex size is dramatically reduced for elastic fluids due to the release of stored 0956-7143/93/020109-08
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@ 1993
non-Newtonian
fluids;
elastic energy. It is not possible to be more specific in this case because of the lack of published studies of expansion flows. In this paper it is hoped to partially redress this situation and show that expansion flows are not as devoid of character as previously assumed. Clearly, both contraction and expansion flows are of great importance in the manufacture and processing of composite materials. Flow through a fibre bed is a highly complex situation involving flow through a complicated network of contraction and expansion flows. Also, the generation and/or enhancement of secondary flows (in the bulk composite or in the resin only) in the vicinity of sharp corners must have a bearing on the properties of a finished product. The composite materials considered in this study are very dilute suspensions of long fibres in a Newtonian suspending fluid. In contrast, most long fibre composites of industrial importance are of relatively high volume fraction. However, an understanding of the former has to be a contributory factor to the fuller understanding of the latter. In fact, certain flow features are common to both dilute and concentrated suspensions. SAMPLE CHARACTERIZATION EXPERIMENTAL PROCEDURE
AND
The fluid samples of principal interest in this study are suspensions of glass fibres in a Newtonian suspending fluid. For the latter a solution of glucose syrup (Cerestar Ltd, UK) and water was used. The water content was adjusted to yield a fluid having a viscosity of 1 Pa s and the resulting density was 1340 kg m-3. Glass fibres Butterworth-Heinemann
Ltd
109
10
Shear rate (l/s) Figure 1 Shear viscosity 9 (0, n ) and first normal stress difference IV, (A, A) as functions of shear rate y for the polymer solutions. Temperature=20”C. Open symbols, 0.05% by weight polyacrylamide in glucose syrup/water; closed symbols, 0.1% Xanthan gum in glucose syrup/water
Smoothing
vessel
400 Valve ii
itI
Geometry
240
Pump 468
I
I
a
Collection
vessel
b
Figure 2 (a) Schematic diagram of the experimental set-up; (b) schematic drawing and dimensions of the contraction/expansion geometry. All dimensions are in millimetres
(supplied by ICI plc, UK) were added at various volume fractions to the suspending fluid. The fibres were supplied in 6.4 mm lengths and had a nominal diameter of 13 ,um. This implies that the nominal aspect ratio of the fibres was 492. However, due to the practical difficulties involved in producing low concentration suspensions of individual fibres, the effective aspect ratio was estimated to be approximately 80. This estimate was obtained by an inspection of the fibre bundles that resulted in the prepared suspension. To put into perspective the special nature of long fibre suspensions, other more conventional fluid samples were also studied. As well as the Newtonian suspending fluid described above, a Boger-type fluid* was also studied. *For present purposes a Boger fluid may be regarded as one which, in a steady simple shear flow, exhibits a constant shear viscosity and a quadratic first normal stress difference, iVI.
110
This was produced by adding a small concentration (0.05% by weight) of polyacrylamide (ElO, Allied Colloids Ltd, UK) to the same Newtonian sample. The resulting fluid is highly elastic but exhibits a shear viscosity that does not differ significantly from 1 Pa s. Finally, a 0.1% solution of Xanthan gum (Keltrol T, Kelco International Ltd, UK) was prepared using the same Newtonian solvent. This fluid is slightly shear thinning and considerably less elastic than the preceding Boger fluid. The shear properties of the non-Newtonian samples were measured using a Weissenberg Rheogoniometer and are presented in Figure 1. Visualization experiments were carried out by pumping the samples through a closed-loop circuit, incorporating an axisymmetric contraction/expansion section constructed from Perspex. The experimental set-up and geometrical parameters are shown schematically in Figure 2. To aid with the visualization, a small quantity of polystyrene dust particles was added to each sample and a split laser beam was used to illuminate a plane of symmetry in the flow. At each flow rate, Q, the flow fields in the contraction and expansion regions were photographed simultaneously. The flow rates were varied to yield a range of Reynold’s number from 0.1 to 10, approximately.
