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The drag on a sphere in a power-law fluid
Journal of Non -Newtonian Fluid Mechanics, 17 (1985) 1-12 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
THE DRAG ON A SPHERE IN A POWER-LAW FLUID
GU DAZHI
* and R.I. TANNER
Department of Mechanical Engineering, The University of Sydney, N.S. W. 2006 (Australia) (Received
February
21, 1984)
Summary Using a finite-element program, the drag on an unbounded fluid (inelastic) power-law fluid is estimated. Comparison with upper and bounds, and with experimental data is given. Some remarks on wall are also made, and it is shown that wall effects are negligible for n G
in an lower effects 0.5.
1. Introduction The drag on a falling sphere is of great theoretical and practical interest in both Newtonian and non-Newtonian fluids. The present paper addresses the problem for creeping flow, and we investigate numerically the drag on a sphere in a power-law fluid. It is surprising that we have not been able to find a satisfactory previous tabulation of the solution to this problem especially for highly non-Newtonian cases. We also make some observations on wall-effects in shear-thinning fluids. The drag D on a sphere, radius R,, moving at speed U in an unbounded power-law fluid, with power-law parameter k, index n (see eqn. (8)), can be written, for creeping flow, as n-1
UR,X(n).
This result follows from dimensional reasoning. When n = 1 (Newtonian case), the value of the function following we shall calculate X(n) for 0.1 < n f 1. When * Permanent address: Technology, Chengdu, 0377-0257/85/$03.30
The Polymer Research Sichuan, China.
Institute,
Chengdu
Q 1985 Elsevier Science Publishers
B.V.
University
X is 1. In the inertia is not of Science
and
2 neglected, we can write, for a fluid of density p Cd =
D
+rpU=R;
=f(Re’,n),
(2)
where Re’, the generalized Reynolds number, is given by Re’ = pU2-“(2R,,)“/k
(3)
In this case one can define the Stokes drag coefficient, from eqns. (1) and (2), as Cdst= 24/Re’,
and then define the relation
Cd= XC&t
7
(5)
where X has, in the creeping flow limit, the value given in eqn. (1). In surveying the literature, several constitutive models have been used for the sphere problem, and the most widely studied model is the power-law fluid [l-9]. Some viscoelastic models have been studied too, such as the Oldroyd fluid [lo], the Maxwell fluid [ll-131, the third-order Rivlin-Ericksen fluid [14-181, the second-order fluid [19], and the Carreau fluid [20-223. The methods of approximation can be sorted into perturbation techniques, [10,14,19]; variational principles [1,3,5,21]; finite difference methods [6]; and an Oseen linearization technique [ll]. Most of these studies concern an unbounded fluid, but a few use cylindrical boundaries [22] and spherical boundaries [13]. In this paper, the main interest is on the effects of shearthinning behaviour. We draw special attention to Slattery’s [5] work on upper and lower bounds on the drag coefficient in a power-law fluid, which result from using variational principles for velocity trial functions and stress trial functions respectively. The upper bound is well known as an energy extremum principle, but the physical meaning of the lower bound is not very clear. Recently, Cho and Hartnett [9] reported an improved approach for drag bounds and give a physical explanation for the lower bound. Chhabra [23] recently gave an interesting discussion of Cho and Hartnett’s report and it is clear that no accurate predictions of drag, even for the power-law fluid case, are easily available. Our results reported below are found from numerical solutions of the full equations of motion and continuity by a finite-element method. In our numerical simulation we used a ball-in-sphere configuration with different radius ratios, and used extrapolation to find the finite-boundary correction; the range of power-law index (0.1~ n < 1) was covered. We also used the ball-in-tube configuration for discussion of the wall effect.
3 2. Numerical Simulation The finite-element program AXFINR developed by Nickell, Tanner and Caswell [24] for solving incompressible, viscous flow problems is used to simulate the creeping motion of a non-rotating solid sphere moving through a power-law fluid. A finite-bounded spherical geometry is employed for the outer fluid surface of the flow field. Fig. 1 shows the finite-element mesh used in our main computations. The reference frame for writing the governing equations is a polar coordinate system with the centre of the sphere instantaneously at its origin as shown in Fig. 1. For low Reynolds number flow of an incompressible fluid the motion is governed by the conservation of mass equation v-v=o,
(6)
and the conservation equation of linear momentum. In the case when inertial and body forces can be neglected we have v-r-vp=o,
(‘4
where r is the deviatoric stress tensor and p is the pressure. The rheological equation-of-state of the fluid under consideration, which describes the constitutive relation between the tensors of stress and the rate of strain, needs to be specified. In this research, we study the power-law model; this
Fig. 1. Typical finite-element mesh and coordinate system. Ratio of outer fluid-sphere radius R to solid-sphere radius R, equals to 20, i.e. s = R/R, = 20.
