Entry and exit flows of casson fluids

NOTES

Entry and Exit Flows of Casson Fluids’ T. V. PHAM and E. MITSOULW Department of Chemical Engineering, University of Ottawa, Ottawa, Ontario KIN 6NS Entry and exit flows through abrupt contractions are studied numerically for Casson fluids exhibiting a yield stress. The Casson constitutive equation recommended for describing blood flow is used with an appropriate modification proposed by Papanastasiou, which applies everywhere in the flow field in both yielded and practically unyielded regions. The emphasis is on determining the extent and shape of unyielded/yielded regions along with the swelling ratio of the free stream for planar and axisymmetric contraction flows for the whole range of Casson numbers. The results for pressure are used to determine the excess pressure losses that give rise to entrance, exit, and the total end correction. On Ctudie numtriquement les tcoulements d’entrte et de sortie dans des contractions abruptes pour des fluides de Casson prksentant une contrainte seuil. L’Quation constitutive de Casson recomandh pour d&rire 1’6.coulementsanguin est utiliste avec une modification approprite par Papanastasiou, qui s’applique partout dans le champ d’tcoulement, dans les regions cisaillkes c o m e dans les regions non cisailltes. On s’attache surtout B dtterminer I’ttendue et la forme des regions cisaillkslnon cisailltes ainsi que le rapport de gonflement du courant libre pour des tcoulements dans des contractions planaires et axisymktriques pour toute la gamme de nombre de Casson. Les rbsultats pour la pression servent ii dtterminer I’excks de perte de charge associt aux effets d’entrke, de sortie et aux effets de bout combints. Keywords: yield stress, Casson model, blood flow, contraction flow, viscoplasticity.

A

n important class of non-Newtonian materials exhibits a yield stress, which must be exceeded before significant deformation can occur. A list of several materials exhibiting yield was given in a seminal paper by Bird et al. (1983). The models presented for such so-called viscoplastic materials included the Bingham, Herschel-Bulkley and Casson. Analytical solutions were provided for the Bingham plastic model in simple flow fields. A similar review work with analytical solutions had been undertaken earlier by Shah (1980) with the focus on blood flow described by the viscoplastic Casson model. Since then a renewed interest has developed among several researchers for the study of these materials in non-trivial flows (see recent reviews by Abdali et al., 1992; Mitsoulis et al., 1993). To model the stress-deformation behaviour of blood, the Casson constitutive equation has been proposed (Casson, 1959). In simple shear flow it takes the form (see also Figure 1):

JF

=

Gy+

for Ir]

Iry . . . . . . . . . . . .

- exp(-

Jmi.)] +

. . . . . . . . . (2)

”

1

2

”

3

4

“

”

5

7

6

8

’ 9

10

Shear Rote, ? (s-’)

(1b)

*Author to whom correspondence should be addressed. *Dedicated to the memory of Professor Tasos Papanastasiou of the Department of Chemical Engineering, Aristole University of Thessaloniki, Greece. 1080

5 -

0’ 0

where T is the shear stress, i. is the shear rate, 7y is the yield stress, and p is a constant plastic viscosity. Note that when the shear stress 7 falls below rYa solid structure is formed (unyielded). To avoid the discontinuity inherent in any viscoplastic model, Papanastasiou (1987) proposed a modification in the equation by introducing a material parameter, which controls the exponential growth of stress. This way the equation is valid for both yielded and unyielded areas. Papanastasiou’s modification when applied to the Casson model becomes: =

6 -

for 171 > ry ............

y=o

J7

c

7 -

Figure I - Shear stress vs. shear rate according to the modified Casson constitutive Equation (2) for several values of the exponent m. Values of ~,=0.01082Pa and p=0.00276 Paas have been used for representing blood according to Shah (1980).

where rn is the stress growth exponent. As shown in Figure 1, this equation mimics the ideal Casson fluid (for rn 2 100). The exponential modification was previously applied to Bingham plastics and used by Abdali et al. (1992) to study entry and exit flows and determine the shape and extent of yieldedhnyieldedregions, extrudate swell and excess pressure losses through planar and axisymmetric geometries as a function of a dimensionlessyield stress or Bingham number, defined respectively by:

?y*

rH

= Y

PVN

..................................