RESULTS The results of this study will be presented in two ways. First, photographs will be used to illustate the steady-state flow fields that develop in the contraction and expansion regions for the different fluid samples considered. To avoid duplication of essentially similar flow fields, only representative cases will be shown. Second, vortex length ratio versus Reynold’s number data are presented. The vortex length ratio (L,) is defined as the ratio of the vortex length (1,) to the large geometry
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diameter (d,) (see Figure 2(b)). Reynold’s number (R,) is defined by:
R, = 4pQlwd, where Q is the volumetric flow rate, p is the fluid density and 9 is the fluid viscosity. For convenience a value of rl is taken to be 1 Pa s for all the fluids tested. Any discrepancy introduced because of the non-constant value of the viscosities of the elastic fluids is immaterial in the context of the present study. The vortex length ratios, for the Newtonian sample, are plotted as functions of R, in Figure 3 for both the contraction and the expansion. It is pertinent to point out that both the expansion and the contraction data tend towards the value of 0.17 in the limit of small R,, in agreement with expectation*. Figure 3 also contains the corresponding vortex length data obtained for the 0.1% Xanthan solution and the 0.05% polyacrylamide solution. These illustrate what is expected for typical elastic polymeric solutions. In the contraction vortices generally increase in size with increasing flow rates, at moderate values of R,, although this enhancement is less marked for the Xanthan solution. Notice that the contraction flow vortex for the Xanthan solution is appreciably larger than the corresponding one for the Boger fluid. However, the inertia-dominated vortex that occurs in the expansion for Newtonian fluids is substantially reduced in size in the case of elastic fluids. For the Xanthan solution it is completely suppressed. Also, notice again that the vortex length ratios tend towards the Newtonian value of 0.17 in the limit of small R, for the polyacrylamide solution. This cannot be said of the Xanthan solution for which such a situation must arise (if it does so at all) at very low values of Reynold’s
number. From the above observations it would seem reasonable to conclude that the observed effects do not corrollate directly with elasticity (as measured by Ni) for either the contraction or the expansion. Vortex length data for several concentrations of the glass suspensions are plotted in Figure 4 for flow through the contraction and in Figure 5 for flow through the expansion. It is immediately obvious that the behaviour of the suspensions is qualitatively similar to that exhibited by the polymeric solutions (in particular, the Xanthan solution) in the case of contraction flow, but is qualitatively different in the case of expansion flow. In contraction flow large vortices are created in the salient corner. However, the vortices do not exhibit the enhancement with flow rate that is characteristic of polymeric solutions such as the Boger-type polyacrylamide solution. These observations are in line with the findings of others6-8. In expansion flow the fibre suspensions produce yet another unexpected characteristic. The inertia-dominated vortices are enhanced at low Reynold’s numbers by the presence of the fibres, in contrast to both the Xanthan and Boger-type solutions. As the Reynold’s number is increased it would appear that the fibres may eventually cause the fluid to behave like the polymeric solutions by reducing the size of the vortex to below that of the equivalent inelastic fluid. This has not been confirmed, however. The above observations are exemplified in Figures 6 and 7 which contain selected photographs of flow fields for the contraction and expansion flows. In each figure the top photographs correspond to the contraction and the bottom ones correspond to the expansion. In other words, the flow direction may be considered to be from the top to the bottom of the page. Figure 6 compares the Newtonian fluid (left-hand side) and the Xanthan
1.2
-0.4 0
2
4
6 Reynold’s
Figure 3 Vortex Xanthan solution
length ratio I,, versus (0, n ). Open symbols,
Reynold’s number expansion; closed
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number,
for the Newtonian symbols, contraction
R,
8
fluid
10
12
Re (0,
l
), Boger-type
polyacrylamide
solution
(A,
A)
and
111
1.2
0.8 J. $ E .c
p 0.4 B i P 0
-0.4 0
2
4
6 Reynold’s
Figure 4 Vortex length through the contraction:
ratio L, versus Reynold’s ---e---, C=O; ---
number x ---,
number,
8 R
10
12
e
R, for various concentrations C of glass fibres C=O.O25; ---n---, C=O.O6; ---A---,
suspended in glucose C=O.l; ---o---,
syrup/water, ~~0.2
flowing
syrup/water,
flowing
0.8
0.6 ii-. .g e 0 0.4 ! P 0.2
0 0
2
4
6 Reynold’s
Figure 5 through
Vortex length ratio L, versus Reynold’s the expansion. Symbols as in Figure 4
number
R, for various