4 fluid model can be expressed,
in simple shearing,
as
7 = klY/“,
(8)
where 7 is the shear stress and t is the shear rate; n and k are constant. For a three-dimensional flow we have the proper generalization given by Slattery
[51* The present on the surface assumed
problem is specified by the following boundary of the solid sphere a no-slip velocity boundary
v=iUonr=R,,
conditions: condition is
(9)
where i is a unit vector in the x,-direction and U is a constant speed. On the outer surface of the fluid boundary the normal velocity is zero: VR= 0
(forr=R).
(10)
A slip boundary condition must be satisfied too, meaning there is no tangential traction on the outer spherical fluid surface. This condition minimizes the influence of the outer boundary on the motion of the sphere. The “wall effect” in the present numerical simulations are more important for calculations using a power-law index close to one, the Newtonian case. The data are corrected for this effect by extrapolation with different ratios of R/R,. In the Newtonian case it is not difficult to show, for a sphere of radius R, moving within a sphere of radius R and with a tangential slip boundary condition on the outer sphere, that the drag is increased from the Stokes unbounded D,, to the value
We can use this result to assess the accuracy of our calculations. Two quantities U and R, are used to normalize the parameters appearing in the governing equations. The system of equations (6) (7) and the constitutive equation are solved numerically by the program AXFINR. The normalized drag force D is calculated from the resulting solution of the stress field by numerical integration. The computations span the range of power-law index (n) 0.1-1.0. The error of the numerical approach depends on the grid used and on the number of iterations. Generally speaking, for lower values of the index n more iterations are needed, but each value of n has its own optimum number of iterations (i.e. the lowest number of iterations after which no significant further improvement in the solution occurred), and we use the optimum results. Taking the solutions of the velocity field as an example, the errors (for R/R,, = 20) are listed in Table 1. In this Table we show the values of 100 [z.,,,(Av,)2/z.,,,,v,211’2 where AVn is the change in velocity V, at
5
node n from the velocity at the previous iteration. It is believed these approximations show satisfactory convergence. We are most interested in the unbounded flow around a sphere, but we cannot avoid wall effects. For the Newtonian case, taking wall effects into consideration and comparing with the Fax&-type correction [25], these effects decrease with the increase of R/R, (= s). For s = 20 the error is less than 8%; for s = 50 the error is 3%; for s = 80 the error is less than 2%. Extrapolating to the unbounded case we find a result very close to Stokes’ law. Note that we are not using the non-slip boundary conditions at the outer radius and hence the correction for finite R/R, is less than it would be if the no-slip boundary condition were applied. We estimate the expected drag in the Newtonian case to be about (1 - 1.5 RJR)-‘D,, from eqn. (11). For the case R,/R = 0.05, we found D = 1.0954,, compared with the exact value of l.OSlDs,. thus the magnitude of the error in the drag is about 1% in this case. Generally, if the radial distance between the innermost circles of elements (Fig. 1) was h, we find the calculated drag, Dcalc, to be related to the true drag D for different radial mesh spacings by Dca,c= D - bh’,
(12)
where b is an unknown factor depending on n and R/R,. From this estimate, using h = 0.1, 0.2, and 1, we again find the expected errors in the calculated drag to be of order l-2%, being least accurate for n = 0.1 and most accurate for n = 1. In addition to these convergence and grid-size errors, we have to estimate the wall-effect. Since we are dealing with large R/R, ratios, we assume that the wall-effect is given by the Fax&i-type form [ 111 (ignoring ( R,/R)5 and similar terms), D = D,,(l
- a(n)R,/R)-‘,
(13)
where the coefficient (Yis a function of n. Fig. 2 shows two examples, n = 1 and n = 0.5, which suggests that D,,/D is indeed a linear function of R,/R in the range of values considered (R/R, 2 20). In each case we used a formula like eqn. (13) to extrapolate back to the unbounded sphere drag. It TABLE 1 The estimated errors of the solution of the velocity field Power-law index n
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
Iterations
4
4
4
7
8
9
6
9
15
Error s&i (less than)
0.02
0.2
0.5
0.05
0.1
0.2
2
2
1
0.1
6
POWER
- LAW -
.
