Journal of Non-Newtonian Fluid Mechanics, 45 (1992) 149-169 Elsevier Science Publishers B.V., Amsterdam

149

Flow of viscoplastic fluids in eccentric annular geometries

Peter Szabo and Ole Hassager

Department of Chemical Engineering, The Technical University of Denmark, DK-2800 Lyngby (Denmark)

(Received March 3, 1992)

Abstract

A classification of flowfields for the flow of a Bingham fluid in general eccentric annular geometries is presented. Simple arguments show that a singularity can exist in the stress gradient on boundaries between zones with yielded and un-yielded fluid respectively. A Finite Element code is used to verify this property of the Bingham fluid. An analytical solution for the flowfield in case of small eccentricities is derived.

Keywords: annular gap; Bingham fluid; eccentric annulus, viscoplastic fluids

1. Introduction

in a

In some practical flow situations, such as transportation of drilling fluids deviation drillings, it is often a good approximation to regard the fluid as viscoplastic liquid. This assumption involves some difficulties in the

prediction of flow fields. The computational problem is a consequence of the existence of a yield stress in the fluid. This yield stress will cause the liquid to stiffen if it is not loaded with a stress in excess of the yield stress.

In simple geometries it is usually possible to determine the location of plug zones by simple arguments. For example, for axial flow in a pipe the shear stress varies linearly from zero at the centerline to the maximum value at the wall. Hence, it is simple to determine the plug zone around the centerline. Another example is unidirectional flow in a wide horizontal slot

Correspondence to: 0. Hassager, Department of Chemical Engineering, The Technical University of Denmark, DK-2800 Lyngby, Denmark.

0377-0257/92/$05.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved

150 P. Szabo and 0. Hassager/ J. Non-Newtonian Fluid Mech. 45 (1992) 149 - 169

where the velocity field varies just with the vertical position. In that situation the fluid will be like a solid in the middle of the slot. Besides these simple examples there are some more complicated ones. The flow of a viscoplastic fluid in an eccentric annulus is a relevant example in connection with the transportation of drilling fluids mentioned above. In this geometry it is not simple to predict the actual shape of the possible plug zones.

It has been argued that plug zones cannot exist in complex geometries [l], but later analysis has revealed plug zones in such geometries [2-51. Also in the analysis in the following sections we show that there will exist plug zones in the general eccentric annulus. Furthermore the shape of the plug zones is in some way surprising.

2. Definition of model and geometry

Viscoplastic fluids behave like Generalized Newtonian fluids when a given stress invariant exceeds the yield stress. Below the yield stress one can regard the fluid as a solid. According to this we write the constitutive equation as

T= -ri, for 7>rCJ, (1)

+=O for 7jr0. (21

The Bingham viscosity model is the simplest model containing a yield stress and is written [6] as

,)-/=m for 71r0,

where T” is the yield stress and where f is defined by

(4)

+ = \i(i,: j)/2.

The stress invariant 7 is defined similarly by

(5)

7 = i(T : T)/2 . (6)

We here analyse stationary axial annular flow with a constant pressure gradient. Therefore the momentum equation reduces to

where 9 is a modified pressure, and the annulus is oriented in the z-direction.

P. Szubo and 0. Hassager/J. Non-Newtonian Fluid Mech. 45 (1992) 149-169 1.51

It is convenient to define a characteristic velocity V,, based on the constant pressure gradient, as

d9 R: &)--

dz ~0' (8)

where R, represents the radius of the outer cylinder in the annular geometry.

After non-dimensionalizing eqns. (l)-(4) and (7) we get

v*. [q*v*u,*] = -1

and

where

roR2 Bn = -

TOk;

(9)

(11)

is the Bingham number. The computational domain is shown in Fig. 1, where we have defined the

parameters K( = RJR,) and 6*( = 6/R,) respectively as the inner radius and the center difference in dimensionless variables.

Hence the axial flow in the eccentric annulus will be characterized by the three non-dimensional numbers K, 6” and Bn.

4Y’

Fig. 1. Eccentric annulus described in non-dimensional coordinates (x*, y*), showing radius of inner cylinder K and distance between centers 6*.