solution (right-hand side) at a Reynold’s number of 1.7, illustrating the increased vortex activity in the contraction for the non-Newtonian fluid and the reduced activity in the expansion. Figure 7 compares the Newtonian behaviour (left-hand side) with that exhibited
112
number,
concentrations
8
10
12
Re C of glass fibres
suspended
in glucose
by a 0.025% by weight suspension of glass fibres at a Reynold’s number of 1.l. In this case the increased vortex activity in both the contraction and the expansion for the fibre suspension is very apparent. The technique used for the visualization of the flow
Composites Manufacturing No 2 1993
Figure 6 Photographs of flow fields at R, = 1.7: left-hand side, Newtonian fluid; right-hand side, Xanthan solution; top row, contraction; bottom row, expansion
field has been employed for many years by a large number of researchers. However, some caution should be exercised when interpreting the flow fields that appear in the resulting photographs. For most situations there is no cause for concern, but in the case of composite materials there exists the possibility that the particulate
Composites
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matter may not in fact flow with the bulk material. In Figure 8 a normal photograph is shown of a glass fibre suspension flowing (from top to bottom) in a 488 mm long circular cylinder of internal diameter 44 mm (the upper half of the geometry shown in Figure 2(b)). Several observations can be made. First, it will be noticed that
113
Figure 7 Photographs of flow fields at R, = 1.1: left-hand side, Newtonian fluid; right-hand side, Newtonian fluid+0.025% top row, contraction; bottom row, expansion
at the contraction (bottom of the photograph) the fibres indicate a preferential orientation in the flow direction. However, on exiting at an expansion (top of photograph) the fibres almost immediately ‘flip’ from an axial orientation within the capillary to a tranverse orientation in the cylinder. This behaviour is not unexpected and
114
(by weight) glass fibres;
has also been shown to occur for high volume fraction (25% volume) fibre-reinforced polypropylene melts’. Also of significance is the absence of fibres in the vicinity of the wall. An annulus of thickness approximately equal to the length of the fibres is almost completely devoid of fibres.
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accelerating (contraction, converging) flows may be qualitatively very different in decelerating (expansion, diverging) flows. It is clear that the extensibility of the long molecules, whether they be polymer chains or glass fibres, is of great significance in the explanation of these differences, although the precise physical mechanisms are not clear. It is clearly not valid to extend the conclusions to materials of high volume fraction fibre content because of the complicating influence of fibre/fibre interactions. However, it is also clear that similarities exist, particularly in the way that the fibres adopt their preferred orientation. The occurrence of recirculating regions in expansion flows will clearly depend on the competing influences of elasticity and extensional viscosity. Any significant elasticity in the fibre network is likely to inhibit the appearance of such vortices. The second area of interest is that concerning the numerical simulation of such flows. It has become customary to model long fibre suspensions by means of anisotropic models (see, for example, Lipscomb et a1.7, Chiba et al8 and Givler et al.“), although it is by no means certain that such low concentration fibre suspensions are in fact anisotropic. In most earlier work such as that of Givler et al.“, it is assumed that for dilute fibre suspensions the presence of fibres does not affect the velocity field. It is now known that such an approximation is grossly in error. That assumption was not used by Lipscomb et aL7 who obtained excellent agreement between their numerical and experimental streamline patterns for flow through a contraction. They did, however, make use of the assumption that fibres move affinely with the bulk velocity field, which for contraction flow appears to be a reasonable approximation. It is particularly interesting to remark that Townsend and Walters’ ’ report that, using a model very similar to that used by Lipscomb et al. with the same affinity condition, vortex sizes are enhanced in expansion flow, in qualitative agreement with the experiments reported here. This is surprising and intriguing since it is clear that fibre orientation data would be qualitatively different to observations. However, the vortex enlargement is greater than that observed experimentally and they find that by relaxing the affine condition quantitative agreement is obtained. The principal conclusions from the present study are that long fibre suspensions behave similarly to polymeric solutions in contraction flows but are qualitatively different in expansion flows. Also, it is clear that in general the fibres do not move affinely with the bulk velocity field. Complex flows of dilute suspensions of long fibres can be accurately simulated provided that the affine condition is relaxed.