0.9
- 0.6
-
OD
I
I
I
0.01
0.02
0.03
1 0.04
0.05
Ro/R
Fig. 2. Extrapolations for wall effects of Newtonian using eqn. (13). The ordinate here is Ds,/D.
fluid and power-law
fluid (n = 0.50)
(a)
4
-L
12
20
Cb)
Fig. 3. Numerical simulation of a uniform creeping flow passing finite-element mesh and coordinates, and (b) streamlines.
around
a fixed sphere,
(a)
7 TABLE 2 Wall-effect
factor cu(n) for a sphere moving axially in a long tube
n
a
1.0 0.8 0.65 0.5 0.0
2.104 (exact) 0.98 0.34 0.05 0.00
formula like eqn. (13) to extrapolate back to the unbounded sphere drag. It is apparent from Fig. 2 that the more shear-thinning is the fluid, the less important is the wall-effect, as has been noted experimentally [25]. Fig. 2 is an example of wall-effect using the sphere-in-sphere geometry. In practical experiments we need to know the wall-effect factors 1y in a sphere-in-tube geometry (Fig. 3). These factors are shown in Table 2. These values were found by taking the computed results (corrected for mesh size) and fitting the form (13) to them. Fig. 4 shows the curve constructed from these values. One sees that the
a
0
0.2
0.4
Fig. 4. Wall-effect
0.6
0.6
1.0
n
factor a for sphere-in-tube
geometry
using eqn. (13) and Table results (A).
8
(a)
lI
-20
-12
4
-4. z
(C)
I2
M
9
I
-20
I
I
-12
I
Z
-20
I
-4
I
I
I
I
12
2u
4
12
20
4
12
20
Cd)
-4
-12
I
4
(e)
-20
-4
-12
Z
(f)
Fig. 5. Velocity and streamline details for various n-values. Radial velocity contours: (a) Newtonian; (b) power-law n = 0.5. Axial velocity contours: (c) Newtonian; (d) power-law n = 0.5. Streamlines: (e) Newtonian; (f) Power-law n = 0.5. In the streamline plots the outermost and innermost contours have the values 0.097 and 3.78 (n = 1) and 0.019 and 0.757 (n = 0.5). The other contours are linearly interpolated between these values.
10 wall-effect is negligible for lack of fluid movement far One may note in passing effects for n > 1, one would the large wall effects.
n ,< 0.5. The velocity fields (Fig. 5) confirm the away from the sphere as n decreases below 0.7. that if an attempt were made to compute wall expect great difficulty in extrapolation owing to
3. Results and discussion The flow field around a sphere is not viscometric, but we found that the Newtonian fluid and power-law fluids have a similar flow pattern. Fig. 5 shows the flow patterns of various fluids of different power-law index when the sphere moves slowly through them. We can find the character of this flow field outlined above from these plots. From Figs. 3 and 5 we can see that the shear-thinning behaviour reduces the long-range disturbance caused by the moving sphere; the region of “circular” flow gradually closes on the sphere as the power-law index decreases. Sigli and Coutanceau [13] have reported some flow visualization results which confirm our numerical simulation (see Fig. 2 in reference [13]). We assume the shear-thinning behaviour of the falling-sphere viscosity obeys the power-law eqn. (8). Table 3 lists our numerical calculation of the values of the drag correction factor X, (eqn. 1). The (upper-bound) results of Wasserman and Slattery [5] and of Cho and Hartnett [9] are also listed in Table 3. These results and their lower bounds are plotted in Fig. 6. We also plot in Fig. 6 some available experimental data reported by Cho and Hartnett [26].
TABLE 3 The values of the drag correction factor X for unbounded flow n
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
X Slattery’s upper bound
Cho & Hartnett’s upper bound
present results
1.005 1.140 1.260 1.366 1.445 1.499 1.549 1.617 1.704 -
1.0060 1.1402 1.2594 1.3604 1.4409 1.4992 1.5335 1.5410 1.5175 1.4581
1.002 1.140 1.240 1.320 1.382 1.420 1.442 1.458 1.413 1.354
11 It is clear that the present results are between the upper bound and lower bound of both Slattery and Cho et al., and are much closer to the experimental data than any bound, giving a greatly improved solution to the problem; the accuracy of our results in Table 3 is estimated to be - 1% (n = 1) down to - 2% (n = 0.1). Bush has calculated the drags by a boundary-element method [27] and has confirmed the first four entries of Table 3 at the 1% accuracy level; the result for n = 0.6 did not converge well with the boundary-element technique. Except for a few research workers [8] who reported XC 1, the present work and most previous researches show X > 1. Acharya et al. [28] give a quantitative explanation of this effect and attempt to distinguish between the effects of shear-thinning viscosity and elasticity. Our results are consistent with experimental data for some liquids, at least qualitatively (Fig. 6). The
X
”
Fig. 6. Comparison of the drag corretion factor X(n) of the present results with the upper present results; and lower bounds of [23,26] and some available experimental data. -, ______ , upper and lower bounds of Wasserman and Slattery [5]; -. - ., upper and lower bounds of Cho and Hartnett [26]; 0, Carbopol 960; P, CMC; 0, Polyacrylamide AP-273.