(3)

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 72, DECEMBER, 1994

YIELDED/UNYIELDED R E G I O N S

YIELDED/UNYIELDED R E G I O N S

4.0

3.0

z j = 1.0 (Ca=l8)

2.0 I .o

2

.o -1.0

-2.0 -5.0 -4.0

I

-4.0

-12.0

-10.0

4.0

-6.0

-4.0

-2.0

.O

2.0

4.0

8.0

8.0

10.0

2.0

4.0

8.0

8.0

10.0

z/R 4.0

4.0

3.0

3.0

2.0

2.0

1.o

1.0

I

2

(r

3

.o -1.0

.o -1.0

-2.0

-2.0

-3.0

-3.0

-4.0 -12.0

-10.0

-8.0

-8.0

-1.0

-2.0

.O

2.0

4.0

6.0

8.0

10.0

-4.0 -12.0

-101)

-a0

-4.0

dl)

-2.0

x/H

.o z/R

1

7: = 3.9 (Ca=l,500,000)

-2.0

.O

2.0

4.0

6.0

8.0

10.0

-12.0

-10.0

4.0

-6.0

-4.0

x/H

7D

= A

crv

............................

(4)

-

-

-

= 7.77

...................................

(5)

where 7.7 is the apparent viscosity given by I

and 1i.l is the magnitude of the rate-of-strain tensor v v + vST, which is given by

=

. . . . . . . . . . (7) L

2.0

4.0

6.0

8.0

10.0

Figure 3 - Progressive growth of the unyielded zone (shaded) for entry flow of Casson fluids in a 4:1 axisymmetric abrupt contraction.

q.

where H is a characteristic length (haif the channel width of radius R), D is the diameter or channel width (2H), V,,, is a characteristic speed taken as the average velocity of a corresponding Newtonian liquid with viscosity p at the same pressure gradient, V is the average velocity of the Bingham or Casson fluid, Bi is the Bingham number and Cu is the Casson number. In all cases, the Newtonian fluid corresponds to T ~ X = Bi = Cu = 0. However, at the other extreme of OD, while 7; reaches a dimensionan unyielded solid, Cu less pressure gradient having the value of 3 for planar and 4 for axisymmetric geometries (see Abdali et al., 1992). In full tensorial form the constitutive Equation (2) of the Casson fluid is then written as: 7

.O

z/R

Figure 2 - Progressive growth of the unyielded zone (shaded) for entry flow of Casson fluids in a 4:l planar abrupt contraction.

Bi = Cu

-2.0

where ZI.; is the second invariant of To track down yieldedhnyielded regions, we shall employ the criterion that the material flows (yields) only when the magnitude of the extra stress tensor 171 exceeds the yield stress 7,,, i.e.,

unyielded 14

5 7,.

...........................

(8b)

Results and discussion ENTRYFLOWS OF CASSON FLUIDS The above constitutive equation is solved together with the conservation equations using the Finite Element Method (FEM) as explained in the previous papers by Abdali et al. (1992) and Mitsoulis et al. (1993). The same finite element grids have been used and the rest of conditions so that the whole range of Casson numbers 0 I Cu < 00 could be studied. The results for the progressive growth of the unyielded zone (shaded) for the entry flow of Casson fluids in a 4:l planar abrupt contraction are shown in Figure 2, while the axisymmetric results for flow through capillaries are shown in Figure 3. Differences with the equivalent results for Bingham fluids are found, as more rounded unyielded regions are obtained in the reservoir due to the stronger non-linearity of the Casson equation. Also the differences between planar and axisymmetric patterns are shown for the first time and give the relative growth of the unyielded region for the same value of 7;.

PHE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 72, DECEMBER, 1994

1081

C

u C

C

0 ._ 42

0

P L

0

1:: 2.0

1

YIELDED/UNYIELDED REGIONS 2.0

I

I

-ao

-6.0

-4.0

-20

.o

2.0

4.0

6.0

8.0

x/H 20

1

I

0

a

0 C 0

L

42

1.5 I

-20 4.0

1 .0

4.0

-2.0

-6.0

-4.0

-2.0

C

W

I

I -8.0

.O x/H

20

4.0

6.0

6.0

2.0

4.0

8.0

8.0

N

0.5 N

I

-20 4.0

Casson Number, Ca

I

I .O

x/H

Figure 4 - Entrance correction vs. Ca for Casson fluids flowing in a 4: 1 abrupt contraction (N corresponds to Newtonian result for Ca = 0). Symbols correspond to numerical simulations. The continuous curves are drawn from a non-linear regression analysis on the numerical simulation data according to Equations (9a) and (9b).