Figure 8 Ph otograph of 0.2% (by weight) suspension of gl ass fibres in glucosi e syrup, flowing (from top to bottom) in a circxlar CY,linder
ACKNOWLEDGEMENTS CON< 3L1 JSIONS The reslults; presented in this paper are interes iting fcJr two princip al reasons. First, they illustrate tha ,t nonNewtor niar I fluids that are qualitatively veryy sirni liar in
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The authors are indebted to Professors K. Walters and P. Townsend, Dr R. Baily and Mr R. Evans for their assistance in various aspects of this work. Support from ICI plc (UK) and SERC (UK) is also gratefully acknowledged.
115
White, S.A., Gotsis, A.D. and Baird, D.G. ‘Review of the entry flow problem: experimental and numerical’ J non-Newtonian Fluid Mech 25 (1987) pp 121-160 Boger, D.V. ‘Viscoelastic flows through contractions’ Ann Rev Fluid Mech 19 (1987) pp 157-182 Binding, D.M. ‘An approximate analysis for contraction and converging flows’ J non-Newtonian Fluid Mech 21 (1988) pp 173-189 Binding, D.M. ‘Further considerations of axisymmetric contraction flows’ J non-Newtonian Fluid Mech 41 (1991) pp 27-42 Debbaut, B. and Crochet, M.J. ‘Extensional effects in complex flows’ J non-Newtonian Fluid Mech 30 (1988) pp 169-184 Binnington, R.J. and Boger, D.V. ‘Entry flow bf semi-rigid rod solutions’ J non-Newronian Fluid Mech 26 (1987) DD 115-123 Lipscomb II, G.G., Denn, M.M., Hur, DU‘and &er, D.V. ‘The flow of fiber suspensions in complex geometries’ J non-Newtonian Fluid Mech 26 (1988) pp 297-325 Chiba, K., Nakamura, K. and Boger, D.V. ‘A numerical solution for the flow of dilute fiber suspensions through an axisymmetric contraction’ J non-Newtonian Fluid Mech 35 (1990) pp 1-14
Binding, D.M. and Sindelar, M. ‘Visuahsation studies of contraction flow of fibre filled polypropylene’ in Theoretical and Applied Rheology edited by P. Moldenaers and R. Keunings (Elsevier Science Publishers, 1992) pp 841-843 Givler, R.C., Crochet, M.J. and Pipes, R.B. ‘Numerical prediction of fiber orientation in dilute suspensions’ J Composite Mater 17 (1983) pp 330-343 Townsend, P. and Walters, K. (private communication, in press, 1993)
AUTHORS D.M. Binding and T. Abdul-Karem are with the Department of Mathematics, University of Wales, Aberystwyth, Dyfed SY23 3BZ, UK and M. Sindelar is with the Institute of Hydrodynamics, Podbabska 13, Praha 6,166 12 Czechoslovakia. Correspondence should be directed to Dr D.M. Binding. (Received 8 February 1993; revised 17 June 1993)
.
116
Composites
Manufacturing
No
2 1993
D.M. Binding and M. Sindelar
An experimental study is reported concerning the flow of non-Newtonian fluids through contractions and expansions, with emphasis on the latter. Flow visualization is used to illustrate how the flow fields are affected by changing the non-Newtonian character of the fluid. In particular, it will be shown that for the range of conditions covered by the study, elastic fluids exhibit vortex enhancement in contraction flow but vortices are inhibited in expansion flows. The effect of inertia is opposite: vortex motion in the contraction is reduced but in an expansion inertia causes vortex enhancement. A different scenario is obtained by adding a very small concentration of inextensible long fibres to a Newtonian fluid. In this case both contraction and expansion flows are characterized by the occurrence of large vortices (although vortex enhancement is not significant). Thus, dilute fibre suspensions behave similarly to polymeric solutions in contractions but not in expansions. The study also indicates that the fibres fail to flow affinely with the bulk flow. This is particularly apparent in expansion flow where the fibres tend to an orientation which is transverse to the flow direction and also close to boundary walls where substantial fibre-free regions develop. Keywords: contraction flows; expansion vortex enhancement; flow visualization
flows;
long fibre
suspensions;