12 effect of elasticity is small for Weissenberg numbers Wi (AU/R, = Wi, where X is the fluid relaxation time) for Wi c 1. This has been shown for an upper-convected Maxwell fluid in several papers [29,30] and is confirmed by our own calculations for a Maxwell fluid; changes of drag of order of 1% are observed at Wi = 1 for the Maxwell fluid. Hence the change of viscosity is often dominant. One must note the inability of power-law models to model real fluids in complex flows. Additionally, it is a fact that for most polymer fluids the nonlinear viscosity and elasticity are coupled to each other, and it is difficult to get a purely viscous shear-thinning fluid or a purely elastic fluid without a shear thinning effect, so it is difficult to distinguish these two effects experimentally. It is however believed that the shear-thinning behaviour is usually the more important effect. References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Y. Tomita, Bull. JSME, 2 (1959) 469. A. Faraoui and R.C. Kintner, Trans. Sot. Rheol., 5 (1961) 369. J.C. Slattery, AIChE J., 8 (1962) 663. G.C. Wallick, J.G. Savins and D.R. Artorburn, Phys. Fluids, 5 (1962) 367. M.L. Wasserman and J.C. Slattety, AIChE J., 10 (1964) 383. K. Ada&i, N. Yoshioka and K. Yamamoto, Chem. Eng. Sci., 28 (1973) 2033. P.H.T. Uhlherr, T.N. Le and Tiu, Can. J. Chem. Eng., 54 (1976) 497. K. Ada&i, N. Yoshioka and K. Sakai, J. NonONewtonian Fluid Mech., 3 (1977/78) 107. Y.I. Cho and J.P. Hartnett, J. Non-Newtonian Fluid Mech. 12 (1983) 243. F.M. Leslie, Quart. J. Mech. and Appl. Math., 14 (1961) 36. J.S. Ultman and M.M. Denn, Chem. Eng. J., 2 (1971) 81. R.Y.S. Lai, Int. J. Eng. Sci., 12 (1974) 645. D. Sigli and M. Coutanceau, J. Non-Newtonian Fluid Mech., 2 (1977) 1. B. Caswell and W.H. Schwartz, J. Fluid Mech., 13 (1962) 417. H. Giesekus, Rheol. Acta, 3 (1963) 59. B. Mena and B. Caswell, Chem. Eng. J., 8 (1974) 125. J.M. Broadbent and B. Mena, Chem. Eng. J., 8 (1974) 11. B. Caswell, Chem. Eng. Sci., 25 (1970) 1167. P. Brunn, Rheol. Acta, 15 (1976) 15. R.P. Chhabra and P.H.T. Uhlherr, Rheol. Acta, 19 (1980) 187. R.P. Chhabra, P.H.T. Uhlherr and D.V. Boger, J. Non-Newtonian Fluid Mech., 6 (1980) 187. R.P. Chhabra, C. Tiu and P.H.T. Uhlherr, Can. J. Chem. Eng. 59 (1981) 771. R.P. Chhabra, J. Non-Newtonian Fluid Mech., 13 (1983) 225. R.E. Nickell, RI. Tanner, B. Caswell, J. Fluid Mech., 65 (1974) 189. RI. Tanner, Chem. Eng. Sci. 19 (1964) 349. Y.I. Cho and J.P. Hartnett, J. Non-Newtonian Fluid Mech., 13 (1983) 229. M.B. Bush & R.I. Tanner, Int. J. Num. Meth. Fluids, 3 (1983) 71. A. Acharya, R.A. Mashelkar and J. Ulbrecht, Rheol. Acta, 15 (1976) 454. G. Tiefenbruck and L.G. LeaI, J. Non-Newtonian Fluid Mech., 10 (1982) 115. 0. Hassager and C. Bisgaard, J. Non-Newtonian Fluid Mech., 12 (1983) 153.