152 P. Szabo and 0. Hassager / J. Non-Newtonian Fluid Mech. 45 (1992) 149 - 169

3. Analytical solution for small eccentricities

It has previously been shown [6] that (9) can be solved analytically when 6” = 0. In the present notation the solution becomes

L:;([)=f[-2BtZ(~-K)-~(~2-K2)+~2 hl(c$/K)] fOrKs(<@-,

q,, = %+yP-> = GYP+) forB_5SrB+

u:(t) = +[ -2Bn(l- 6) + $(l - 5’) + B* In(t)] for B+S[S 1,

(12)

where B* is a constant of integration which fulfils

p’=p+(p+- 2Bn) = p-(6_+ 2Bn). (13)

The [( = r/R2) values in the interval [B _; B +] give the plug zone where the fluid behaves like a solid. Equation (121 above refers to a cylindrical coordinate system (Y, 8, z> with L’; = U,(Y).

The values of (B, B+, B-1 are determined by the three coupled equa- tions in (13). It is possible to eliminate variables in (12) so as to obtain a single equation for B+:

0 = 2B+(B+- 2Bn) In - 1 f (2Bn + K)” + 4Bn(l -B+).

The corresponding dimensionless volumetric flow is given [2] by

Q* = ; [(1 -K”) - 2B+(B+- 2Bn)(l -K’)

- :(I + K~)B~ + y(p+- B+~B~],

where Q* = Q/<V&).

(15)

We now wish to show that the above solution for 6* = 0 may be modified to give an exact analytical solution for 6 * smaller than some critical value 8,: to be determined later. To show this we note that an overall force balance on the plug requires

dg (Area of plug zone) - x

( 1 = r,(Length of lY+ and lY_), (16)

where r+ and r_ refer to the boundaries shown in Fig. 2. In the non-dimensional notation this becomes

B:-B”_= 2Bn(P++p_). (17)

P. Szabo and 0. Hassager / J. Non-Newtonian Fluid Mech. 45 (1992) 149 - 169 153

Fig. 2. The eccentric annular geometry in a situation where there exists an analytical solution. The shaded area shows the plug zone. Bn = 0.25; K = 0.4; 6* = 0.17.

However, this force balance is already contained in (13) and is not depen- dent on the cylinders being coaxial. Hence for a given Bn there may exist solutions to the eccentric annular flow with inner and outer flow regions identical to those for concentric cylinders, but where the plug width is changing in the angular direction.

Hence if we introduce a displaced coordinate system (X’, Y ‘) with origin 0’ at the center of the inner cylinder, as shown in Fig. 2, we may express such solutions as

u%([‘) = i{ -2Bn(5’- K) - t[(s’)‘- K’] +p* ln([‘/K))

for K 15’rp_,

u~,,=~:(S’=p_)=u,*(5=p+) for t’>p- and [<p+, (18)

u,*(t) = +[ -2Bn(l-[) + +(l -t*) +p* In([)] for p+I 5 5 1,

where p* is given in (13) as before but where (p_, 5’) refers to (X’, Y’)- coordinates and (p+, 5) refers to (X, Y)-coordinates in Fig. 2.

In summary, (18) is an exact solution to (9) and (10) with boundary conditions corresponding to the eccentric annular geometry in Fig. 2. Thus the fluid velocity is identical to the plug velocity and the shear stress is equal to the yield stress at I+ and I_. In addition a force balance for the plug is obeyed. In the areas with flow, the solution is identical to the concentric flow solution, the cross-sectional area of the plug is also identi- cal to that for the concentric annulus, but the plug shape is of course different.

The existence of such solutions is dependent on the interior stress in the plug not exceeding the yield stress. Hence the eccentricity S” must be

1.54 P. Szabo and 0. Hassager/ J. Non-Newtonian Fluid Mech. 45 (1992) 149-169

020:

/

/’ (IV)

0 ,/ 00 71, 2 I I, 3 1 ), ,, I, r,, ,z ‘I I 0 00 0 20 0 40 0 60 0 80 0 00 0 20 0 40 0 60

Bfl Bn

000 -11,1.1,-=!~,~,,%~~~ ~~~“,~~~I”“““‘I 0 00 0 10 0 20 0.30 0 40

Bn Bn

Fig. 3. Phase plots for different K values: (a) K = 0.2; (b) K = 0.4; (c) K = 0.6; (d) K = 0.8. The roman numbers describe the areas: (I) no flow at all; (II) one moving plug zone plus one stationary zone; (III) two moving plug zones: (IV) one moving plug zone.

below some critical value SC;, the determination of which will be the object of the following considerations.