INTRODUCTION The flow of non-Newtonian fluids through expansions has received scant attention in the literature compared with the vast amount of space that has been devoted to studies of contraction flows. This situation has arisen primarily because of the fact that contraction flows have provided researchers with such a variety of flow characteristics that expansion flows have appeared comparatively uninteresting. One of the most important features of abrupt contraction flows is the appearance of a vortex in the upstream region, attached to the contraction plane. The size of the recirculating zone and its behaviour with flow rate are linked in a complex way to the rheometric properties of the fluid as well as to the dimensions of the geometry ’ ~4 . In general terms, the size of the vortex is associated closely with the Trouton ratio, i.e., the ratio of the extensional to shear viscosities of the fluid. Vortex enhancement (increasing vortex size with increasing flow rate) is generally observed whenever the Trouton ratio is an increasing function of strain rate. A fluid with a large but constant value for the Trouton ratio might be expected to exhibit a large vortex but not vortex enhancement. The influence of elasticity, as measured by the Weissenberg number, for example, is more difficult to quantify, although some numerical results5 suggest that it causes a reduction in vortex size. Fluid inertia invariably has the effect of reducing the vortex size. In expansion flows it is known that inertia will increase the size of vortices in the downstream corners. It is also generally accepted that the vortex size is dramatically reduced for elastic fluids due to the release of stored 0956-7143/93/020109-08
Composites Manufacturing Vol 4 No 2 1993
@ 1993
non-Newtonian
fluids;
elastic energy. It is not possible to be more specific in this case because of the lack of published studies of expansion flows. In this paper it is hoped to partially redress this situation and show that expansion flows are not as devoid of character as previously assumed. Clearly, both contraction and expansion flows are of great importance in the manufacture and processing of composite materials. Flow through a fibre bed is a highly complex situation involving flow through a complicated network of contraction and expansion flows. Also, the generation and/or enhancement of secondary flows (in the bulk composite or in the resin only) in the vicinity of sharp corners must have a bearing on the properties of a finished product. The composite materials considered in this study are very dilute suspensions of long fibres in a Newtonian suspending fluid. In contrast, most long fibre composites of industrial importance are of relatively high volume fraction. However, an understanding of the former has to be a contributory factor to the fuller understanding of the latter. In fact, certain flow features are common to both dilute and concentrated suspensions. SAMPLE CHARACTERIZATION EXPERIMENTAL PROCEDURE
AND
The fluid samples of principal interest in this study are suspensions of glass fibres in a Newtonian suspending fluid. For the latter a solution of glucose syrup (Cerestar Ltd, UK) and water was used. The water content was adjusted to yield a fluid having a viscosity of 1 Pa s and the resulting density was 1340 kg m-3. Glass fibres Butterworth-Heinemann
Ltd
109
10
Shear rate (l/s) Figure 1 Shear viscosity 9 (0, n ) and first normal stress difference IV, (A, A) as functions of shear rate y for the polymer solutions. Temperature=20”C. Open symbols, 0.05% by weight polyacrylamide in glucose syrup/water; closed symbols, 0.1% Xanthan gum in glucose syrup/water
Smoothing
vessel
400 Valve ii
itI
Geometry
240
Pump 468
I
I
a
Collection
vessel
b
Figure 2 (a) Schematic diagram of the experimental set-up; (b) schematic drawing and dimensions of the contraction/expansion geometry. All dimensions are in millimetres
(supplied by ICI plc, UK) were added at various volume fractions to the suspending fluid. The fibres were supplied in 6.4 mm lengths and had a nominal diameter of 13 ,um. This implies that the nominal aspect ratio of the fibres was 492. However, due to the practical difficulties involved in producing low concentration suspensions of individual fibres, the effective aspect ratio was estimated to be approximately 80. This estimate was obtained by an inspection of the fibre bundles that resulted in the prepared suspension. To put into perspective the special nature of long fibre suspensions, other more conventional fluid samples were also studied. As well as the Newtonian suspending fluid described above, a Boger-type fluid* was also studied. *For present purposes a Boger fluid may be regarded as one which, in a steady simple shear flow, exhibits a constant shear viscosity and a quadratic first normal stress difference, iVI.