Figure 6 - Progressive growth of the unyielded zone (shaded) in planar exit flow of Casson fluids. instructive to give analytical relations for these results for quick reference. For this purpose, a non-linear regression analysis was performed on the data and gave the following relationship for the entrance correction (continuous curves on the graphs):

cu

(planar geometry) nen = 0.381 + 0.33[10g(Cu)

+ 0.41 CU - 0.4 .

(axisymmetric geometry) nen = 0.579 + 0.52[10g(Cu)

+ 0.41 Ca

Ca

- 0.4

(94

. (9b)

The present results indicate that increasing the yield stress increases substantially the excess pressure losses and thus the entrance correction.

EXITFLOWS OF CASSON FLUIDS

10-2

10-1

loo

10'

lo2

lo3

lo4

lo5

Casson Number, Ca

Figure 5 - Determination of the free surface swell vs Ca for Casson fluids flowing through planar channels and axisymmetric capillaries (N corresponds to Newtonian result for Ca = 0). Symbols correspond to numerical simulations. The dashed line corresponds to no swelling. The pressure drop in the system gives rise to the dimensionless entrance correction nen (Abdali et al., 1992), over and above the fullydeveloped values obtained in the absence of the contraction. The entrance corrections for the two geometries are shown in Figure 4 as a function of Cu. A linear behaviour is obtained for Casson fluids for Ca > 1, whereas a sigmoidal relationship was obtained for Bingham plastics in the previous work by Abdali et al. (1992). It is 1082

Calculations were carried out for the same range of Ca values as in the contraction flows. The results for the swelling ratio of the free stream as it exits from the channel are given in Figure 5 . Again a slight decrease of the free stream is obtained for some intermediate range of the dimensionless numbers as was the case in the previous work for Bingham plastics, but now this decrease is even smaller reaching about 2%. The progressive growth of the unyielded zone (shaded) in planar exit flow of Casson fluids is shown in Figure 6, while the corresponding axisymmetric results are show in Figure 7. The pressure drop in the system gives rise to the dimensionless exit correction nex (Abdali et al., 1992), over and above the fully-developed values obtained in the absence of the free stream. The exit corrections for the two geometries are shown in Figure 8 as a function of Ca. A sigmoidal behaviour is obtained. The present results indicate that increasing the yield stress decreases substantially the excess pressure losses at the exit, reducing them to zero for a totally unyielded material.

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 72, DECEMBER, 1994

YIELDED/UNYIELDED REGIONS 20

Axisymmetric

1

T,:= 15 fC0=30.2)

u C -2.0

I

I

L

-8.0

-6.0

-4.0

.O

-2.0

2.0

4.0

6.0

8.0

z/R 2.0

I

r.: =2.5

I

(Co=262)

-2.0

-8.0

-6.0

-4.0

-2.0

.O z/R

2.5

0 .-

1

2.0

1.0

(1.0

3.0

2.0

1

1

J

/

8.0

i

2.0

0.5N 0.0

I

10-2

lo-'

loo

10'

lo2

103

lo4

lo5

I

-2.0 -8.0

-6.0

-4.0

.O

-2.0

2.0

4.0

6.0

(1.0

Casson Number, Ca

Z/R

Figure 7 - Progressive growth of the unyielded zone (shaded) in axisymmetric exit flow of Casson fluids.

0 ~ -Planar -

Figure 9 - End (Casson) correction vs. Cu for Casson fluids flowing out of a planar channel or an axisymmetric capillary (N corresponds to Newtonian result for Cu = 0). Symbols correspond to numerical simulations. The continuous curves are drawn from a non-linear regression analysis on the numerical simulation data according to Equations (1 la) and (1 1b).

(planar geometry) nc = 0.532 + 0.3l[log(Ca)

0.30

+ 0.51

Ca (1 la) CU - 0.32

(axisymmetric geometry) nc = 0.815 + 0.47[10g(C~)+ 0.51

Ca (1 lb) Ca - 0.32

X 0,

C C

0 .+.J 0

2

The present results indicate that increasing the yield stress increases susbstantially the excess pressure losses in the system as the Casson number increases.