4. Development of the force-balance analysis

The S,*, values determined in this section can be drawn as a function of the Bingham number Bn. The final diagrams for some different K values are shown in Figs. 3(a)-3(d).

In area (I) there is no flow at all since the Bingham number is too high. In area (II) at smaller Bingham numbers but high eccentricity there is flow, but only in the wide part of the annulus. Thus in this region there is a plug attached to the walls in the narrow part of the annulus, and a moving plug surrounded by shearing fluid in the wide part of the annulus. In area (III) there are two separate moving plugs, one in the narrow part of the annulus,

P. Szabo and 0. Hassager/ J. Non-Newtonian Fluid Mech. 45 (1992) 149-169 155

Fig. 4. The variables for analyzing the fi function

and one in the wide part. This situation is promoted by a lower eccentricity. Finally, in the near-concentric domain (IV) the inner cylinder is sur- rounded by a single moving plug. In this area the velocity profile is expressed by the simple analytical solution in Section 3.

The lines separating the different regions are determined by simple force balances, as described in the remainder of this section.

The separation between areas (I) and (IV) is given by the line Bn = (1 - ~)/2 as is obtained from setting p_= K and /3 + = 1 in eqn. (131. This also corresponds to a force balance equating the axial pressure forces on the plug to the shear stress at the surfaces of the annulus, thus giving the maximum Bingham number for any flow in the concentric and near concentric situation.

The separation between areas (III) and (IV) may be obtained from the following force balance on the area A in Fig. 4:

+ (s_+s+)ro + 17,. (19)

where s_ and s + are the length of the boundaries A n r_ and A n I- + respectively. This equation then gives the stress 7, along the line I defined by s_ and s,.

The existence of the one-plug situation demands that the stress r, be less than or equal to the yield stress T(). Therefore, we get a split-up criterion if we seek eccentricities for which T, is equal to r(,. In that situation the force balance will look like

O=(s_+s++Z)rO-A (20)

which makes it convenient to define a helping function which we shall write as

F(s_, s+) = (s_+s++ Qr” -A (21)

1.56 P. Szabo and 0. Hassager/ J. Non-Newtonian Fluid Mech. 45 (1992) 149 - 169

The function above has the useful property that if the minimum of F(s_, s+> is equal to zero for some eccentricity 6* then we have found the value of 6:. In a more schematic form the criterion, therefore, becomes

Fmin > 0 : Sin& plug situation.

F,,, = 0 : Critical situation (Q-~ = 7()). (22) Fmin < 0 : Double plug situation.

where Fmin = min{F(s_, s,) 10 <S-S rp_, 0 IS+< r/3+1. The determination of S,T is a straightforward process containing a

two-dimensional Newton iteration on dF/as _ = 0 and aF/ds + = 0. The separation between areas (I) and (II) may be obtained from a

similar force balance. This situation is in a way easy to analyze because it is known that /3+= 1 and p_ = K. After this point we can build up the same function F(s_, s+) as above. The S,T values which we get in this analysis determine the boundary between a situation with no flow at all and another situation with a stationary fluid zone in the narrow part of the eccentric annulus as well as a moving plug zone in the wide part.

To obtain a complete description of the possible flow domains, it is necessary to determine the separation between the attached plug zone in area (II) and the moving plug zone in area (III). This separation is located only in an approximate manner. In the present analysis we make an assumption on the shape of the ending boundary I in Fig. 5. For simplicity we approximate the ending boundary 1 by a straight line. Then we can construct a function G which is similar to F with one exception. This exception is a consequence of the flow geometry and the rheology of the fluid. It is natural that the velocity component, LJ=, has a maximum in the wide part of the eccentric annulus and a minimum in the narrow part. This means that the fluid is pulling the ending boundary and that the fluid is pulled by the boundaries s_ and s,. A consequence of this is a negative contribution to the force balance on the line 1.

’ R,-! I-

Fig. 5. The variables for analyzing the G function.