110
This was produced by adding a small concentration (0.05% by weight) of polyacrylamide (ElO, Allied Colloids Ltd, UK) to the same Newtonian sample. The resulting fluid is highly elastic but exhibits a shear viscosity that does not differ significantly from 1 Pa s. Finally, a 0.1% solution of Xanthan gum (Keltrol T, Kelco International Ltd, UK) was prepared using the same Newtonian solvent. This fluid is slightly shear thinning and considerably less elastic than the preceding Boger fluid. The shear properties of the non-Newtonian samples were measured using a Weissenberg Rheogoniometer and are presented in Figure 1. Visualization experiments were carried out by pumping the samples through a closed-loop circuit, incorporating an axisymmetric contraction/expansion section constructed from Perspex. The experimental set-up and geometrical parameters are shown schematically in Figure 2. To aid with the visualization, a small quantity of polystyrene dust particles was added to each sample and a split laser beam was used to illuminate a plane of symmetry in the flow. At each flow rate, Q, the flow fields in the contraction and expansion regions were photographed simultaneously. The flow rates were varied to yield a range of Reynold’s number from 0.1 to 10, approximately.
RESULTS The results of this study will be presented in two ways. First, photographs will be used to illustate the steady-state flow fields that develop in the contraction and expansion regions for the different fluid samples considered. To avoid duplication of essentially similar flow fields, only representative cases will be shown. Second, vortex length ratio versus Reynold’s number data are presented. The vortex length ratio (L,) is defined as the ratio of the vortex length (1,) to the large geometry
Composites Manufacturing
No 2 1993
diameter (d,) (see Figure 2(b)). Reynold’s number (R,) is defined by:
R, = 4pQlwd, where Q is the volumetric flow rate, p is the fluid density and 9 is the fluid viscosity. For convenience a value of rl is taken to be 1 Pa s for all the fluids tested. Any discrepancy introduced because of the non-constant value of the viscosities of the elastic fluids is immaterial in the context of the present study. The vortex length ratios, for the Newtonian sample, are plotted as functions of R, in Figure 3 for both the contraction and the expansion. It is pertinent to point out that both the expansion and the contraction data tend towards the value of 0.17 in the limit of small R,, in agreement with expectation*. Figure 3 also contains the corresponding vortex length data obtained for the 0.1% Xanthan solution and the 0.05% polyacrylamide solution. These illustrate what is expected for typical elastic polymeric solutions. In the contraction vortices generally increase in size with increasing flow rates, at moderate values of R,, although this enhancement is less marked for the Xanthan solution. Notice that the contraction flow vortex for the Xanthan solution is appreciably larger than the corresponding one for the Boger fluid. However, the inertia-dominated vortex that occurs in the expansion for Newtonian fluids is substantially reduced in size in the case of elastic fluids. For the Xanthan solution it is completely suppressed. Also, notice again that the vortex length ratios tend towards the Newtonian value of 0.17 in the limit of small R, for the polyacrylamide solution. This cannot be said of the Xanthan solution for which such a situation must arise (if it does so at all) at very low values of Reynold’s
number. From the above observations it would seem reasonable to conclude that the observed effects do not corrollate directly with elasticity (as measured by Ni) for either the contraction or the expansion. Vortex length data for several concentrations of the glass suspensions are plotted in Figure 4 for flow through the contraction and in Figure 5 for flow through the expansion. It is immediately obvious that the behaviour of the suspensions is qualitatively similar to that exhibited by the polymeric solutions (in particular, the Xanthan solution) in the case of contraction flow, but is qualitatively different in the case of expansion flow. In contraction flow large vortices are created in the salient corner. However, the vortices