L

0

0 J ..I_

X

W

Conclusions

1

0.05 0.00 10-2

loo

10'

lo2

lo3

lo4

lo5

Casson Number, Ca Figure 8 - Exit correction vs. Ca for Casson fluids flowing out of a planar channel or an axisymmetric capillary (N corresponds to Newtonian result for Ca = 0).Symbols correspond to numerical simulations.

The sum of entry and exit pressure losses gives rise to the end (or Casson) correction nc, nc

=

n,

+ nex . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)

The end corrections for the two geometries are shown in Figure 9 as a function of Cu. A linear relationship is obtained for Ca > 1, while a sigmoidal behaviour was obtained in the previous work for Bingham plastics. Again a non-linear regression analysis was performed on the data and gave the following relationship for the end (or Casson) correction (continuous curves on the graphs):

Finite element simulations have been undertaken for entry and exit flows of Casson fluids through planar and axisymmetric channels (capillaries). The Casson constitutive equation for blood flow has been modified as proposed by Papanastasiou (1987) with an exponential growth term to make it valid in both the yielded and unyielded regions, thus eliminating the need for tracking the location of yield surfaces. The present results confirm earlier studies on Bingham plastics regarding reduction of the free surface swelling as the dimensionless yield stress or Casson number increases. Furthermore, the extent and shape of yieldedhnyielded regions have been determined using the criterion of the magnitude of the extra stress tensor exceeding the yield stress. New results include the determination of the entrance, exit, and end corrections as a function of a dimensionlessCasson number (Cu). The entrance correction increases as well as the end correction, while the exit correction goes to zero as Ca QO.

-

Acknowledgement Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged.

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 72, DECEMBER, 1994

1083

Nomenclature

Superscripts

Bi

=

T

Cu

= = = = = = =

D H If rn

nc nen

“ex

R V

Bingham number = r , D / p V (Equation (4)) Casson number = r , D / p V (Equation (4)) capillary die diameter (m) channel half gap (m) second invariant of a tensor stress growth exponent (s) end (Casson) correction, dimensionless entrance correction = (AP-APrp,-APo)/2~,, dimensionless = exit correction = (AP-AP0)/27,, dimensionless = capillary die radius (m) = average velocity (m/s)

Greek letters rate-of-strain tensor (s -‘I shear rate (s -’) overall pressure drop in the system (Pa) pressure drop in the reservoir (Pa) pressure drop in the channel (Pa) = apparent viscosity (Pa.s) = plastic viscosity (Pa-s) = extra stress tensor (Pa) = shear stress (Pa) = wall shear stress (Pa) = yield stress (Pa) = dimensionless yield stress = r,H/pVN (Equation (3))

= = = = =

*

= transpose of a vector or a matrix = dimensionless quantity

References Abdali, S. S., E. Mitsoulis and N. C. Markatos, “Entry and Exit Flows of Bingham Fluids”, J. Rheol. 36, 389-407 (1992). Bird, R. B., G. C. Doi and B. J. Yarusso, “The Rheology and Flow of Viscoplastic Materials”, Rev. Chem. Eng. 1, 1-70, (1983). Casson, N., “Rheology of Disperse Systems”, C. C. Mill, ed., Pergamon Press, New York (1959), p. 84. Mitsoulis, E., S. S. Abdali and N. C. Markatos, “Flow Simulation of Herschel-Bulkley Fluids through Extrusion Dies”, Can. J. Chem. Eng. 71, 147-160 (1993). Papanastasiou, T. C., “Flow of Materials with Yield”, J. Rheol. 31, 385-404 (1987). Shah, V. L., “Blood Flow”, in “Advances in Transport Processes”, Vol. I, A. S. Mujumdar and R. A. Mashelkar, eds., Halsted Press, Wiley Eastern Ltd., New Delhi (1980), p. 1.

Subscripts C

en ex

N res w

0

1084

= = = = = = =

Casson entry exit Newtonian reservoir wall channel

Manuscript received April 15, 1994; revised manuscript received July 6,1994; accepted for publication July 24, 1994.

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 72, DECEMBER, 1994