P. Szabo and 0. Hassager/ J. Non-Newtonian Fluid Mech. 45 (1992) I49- 169 157

The G-function can now be written as

G(L s+)=(s_+s + (23)

which obviously can be analyzed precisely as F. The difference between the analysis of G and the previous analysis of F is that we are now seeking values of 6 * so that G = 0 is a maximum.

5. Finite element analysis of Bingham flow

In this section we shall analyze the flow of a Bingham fluid by using two fundamentally different schemes of the finite-element method (FEM).

The first method considered is a pseudo-Bingham approach where we approximate the infinite viscosity for i, equal to zero with some finite but very large viscosity when + becomes small.

In the second implementation we use the exact Bingham viscosity model but with the domain of solution restricted to the area where + is non-zero.

5.1 The Galerkin formulation

Before the description of the Bingham model implementations we shall give a brief introduction to the Galerkin formulation of the FEM equa- tions.

Let

M

c,= Cu,N, (24) i=l

denote a function which approximates u:. The set N,(i = 1,. . . , M) consists of bilinear interpolation functions.

The Galerkin equations for ~7, are then

o = j--4(“*. [q*V*Q] + 1) da. (25)

The expression in (25) can, after some manipulations and with the use of the Gauss-Ostrogradskii theorem [6], be written as

where n is a unit vector directed outward normal to the domain fi and where we identify -q *(n . V * 6,) with the normal stress component r,*, = [(n . T) * S,]/(~oV,/R,) at the boundary of a.

158 P. Szabo and 0. Hassager /J. Non-Newtonian Fluid Mech. 45 (1992) 149- 169

The combination of (24) and (261 gives a symmetric system of M non-linear algebraic equations

+

i3N, dN, ~~ ay* ay*

I 1 dx” dy” u,

= N, dx* dy* - N,r,*, dl?. / R / l-

(27)

Because the shape functions are exactly zero at the boundaries, where we have an essential boundary condition (i.e. L!=* is known), we can simplify the expression above. The simplification follows from there being only non-zero contributions to the boundary integral on the boundary I,, which is the boundary where we have a natural boundary condition (i.e. r,*, is known).

Due to the symmetry about the x-axis (see Fig. l), it is necessary only to discretize the upper part (y 2 0) of the annulus. In the discretization we subdivide the domain of solution, CI, in isoparametric quadrilateral ele- ments [7,8] in which we use locally defined shape functions.

5.2 The pseudo-Bingham model implementation

This modified Bingham model is given by

Bn ~*=l+~ forj*>g*,

Y

Bn 7 *=l+, forj*lg*,

g (29)

where g” is some very small number. In practical calculations we have seen convergence to four significant

figures in the volumetric flow when we have used g* values below 10P4. That the method converges for g* -+ 0 does not, of course, guarantee the accuracy of the calculated volumetric flow. Therefore, we have done some more quantitative convergence tests which will be described in section 5.2.2.

5.2.1 The domain of solution and boundary conditions In the pseudo-Bingham model implementation we discretize the entire

area between the two cylinders in the annulus for y 2 0. Hence we have essential boundary conditions <u,* = 01 at the surfaces of the inner and the outer cylinders. The natural boundary conditions in this implementation

P. Szabo and 0. Hassager/J. Non-Newtonian Fluid Mech. 45 (1992) 149-169 159

are (n * V* u,*, = 0 at y = 0 in the annulus. Therefore we can set the boundary integral in (27) equal to zero in the pseudo-Bingham formulation.

5.2.2 Convergence of the FE44 discretization In the process of testing the convergence effectively under mesh refine-

ment, we have used a number of analytical solutions for the laminar unidirectional flow in the annulus.

We have studied the convergence for both the pseudo-Bingham and the Carreau (see the Appendix) viscosity models besides the simple Newtonian model. We shall consider only the non-linear examples here.

First we shall describe the norm function which measures the difference between the exact and the calculated solution.

We define a local deviation function, AP_l

Ap,r = v, - Q,r 7

where vi is the exact velocity in a node i, which approximate velocity component Spr. In APl and indicates a particular mesh. The definition in (30) define a global deviation function as

i

has the corresponding in tjP,i, the subscript p makes it reasonable to

(31)

where Nno& is the total number of nodes in the mesh. We expect the FEM discretization to have h* convergence, that is we

expect

II v - ti, II = ch;, (32)

where c is some constant and where the double lines indicate an L2 norm. In our tests of h2 convergence we have approximated the L* norm by the global deviation function expressed in (31). The variable h, is a character- istic length in an element. We have performed two series of computations which show h* convergence: a Carreau model simulation and a pseudo- Bingham model simulation.