do not exhibit the enhancement with flow rate that is characteristic of polymeric solutions such as the Boger-type polyacrylamide solution. These observations are in line with the findings of others6-8. In expansion flow the fibre suspensions produce yet another unexpected characteristic. The inertia-dominated vortices are enhanced at low Reynold’s numbers by the presence of the fibres, in contrast to both the Xanthan and Boger-type solutions. As the Reynold’s number is increased it would appear that the fibres may eventually cause the fluid to behave like the polymeric solutions by reducing the size of the vortex to below that of the equivalent inelastic fluid. This has not been confirmed, however. The above observations are exemplified in Figures 6 and 7 which contain selected photographs of flow fields for the contraction and expansion flows. In each figure the top photographs correspond to the contraction and the bottom ones correspond to the expansion. In other words, the flow direction may be considered to be from the top to the bottom of the page. Figure 6 compares the Newtonian fluid (left-hand side) and the Xanthan
1.2
-0.4 0
2
4
6 Reynold’s
Figure 3 Vortex Xanthan solution
length ratio I,, versus (0, n ). Open symbols,
Reynold’s number expansion; closed
Composites Manufacturing No 2 1993
number,
for the Newtonian symbols, contraction
R,
8
fluid
10
12
Re (0,
l
), Boger-type
polyacrylamide
solution
(A,
A)
and
111
1.2
0.8 J. $ E .c
p 0.4 B i P 0
-0.4 0
2
4
6 Reynold’s
Figure 4 Vortex length through the contraction:
ratio L, versus Reynold’s ---e---, C=O; ---
number x ---,
number,
8 R
10
12
e
R, for various concentrations C of glass fibres C=O.O25; ---n---, C=O.O6; ---A---,
suspended in glucose C=O.l; ---o---,
syrup/water, ~~0.2
flowing
syrup/water,
flowing
0.8
0.6 ii-. .g e 0 0.4 ! P 0.2
0 0
2
4
6 Reynold’s
Figure 5 through
Vortex length ratio L, versus Reynold’s the expansion. Symbols as in Figure 4
number
R, for various
solution (right-hand side) at a Reynold’s number of 1.7, illustrating the increased vortex activity in the contraction for the non-Newtonian fluid and the reduced activity in the expansion. Figure 7 compares the Newtonian behaviour (left-hand side) with that exhibited
112
number,
concentrations
8
10
12
Re C of glass fibres
suspended
in glucose
by a 0.025% by weight suspension of glass fibres at a Reynold’s number of 1.l. In this case the increased vortex activity in both the contraction and the expansion for the fibre suspension is very apparent. The technique used for the visualization of the flow
Composites Manufacturing No 2 1993
Figure 6 Photographs of flow fields at R, = 1.7: left-hand side, Newtonian fluid; right-hand side, Xanthan solution; top row, contraction; bottom row, expansion
field has been employed for many years by a large number of researchers. However, some caution should be exercised when interpreting the flow fields that appear in the resulting photographs. For most situations there is no cause for concern, but in the case of composite materials there exists the possibility that the particulate
Composites
Manufacturing
No 2 1993
matter may not in fact flow with the bulk material. In Figure 8 a normal photograph is shown of a glass fibre suspension flowing (from top to bottom) in a 488 mm long circular cylinder of internal diameter 44 mm (the upper half of the geometry shown in Figure 2(b)). Several observations can be made. First, it will be noticed that
113
Figure 7 Photographs of flow fields at R, = 1.1: left-hand side, Newtonian fluid; right-hand side, Newtonian fluid+0.025% top row, contraction; bottom row, expansion
at the contraction (bottom of the photograph) the fibres indicate a preferential orientation in the flow direction. However, on exiting at an expansion (top of photograph) the fibres almost immediately ‘flip’ from an axial orientation within the capillary to a tranverse orientation in the cylinder. This behaviour is not unexpected and
114
(by weight) glass fibres;
has also been shown to occur for high volume fraction (25% volume) fibre-reinforced polypropylene melts’. Also of significance is the absence of fibres in the vicinity of the wall. An annulus of thickness approximately equal to the length of the fibres is almost completely devoid of fibres.