THE DRAG ON A SPHERE IN A POWER-LAW FLUID
GU DAZHI
* and R.I. TANNER
Department of Mechanical Engineering, The University of Sydney, N.S. W. 2006 (Australia) (Received
February
21, 1984)
Summary Using a finite-element program, the drag on an unbounded fluid (inelastic) power-law fluid is estimated. Comparison with upper and bounds, and with experimental data is given. Some remarks on wall are also made, and it is shown that wall effects are negligible for n G
in an lower effects 0.5.
1. Introduction The drag on a falling sphere is of great theoretical and practical interest in both Newtonian and non-Newtonian fluids. The present paper addresses the problem for creeping flow, and we investigate numerically the drag on a sphere in a power-law fluid. It is surprising that we have not been able to find a satisfactory previous tabulation of the solution to this problem especially for highly non-Newtonian cases. We also make some observations on wall-effects in shear-thinning fluids. The drag D on a sphere, radius R,, moving at speed U in an unbounded power-law fluid, with power-law parameter k, index n (see eqn. (8)), can be written, for creeping flow, as n-1
UR,X(n).
This result follows from dimensional reasoning. When n = 1 (Newtonian case), the value of the function following we shall calculate X(n) for 0.1 < n f 1. When * Permanent address: Technology, Chengdu, 0377-0257/85/$03.30
The Polymer Research Sichuan, China.
Institute,
Chengdu
Q 1985 Elsevier Science Publishers
B.V.
University
X is 1. In the inertia is not of Science
and
2 neglected, we can write, for a fluid of density p Cd =
D
+rpU=R;
=f(Re’,n),
(2)
where Re’, the generalized Reynolds number, is given by Re’ = pU2-“(2R,,)“/k
(3)
In this case one can define the Stokes drag coefficient, from eqns. (1) and (2), as Cdst= 24/Re’,
and then define the relation
Cd= XC&t
7
(5)
where X has, in the creeping flow limit, the value given in eqn. (1). In surveying the literature, several constitutive models have been used for the sphere problem, and the most widely studied model is the power-law fluid [l-9]. Some viscoelastic models have been studied too, such as the Oldroyd fluid [lo], the Maxwell fluid [ll-131, the third-order Rivlin-Ericksen fluid [14-181, the second-order fluid [19], and the Carreau fluid [20-223. The methods of approximation can be sorted into perturbation techniques, [10,14,19]; variational principles [1,3,5,21]; finite difference methods [6]; and an Oseen linearization technique [ll]. Most of these studies concern an unbounded fluid, but a few use cylindrical boundaries [22] and spherical boundaries [13]. In this paper, the main interest is on the effects of shearthinning behaviour. We draw special attention to Slattery’s [5] work on upper and lower bounds on the drag coefficient in a power-law fluid, which result from using variational principles for velocity trial functions and stress trial functions respectively. The upper bound is well known as an energy extremum principle, but the physical meaning of the lower bound is not very clear. Recently, Cho and Hartnett [9] reported an improved approach for drag bounds and give a physical explanation for the lower bound. Chhabra [23] recently gave an interesting discussion of Cho and Hartnett’s report and it is clear that no accurate predictions of drag, even for the power-law fluid case, are easily available. Our results reported below are found from numerical solutions of the full equations of motion and continuity by a finite-element method. In our numerical simulation we used a ball-in-sphere configuration with different radius ratios, and used extrapolation to find the finite-boundary correction; the range of power-law index (0.1~ n < 1) was covered. We also used the ball-in-tube configuration for discussion of the wall effect.
3 2. Numerical Simulation The finite-element program AXFINR developed by Nickell, Tanner and Caswell [24] for solving incompressible, viscous flow problems is used to simulate the creeping motion of a non-rotating solid sphere moving through a power-law fluid. A finite-bounded spherical geometry is employed for the outer fluid surface of the flow field. Fig. 1 shows the finite-element mesh used in our main computations. The reference frame for writing the governing equations is a polar coordinate system with the centre of the sphere instantaneously at its origin as shown in Fig. 1. For low Reynolds number flow of an incompressible fluid the motion is governed by the conservation of mass equation v-v=o,
(6)
and the conservation equation of linear momentum. In the case when inertial and body forces can be neglected we have v-r-vp=o,
(‘4
where r is the deviatoric stress tensor and p is the pressure. The rheological equation-of-state of the fluid under consideration, which describes the constitutive relation between the tensors of stress and the rate of strain, needs to be specified. In this research, we study the power-law model; this
Fig. 1. Typical finite-element mesh and coordinate system. Ratio of outer fluid-sphere radius R to solid-sphere radius R, equals to 20, i.e. s = R/R, = 20.