In the full-annular domain discretization we can map the upper annular area on to a rectangular domain. If we consider an equidistant N,, XIV,,

domain decomposition, the h, value is proportional to l/N,. Hence we plot the global deviation function, EP, as a function of N, in an N, X Np

domain decomposition. The results are visualized in Figs. 6 and 7 for the Carreau and for the pseudo-Bingham models respectively. The Carreau model calculations show almost exact agreement with the h” convergence (a,, a IV;*.“). In the pseudo-Bingham calculations there seems to be good

160 P. Szabo and 0. Hassager / J. Non-Newtonian Fluid Mech. 45 (1992) 149 - 169

10

G

Fig. 6. Test for h2 convergence in the Carreau model implementation: De = 16; n = 0.5; qJqo = 0; K = 0.4; s* = 0.

10 -6: i 10 100

NP Fig. 7. Test for h’ convergence in the pseudo-Bingham model implementation: Bn = 0.1;

g *=10-‘0. K=04.6*=0 . >

agreement except for very fine meshes. For the very fine meshes we can observe a slight falloff from the h2 convergence. The reason for this may be that the accuracy of the numerical solution is so high that the pseudo-Bing- ham approximation becomes relatively inaccurate; in other words: we approach the limit of the pseudo-Bingham approximation.

5.2.3 Volumetric flow-rate calculations In Section 4 we have made qualitative phase plots of the possible flow

situations. These illustrations are helpful for understanding the flow condi- tions, but give no direct information about the volumetric flow rate. A representation for the volumetric flow rate is shown in Fig. 8, which has

P. Szabo and 0. Hassager/ J. Non-Newtonian Fluid Mech. 45 (1992) 149-169 161

0 25

0 30

0 35

Fig. 8. Volumetric flow as a function of the eccentricity 6* and the Bingham number Bn, for a particular mesh and geometry: N, x N,, = 72 x 72; K = 0.4; g * = lo-“.

been obtained from the pseudo-Bingham implementation. The figure shows Q/QN as a function of the eccentricity 6* with the Bingham number Bn as a parameter. Here QN is the Newtonian flow rate for a concentric annulus, given by [9]

Q Newtonian,concentrlc = f (1 -fc”) - &;I:; 1 Iq?;. (33)

The agreement between Fig. 3(b) and Fig. 8 is very good. Both figures correctly show that the volumetric flow approaches zero for Bn = 0.30 when 6* + 0.2 from above. In addition to this, we can observe a large increase in the volumetric flow rate when the annular geometry becomes eccentric.

5.3 The exact Bingham model implementation

In the pseudo-Bingham model implementation we have solved the flow problem in the entire annular cross-section. This method has the disadvan- tage of using an approximate viscosity model in those areas where the shear rate, +, is less than some limit value, g. In situations where one is interested in locating the yield surfaces this method will not be precise enough. We turn, therefore, in this section to the determination of the yield surfaces based on the exact Bingham model.

As described in Section 4 there are four different kinds of flow situations (I-IV) dependent on the geometry (see Fig. 3). These figures show that it is necessary to find a numerical solution in the areas (II) and (III) only,

162 P. Szabo and 0. Hassager/ J. Non-Newtonian Fluid Mech. 45 (1992) 149 - 169

X

Fig. 9. The shaded area shows the initial domain of discretization in the exact Bingham model implementation before iteration.

because we have an analytical field in the areas (I) and (IV).

Because the implementation parameter areas (II and III) double-plug situation (III).

solution (18) which expresses the velocity

principles seem to be similar in the two we have chosen one of them: the free

5.3.1 The domain of solutiorl and related (boundary) conditions The key difference between the pseudo-Bingham and the exact Bingham

model implementations is that we do not discretize the whole annular cross-section in the latter method.

Because we have chosen the free double-plug situation for our simula- tions, we discretize an assumed domain similar to the accentuated area in Fig. 9. This method will require a sequence of iterations to determine the exact boundary positions. The determination of these boundaries will be described in Section 5.3.2.