Composites
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accelerating (contraction, converging) flows may be qualitatively very different in decelerating (expansion, diverging) flows. It is clear that the extensibility of the long molecules, whether they be polymer chains or glass fibres, is of great significance in the explanation of these differences, although the precise physical mechanisms are not clear. It is clearly not valid to extend the conclusions to materials of high volume fraction fibre content because of the complicating influence of fibre/fibre interactions. However, it is also clear that similarities exist, particularly in the way that the fibres adopt their preferred orientation. The occurrence of recirculating regions in expansion flows will clearly depend on the competing influences of elasticity and extensional viscosity. Any significant elasticity in the fibre network is likely to inhibit the appearance of such vortices. The second area of interest is that concerning the numerical simulation of such flows. It has become customary to model long fibre suspensions by means of anisotropic models (see, for example, Lipscomb et a1.7, Chiba et al8 and Givler et al.“), although it is by no means certain that such low concentration fibre suspensions are in fact anisotropic. In most earlier work such as that of Givler et al.“, it is assumed that for dilute fibre suspensions the presence of fibres does not affect the velocity field. It is now known that such an approximation is grossly in error. That assumption was not used by Lipscomb et aL7 who obtained excellent agreement between their numerical and experimental streamline patterns for flow through a contraction. They did, however, make use of the assumption that fibres move affinely with the bulk velocity field, which for contraction flow appears to be a reasonable approximation. It is particularly interesting to remark that Townsend and Walters’ ’ report that, using a model very similar to that used by Lipscomb et al. with the same affinity condition, vortex sizes are enhanced in expansion flow, in qualitative agreement with the experiments reported here. This is surprising and intriguing since it is clear that fibre orientation data would be qualitatively different to observations. However, the vortex enlargement is greater than that observed experimentally and they find that by relaxing the affine condition quantitative agreement is obtained. The principal conclusions from the present study are that long fibre suspensions behave similarly to polymeric solutions in contraction flows but are qualitatively different in expansion flows. Also, it is clear that in general the fibres do not move affinely with the bulk velocity field. Complex flows of dilute suspensions of long fibres can be accurately simulated provided that the affine condition is relaxed.
Figure 8 Ph otograph of 0.2% (by weight) suspension of gl ass fibres in glucosi e syrup, flowing (from top to bottom) in a circxlar CY,linder
ACKNOWLEDGEMENTS CON< 3L1 JSIONS The reslults; presented in this paper are interes iting fcJr two princip al reasons. First, they illustrate tha ,t nonNewtor niar I fluids that are qualitatively veryy sirni liar in
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The authors are indebted to Professors K. Walters and P. Townsend, Dr R. Baily and Mr R. Evans for their assistance in various aspects of this work. Support from ICI plc (UK) and SERC (UK) is also gratefully acknowledged.
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White, S.A., Gotsis, A.D. and Baird, D.G. ‘Review of the entry flow problem: experimental and numerical’ J non-Newtonian Fluid Mech 25 (1987) pp 121-160 Boger, D.V. ‘Viscoelastic flows through contractions’ Ann Rev Fluid Mech 19 (1987) pp 157-182 Binding, D.M. ‘An approximate analysis for contraction and converging flows’ J non-Newtonian Fluid Mech 21 (1988) pp 173-189 Binding, D.M. ‘Further considerations of axisymmetric contraction flows’ J non-Newtonian Fluid Mech 41 (1991) pp 27-42 Debbaut, B. and Crochet, M.J. ‘Extensional effects in complex flows’ J non-Newtonian Fluid Mech 30 (1988) pp 169-184 Binnington, R.J. and Boger, D.V. ‘Entry flow bf semi-rigid rod solutions’ J non-Newronian Fluid Mech 26 (1987) DD 115-123 Lipscomb II, G.G., Denn, M.M., Hur, DU‘and &er, D.V. ‘The flow of fiber suspensions in complex geometries’ J non-Newtonian Fluid Mech 26 (1988) pp 297-325 Chiba, K., Nakamura, K. and Boger, D.V. ‘A numerical solution for the flow of dilute fiber suspensions through an axisymmetric contraction’ J non-Newtonian Fluid Mech 35 (1990) pp 1-14
Binding, D.M. and Sindelar, M. ‘Visuahsation studies of contraction flow of fibre filled polypropylene’ in Theoretical and Applied Rheology edited by P. Moldenaers and R. Keunings (Elsevier Science Publishers, 1992) pp 841-843 Givler, R.C., Crochet, M.J. and Pipes, R.B. ‘Numerical prediction of fiber orientation in dilute suspensions’ J Composite Mater 17 (1983) pp 330-343 Townsend, P. and Walters, K. (private communication, in press, 1993)
AUTHORS D.M. Binding and T. Abdul-Karem are with the Department of Mathematics, University of Wales, Aberystwyth, Dyfed SY23 3BZ, UK and M. Sindelar is with the Institute of Hydrodynamics, Podbabska 13, Praha 6,166 12 Czechoslovakia. Correspondence should be directed to Dr D.M. Binding. (Received 8 February 1993; revised 17 June 1993)
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