4 fluid model can be expressed,
in simple shearing,
as
7 = klY/“,
(8)
where 7 is the shear stress and t is the shear rate; n and k are constant. For a three-dimensional flow we have the proper generalization given by Slattery
[51* The present on the surface assumed
problem is specified by the following boundary of the solid sphere a no-slip velocity boundary
v=iUonr=R,,
conditions: condition is
(9)
where i is a unit vector in the x,-direction and U is a constant speed. On the outer surface of the fluid boundary the normal velocity is zero: VR= 0
(forr=R).
(10)
A slip boundary condition must be satisfied too, meaning there is no tangential traction on the outer spherical fluid surface. This condition minimizes the influence of the outer boundary on the motion of the sphere. The “wall effect” in the present numerical simulations are more important for calculations using a power-law index close to one, the Newtonian case. The data are corrected for this effect by extrapolation with different ratios of R/R,. In the Newtonian case it is not difficult to show, for a sphere of radius R, moving within a sphere of radius R and with a tangential slip boundary condition on the outer sphere, that the drag is increased from the Stokes unbounded D,, to the value
We can use this result to assess the accuracy of our calculations. Two quantities U and R, are used to normalize the parameters appearing in the governing equations. The system of equations (6) (7) and the constitutive equation are solved numerically by the program AXFINR. The normalized drag force D is calculated from the resulting solution of the stress field by numerical integration. The computations span the range of power-law index (n) 0.1-1.0. The error of the numerical approach depends on the grid used and on the number of iterations. Generally speaking, for lower values of the index n more iterations are needed, but each value of n has its own optimum number of iterations (i.e. the lowest number of iterations after which no significant further improvement in the solution occurred), and we use the optimum results. Taking the solutions of the velocity field as an example, the errors (for R/R,, = 20) are listed in Table 1. In this Table we show the values of 100 [z.,,,(Av,)2/z.,,,,v,211’2 where AVn is the change in velocity V, at
5
node n from the velocity at the previous iteration. It is believed these approximations show satisfactory convergence. We are most interested in the unbounded flow around a sphere, but we cannot avoid wall effects. For the Newtonian case, taking wall effects into consideration and comparing with the Fax&-type correction [25], these effects decrease with the increase of R/R, (= s). For s = 20 the error is less than 8%; for s = 50 the error is 3%; for s = 80 the error is less than 2%. Extrapolating to the unbounded case we find a result very close to Stokes’ law. Note that we are not using the non-slip boundary conditions at the outer radius and hence the correction for finite R/R, is less than it would be if the no-slip boundary condition were applied. We estimate the expected drag in the Newtonian case to be about (1 - 1.5 RJR)-‘D,, from eqn. (11). For the case R,/R = 0.05, we found D = 1.0954,, compared with the exact value of l.OSlDs,. thus the magnitude of the error in the drag is about 1% in this case. Generally, if the radial distance between the innermost circles of elements (Fig. 1) was h, we find the calculated drag, Dcalc, to be related to the true drag D for different radial mesh spacings by Dca,c= D - bh’,
(12)
where b is an unknown factor depending on n and R/R,. From this estimate, using h = 0.1, 0.2, and 1, we again find the expected errors in the calculated drag to be of order l-2%, being least accurate for n = 0.1 and most accurate for n = 1. In addition to these convergence and grid-size errors, we have to estimate the wall-effect. Since we are dealing with large R/R, ratios, we assume that the wall-effect is given by the Fax&i-type form [ 111 (ignoring ( R,/R)5 and similar terms), D = D,,(l
- a(n)R,/R)-‘,
(13)
where the coefficient (Yis a function of n. Fig. 2 shows two examples, n = 1 and n = 0.5, which suggests that D,,/D is indeed a linear function of R,/R in the range of values considered (R/R, 2 20). In each case we used a formula like eqn. (13) to extrapolate back to the unbounded sphere drag. It TABLE 1 The estimated errors of the solution of the velocity field Power-law index n
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
Iterations
4
4
4
7
8
9
6
9
15
Error s&i (less than)
0.02
0.2
0.5
0.05
0.1
0.2
2
2
1
0.1
6
POWER
- LAW -
.