In addition to the boundary conditions mentioned in section 5.2.1 there will be a natural condition at the plug-zone boundaries. The natural condition is a consequence of the Bingham fluid which demands the shear stress, r, to be equal to the yield-stress at the plug boundaries. That is formally

n-7= fT$i*, (34)

where 6; is a unit vector in the z-direction. The consequence of (34) is that rnZ = -L-r0 or T,*, = ,Bn. The sign on 7O in (34) is explained in Section 4 and is given by

bi,, bZx7 b,, and bZr7 > (35)

where the boundaries, bka, are shown in Fig. 9.

P. Szabo and 0. Hassager/ J. Non-Newtonian Fluid Mech. 45 (1992) 149- I69 163

The boundary integral in (27) can now be determined as

Fig. 10. The shape of the plug zones for the parameters: K = 0.4; Bn = 0.1. (a) 6* = 0.15; (b) 6 * = 0.04.

164 P. Szabo and 0. Hassager / J. Non-Newtonian Fluid Mech. 45 (1992) 149 - 169

The exact Bingham model flow is then solved by the following algorithm: 1. For a given location of plug zones, solve for LIJX, y) outside the plug

zones with the condition (36) on the yield surfaces. The initial location must be qualitatively correct, as indicated, e.g. in Fig. 9. The resulting solution will not necessarily have U, constant on the yield surfaces, and may not fulfil the force balances (21) and (23).

2. The yield surfaces are relocated in a way to obtain constant U, on the yield surfaces and to fulfil (21) and (23). The relocation is based on qualitative arguments (e.g. in the gap, between a stationary wall and a plug, the velocity at the plug boundary will increase if the gap width increases and vice versa>. The iteration then returns to 1. until the yield surfaces have converged to a desired accuracy.

This algorithm has been found to converge in the annular geometry, but it would be useful with a more general algorithm for relocating the plug surfaces.

5.3.2 The process of shaping plug zones The force balance on a plug zone can generally be expressed as a

function of one variable and a set of parameters which we can manipulate . in the iteration loops. Therefore F and G can be written as

F = F(s, cu) and G = G(t, p) (37)

Fig. 11. An example of the final mesh. (Compare with Figs. 9 and 10(b).)

P. Szabo and 0. Hassager / J. Non-Newtonian Fluid Mech. 45 (1992) 149 - 169 16.5

where s and t are variables that describe the areas of the plug zones and (Y and p contain a set of spine vectors.

The intention with the formulation above is that, during an iteration, we shall manipulate only relative quantities. We can then determine the values of s and t after an iteration loop which fulfils the equations F = G = 0. This determination has to be iterative too, but it can be done with a Newton scheme. The final plug zones are visualized in Figs. 10 and 11.

6. Conclusions and discussion

We believe that we have revealed some surprising new properties of a yield-stress fluid, exemplified by the Bingham fluid.

First, the plug in the narrow part of the annulus is interesting in as much as there will be a condition rn2 = +T,, on part of the surface, and r,,2 = -TV on another part of the surface. The reason for this is that fluid will be pulling the plug near the opening to the wider part of the annulus Cr.,, = +Q-& while the plug will be pulling the fluid near the annular walls (T,, = --~a). The transition from the condition 7nZ = +~a to T,* = -TV must necessarily occur at a point, i.e. with a singularity in the stress gradient. We see from Fig. 10 that this transition takes place at a sharp corner. We believe the demonstration of such transitions in yield-stress fluids to be novel.

Second we may compare our results with those obtained in the analysis by Walton and Bittleston [lo] who use a slot approximation to describe a narrow eccentric annulus (for K > 0.7). In Fig. 12 we have plotted the results from the slot approximation together with the phase plot from Section 4 for a K value of 0.8. The figure shows very good agreement

c 20

Fig. 12. Comparison ofR;he present analysis ( -1 with that of Walton and Bittleston [lo]

(- - --> in the valid interval of eccentricity.

166 P. Szabo and 0. Hassager / J. Non-Newtonian Fluid Mech. 45 (1992) 149 - 169

Fig. 13. Comparison of the present analysis (- ) with that of Walton and Bittleston [lo]

(-- - -_) for an eccentricity where the slot approximation breaks down.

between the two methods for the transition between the free double-plug situation and a double plug with one stationary zone. The two methods also give similar predictions for the transition curve separating the area with no flow at all from the area with a double plug with stationary zone. In this prediction, the methods seem to diverge for decreasing K values. This can be observed in Fig. 13, where we have done a similar comparison as in Fig. 12, but with K = 0.6. This is strictly not in the valid area of the slot approximation developed in Ref. 10, but it is interesting to observe the similarity in the predictions of the transition between the areas (II) and (III).