0.9
- 0.6
-
OD
I
I
I
0.01
0.02
0.03
1 0.04
0.05
Ro/R
Fig. 2. Extrapolations for wall effects of Newtonian using eqn. (13). The ordinate here is Ds,/D.
fluid and power-law
fluid (n = 0.50)
(a)
4
-L
12
20
Cb)
Fig. 3. Numerical simulation of a uniform creeping flow passing finite-element mesh and coordinates, and (b) streamlines.
around
a fixed sphere,
(a)
7 TABLE 2 Wall-effect
factor cu(n) for a sphere moving axially in a long tube
n
a
1.0 0.8 0.65 0.5 0.0
2.104 (exact) 0.98 0.34 0.05 0.00
formula like eqn. (13) to extrapolate back to the unbounded sphere drag. It is apparent from Fig. 2 that the more shear-thinning is the fluid, the less important is the wall-effect, as has been noted experimentally [25]. Fig. 2 is an example of wall-effect using the sphere-in-sphere geometry. In practical experiments we need to know the wall-effect factors 1y in a sphere-in-tube geometry (Fig. 3). These factors are shown in Table 2. These values were found by taking the computed results (corrected for mesh size) and fitting the form (13) to them. Fig. 4 shows the curve constructed from these values. One sees that the
a
0
0.2
0.4
Fig. 4. Wall-effect
0.6
0.6
1.0
n
factor a for sphere-in-tube
geometry
using eqn. (13) and Table results (A).
8
(a)
lI
-20
-12
4
-4. z
(C)
I2
M
9
I
-20
I
I
-12
I
Z
-20
I
-4
I
I
I
I
12
2u
4
12
20
4
12
20
Cd)
-4
-12
I
4
(e)
-20
-4
-12
Z
(f)
Fig. 5. Velocity and streamline details for various n-values. Radial velocity contours: (a) Newtonian; (b) power-law n = 0.5. Axial velocity contours: (c) Newtonian; (d) power-law n = 0.5. Streamlines: (e) Newtonian; (f) Power-law n = 0.5. In the streamline plots the outermost and innermost contours have the values 0.097 and 3.78 (n = 1) and 0.019 and 0.757 (n = 0.5). The other contours are linearly interpolated between these values.
10 wall-effect is negligible for lack of fluid movement far One may note in passing effects for n > 1, one would the large wall effects.
n ,< 0.5. The velocity fields (Fig. 5) confirm the away from the sphere as n decreases below 0.7. that if an attempt were made to compute wall expect great difficulty in extrapolation owing to
3. Results and discussion The flow field around a sphere is not viscometric, but we found that the Newtonian fluid and power-law fluids have a similar flow pattern. Fig. 5 shows the flow patterns of various fluids of different power-law index when the sphere moves slowly through them. We can find the character of this flow field outlined above from these plots. From Figs. 3 and 5 we can see that the shear-thinning behaviour reduces the long-range disturbance caused by the moving sphere; the region of “circular” flow gradually closes on the sphere as the power-law index decreases. Sigli and Coutanceau [13] have reported some flow visualization results which confirm our numerical simulation (see Fig. 2 in reference [13]). We assume the shear-thinning behaviour of the falling-sphere viscosity obeys the power-law eqn. (8). Table 3 lists our numerical calculation of the values of the drag correction factor X, (eqn. 1). The (upper-bound) results of Wasserman and Slattery [5] and of Cho and Hartnett [9] are also listed in Table 3. These results and their lower bounds are plotted in Fig. 6. We also plot in Fig. 6 some available experimental data reported by Cho and Hartnett [26].
TABLE 3 The values of the drag correction factor X for unbounded flow n
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
X Slattery’s upper bound
Cho & Hartnett’s upper bound
present results
1.005 1.140 1.260 1.366 1.445 1.499 1.549 1.617 1.704 -
1.0060 1.1402 1.2594 1.3604 1.4409 1.4992 1.5335 1.5410 1.5175 1.4581
1.002 1.140 1.240 1.320 1.382 1.420 1.442 1.458 1.413 1.354
11 It is clear that the present results are between the upper bound and lower bound of both Slattery and Cho et al., and are much closer to the experimental data than any bound, giving a greatly improved solution to the problem; the accuracy of our results in Table 3 is estimated to be - 1% (n = 1) down to - 2% (n = 0.1). Bush has calculated the drags by a boundary-element method [27] and has confirmed the first four entries of Table 3 at the 1% accuracy level; the result for n = 0.6 did not converge well with the boundary-element technique. Except for a few research workers [8] who reported XC 1, the present work and most previous researches show X > 1. Acharya et al. [28] give a quantitative explanation of this effect and attempt to distinguish between the effects of shear-thinning viscosity and elasticity. Our results are consistent with experimental data for some liquids, at least qualitatively (Fig. 6). The
X
”
Fig. 6. Comparison of the drag corretion factor X(n) of the present results with the upper present results; and lower bounds of [23,26] and some available experimental data. -, ______ , upper and lower bounds of Wasserman and Slattery [5]; -. - ., upper and lower bounds of Cho and Hartnett [26]; 0, Carbopol 960; P, CMC; 0, Polyacrylamide AP-273.
12 effect of elasticity is small for Weissenberg numbers Wi (AU/R, = Wi, where X is the fluid relaxation time) for Wi c 1. This has been shown for an upper-convected Maxwell fluid in several papers [29,30] and is confirmed by our own calculations for a Maxwell fluid; changes of drag of order of 1% are observed at Wi = 1 for the Maxwell fluid. Hence the change of viscosity is often dominant. One must note the inability of power-law models to model real fluids in complex flows. Additionally, it is a fact that for most polymer fluids the nonlinear viscosity and elasticity are coupled to each other, and it is difficult to get a purely viscous shear-thinning fluid or a purely elastic fluid without a shear thinning effect, so it is difficult to distinguish these two effects experimentally. It is however believed that the shear-thinning behaviour is usually the more important effect. References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Y. Tomita, Bull. JSME, 2 (1959) 469. A. Faraoui and R.C. Kintner, Trans. Sot. Rheol., 5 (1961) 369. J.C. Slattery, AIChE J., 8 (1962) 663. G.C. Wallick, J.G. Savins and D.R. Artorburn, Phys. Fluids, 5 (1962) 367. M.L. Wasserman and J.C. Slattety, AIChE J., 10 (1964) 383. K. Ada&i, N. Yoshioka and K. Yamamoto, Chem. Eng. Sci., 28 (1973) 2033. P.H.T. Uhlherr, T.N. Le and Tiu, Can. J. Chem. Eng., 54 (1976) 497. K. Ada&i, N. Yoshioka and K. Sakai, J. NonONewtonian Fluid Mech., 3 (1977/78) 107. Y.I. Cho and J.P. Hartnett, J. Non-Newtonian Fluid Mech. 12 (1983) 243. F.M. Leslie, Quart. J. Mech. and Appl. Math., 14 (1961) 36. J.S. Ultman and M.M. Denn, Chem. Eng. J., 2 (1971) 81. R.Y.S. Lai, Int. J. Eng. Sci., 12 (1974) 645. D. Sigli and M. Coutanceau, J. Non-Newtonian Fluid Mech., 2 (1977) 1. B. Caswell and W.H. Schwartz, J. Fluid Mech., 13 (1962) 417. H. Giesekus, Rheol. Acta, 3 (1963) 59. B. Mena and B. Caswell, Chem. Eng. J., 8 (1974) 125. J.M. Broadbent and B. Mena, Chem. Eng. J., 8 (1974) 11. B. Caswell, Chem. Eng. Sci., 25 (1970) 1167. P. Brunn, Rheol. Acta, 15 (1976) 15. R.P. Chhabra and P.H.T. Uhlherr, Rheol. Acta, 19 (1980) 187. R.P. Chhabra, P.H.T. Uhlherr and D.V. Boger, J. Non-Newtonian Fluid Mech., 6 (1980) 187. R.P. Chhabra, C. Tiu and P.H.T. Uhlherr, Can. J. Chem. Eng. 59 (1981) 771. R.P. Chhabra, J. Non-Newtonian Fluid Mech., 13 (1983) 225. R.E. Nickell, RI. Tanner, B. Caswell, J. Fluid Mech., 65 (1974) 189. RI. Tanner, Chem. Eng. Sci. 19 (1964) 349. Y.I. Cho and J.P. Hartnett, J. Non-Newtonian Fluid Mech., 13 (1983) 229. M.B. Bush & R.I. Tanner, Int. J. Num. Meth. Fluids, 3 (1983) 71. A. Acharya, R.A. Mashelkar and J. Ulbrecht, Rheol. Acta, 15 (1976) 454. G. Tiefenbruck and L.G. LeaI, J. Non-Newtonian Fluid Mech., 10 (1982) 115. 0. Hassager and C. Bisgaard, J. Non-Newtonian Fluid Mech., 12 (1983) 153.