,_-/ I

/

,

I

,

,

i;

Fig. 14. Comparison of the qualitative convergence of the two FEM implementations:

( -) pseudo-Bingham; (---_) exact Bingham. Bn = 0.1; 6* = 0.15; K = 0.40; (g* = lo-” in the pseudo-Bingham model).

P. Szabo and 0. Hassager/ J. Non-Newtonian Fluid Mech. 45 (1992) 149- 169 167

Thirdly, in their analysis, Walton and Bittleston have introduced the term pseudo-plug for a region which connects the two real plug zones. Characteristic of the pseudo-plug is an almost constant velocity, which changes slowly in the tangential direction but very little in the radial direction. It is interesting that such plug zones can be clearly observed in the exact Bingham model implementation as may be seen in Fig. 10.

Finally, in the eccentric annular analysis, we have predicted the area (IV) where we have an analytical solution describing the velocity field and the constant volumetric flow rate. This area cannot be predicted from the slot approximation.

In closing we remark that we cannot investigate the convergence of the exact Bingham implementation in regions (II) and (III) as quantitatively as in the pseudo-Bingham implementation due to the lack of an analytical solution. We have, however, illustrated the qualitative convergence of the two implementations in Fig. 14 for a particular choice of parameters.

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Appendix

The Carreau viscosity model

The Carreau model [6] for the viscosity function q(j) is

n(9) - %a

770 - %Z = [ 1 + (*j)y2), (Al)

168 P. Szabo and 0. Hassager / J. Non-Newtonian Fluid Mech. 4.5 (1992) 149 - 169

where 70 = limi.,oT(j) and qrn = limi,,30q(j). The parameter A is a time constant and y1 is the Power Law index when + approaches + ~0.

Analytical solutions for concentric annular jZow

When 7a, = 0 and II = i or n = 0 it can be shown that the velocity field in a concentric annulus can be expressed as simple integral expressions,

/ f=5 f(t)

%w%=1/2= - I=K oe& -[/f2(t) + v!!] dt

and

W)

%%%z=o = -l=, K f=‘f(t) [l _f 2(t)] -112 dt, w

TABLEAl

p values for use in (A4)

De P(De, K)

K = 0.2 K = 0.4 K = 0.6

n=1/2and6*=0

0.0 0.5461 0.6770 0.7915

0.5 0.5455 0.6769 0.7915

1.0 0.5437 0.6766 0.7914

2.0 0.5376 0.6753 0.7912

4.0 0.5262 0.6713 0.7904

8.0 0.5201 0.6669 0.7886

16.0 0.5190 0.6656784 0.7875

32.0 0.5189 0.6655 0.7872

+m 0.5189 0.6655 0.7872

n=OandiS*=O

0.0 0.5461 0.6770 0.7915 0.5 0.5448 0.6768 0.7914 1.0 0.5404 0.6760 0.7913 1.5 0.5295 0.6746 0.7911 2.5 0.4472 0.6674 0.7904 3.5 _ - 0.7890 4.5 - _ 0.7849

Deer 2.5000 3.3333 5.0000

P CT 0.4472 0.6325 0.7746

P. Szabo and 0. Hassager/J. Non-Newtonian Fluid Mech. 45 (1992) 149-169 169

where ,$ = r/R,, De = Al/,/R, and the function f is given by

w

where P(De, K, n) is a constant of integration tabulated in Table Al. The p values have been determined by requiring the integrals in (A2)

and (A3) to be zero, when the upper limit is 5 = 1. When y1= 0 the volumetric flow approaches +m if 1 f(l) I is equal to

unity. This gives rise to a pair of critical values for p and De, that we denote p,, and Deer respectively. These are determined by the conditions

1 f&) 1 = /f,,(l) I = 1, W)

such that

p,, = 6 and Deer = & . @6)

From (A6) it follows naturally that the Carreau model, with qrn = IZ = 0, is usable for De < Deer only.