## Problems

The level of difficulty of problems has been graded: (a): straightforward, (b): somewhat more complex, and (c): most difficult. In any given Chapter the readers are recommended to tackle problems in increasing order of difficulty.

LEVEL

1.1 The following rheological data have been obtained for a liquid at (a) 295.5 K.

Shear rate |
Shear stress |
Shear rate |
Shear stress |

(s-1) |
(Pa) |
(s-1) |
(Pa) |

2.22 |
1.32 |
35.16 |
20.92 |

4.43 |
2.63 |
44.26 |
26.33 |

7.02 |
4.17 |
70.15 |
41.74 |

8.83 |
5.25 |
88.31 |
52.54 |

11.12 |
6.62 |
111.17 |
66.15 |

14 |
8.33 |
139.96 |
83.27 |

17.62 |
10.48 |
176.2 |
104.84 |

22.18 |
13.20 |
221.82 |
132.0 |

27.93 |
16.62 |
279.25 |
166.15 |

By plotting these data on linear and logarithmic scales, ascertain the type of fluid behaviour, e.g. Newtonian, or shear-thinning, or shear-thickening, etc. Also, if the liquid is taken to have power-law rheology, calculate the consistency and flow-behaviour indices respectively for this liquid.

1.2 The following rheological data have been reported for a 0.6% (by (a) weight) carbopol solution in a 1.5% (by weight) NaOH aqueous solution at 292 K.

P (s-1) |
t (Pa) |
P(s-1) |
t (Pa) |

0.356 |
1.43 |
7.12 |
4.86 |

0.449 |
1.54 |
8.96 |
5.40 |

0.564 |
1.65 |
11.25 |
6.03 |

0.712 |
1.79 |
14.17 |
6.62 |

0.896 |
1.97 |
17.82 |
7.45 |

1.13 |
2.17 |
22.47 |
8.40 |

P (s-1 ) |
t (Pa) |
P(s-1) |
t (Pa) |

1.42 |
2.44 |
28.30 |
9.46 |

1.78 |
2.61 |
35.57 |
10.41 |

2.25 |
2.84 |
44.89 |
12.06 |

2.83 |
3.27 |
56.43 |
13.60 |

3.56 |
3.60 |
71.15 |
15.14 |

4.49 |
3.98 |
89.55 |
17.03 |

5.64 |
4.38 |
112.5 |
19.16 |

Plot these data in the form of r — p and y — p on logarithmic coordinates. Evaluate the power-law parameters for this fluid. Does the use of the Ellis fluid (equation 1.15) or of the truncated Carreau fluid (equation 1.14) model offer any improvement over the power-law model in representing these data? What are the mean and maximum % deviations from the data for these three models?

1.3 The following data for shear stress (r) and first normal stress difference (a) (Ni) have been reported for a 2% (by weight) Separan AP-30 solution in water measured at 289.5 K using a cone and plate rheometer.

P (s-1 ) |
t (Pa) |
N (Pa) |
P (s-1) |
T (Pa) |
N (Pa) |

0.004 49 |
0.26 |
- |
4.49 |
19.85 |
57.8 |

0.005 64 |
0.33 |
- |
7.12 |
22.96 |
72.3 |

0.00712 |
0.42 |
- |
11.25 |
26.13 |
90.6 |

0.008 96 |
0.53 |
- |
17.83 |
29.93 |
125.30 |

0.0113 |
0.66 |
- |
28.30 |
34.44 |
154.20 |

0.0142 |
0.75 |
- |
44.9 |
40.38 |
221.70 |

0.0178 |
0.96 |
- |
71.2 |
46.32 |
318.60 |

0.0225 |
1.14 |
- |
112.5 |
53.44 |
424.10 |

0.0356 |
1.65 |
- |
89.6 |
49.88 |
366.30 |

0.0283 |
1.39 |
- |
56.4 |
43.00 |
269.90 |

0.0449 |
1.99 |
- |
35.6 |
37.5 |
192.8 |

0.0564 |
2.30 |
- |
22.5 |
32.1 |
163.8 |

0.0712 |
2.85 |
- |
14.2 |
28 |
120.5 |

0.0896 |
3.33 |
- |
8.96 |
25.2 |
96.4 |

0.113 |
3.83 |
4.82 |
5.64 |
21.1 |
84.8 |

0.178 |
5.50 |
7.23 |
3.56 |
18.2 |
62.7 |

0.283 |
6.94 |
11.60 |
2.25 |
15.6 |
43.4 |

0.449 |
8.37 |
19.80 |
1.42 |
13.3 |
38.6 |

0.712 |
10.16 |
22.20 |
0.896 |
11 |
25.1 |

1.13 |
12.32 |
27.95 |
0.564 |
9.09 |
20.3 |

1.78 |
14.60 |
34.70 |
0.356 |
7.41 |
10.6 |

(i) Plot these shear stress, apparent viscosity and first normal stress difference data against shear rate on log-log scales. Does the shear stress data extend to the lower Newtonian region? What is the value of the zero-shear viscosity for this solution?

(ii) Suggest and fit suitable viscosity models covering the entire range. What is their maximum deviation?

(iii) Calculate and plot the Maxwellian relaxation time (equation 1.25) as a function of shear rate for this polymer solution.

(iv) Is it possible to characterize the visco-elastic behaviour of this solution using a single characteristic time (using equation 1.26) in the higher shear rate region? At what shear rate does it coincide with the Maxwellian relaxation time calculated in part (iii)?

1.4 The following rheological data for milk chocolate at 313 K are (a) available. Determine the Bingham plastic (equation 1.16) and Casson model (equation 1.18) parameters for this material. What are the mean and maximum deviations for both these models?

Y (s-1) |
t (Pa) |
Y (s-1) |
t (Pa) |

0.099 |
28.6 |
6.4 |
123.8 |

0.14 |
35.7 |
7.9 |
133.3 |

0.20 |
42.8 |
11.5 |
164.2 |

0.39 |
52.4 |
13.1 |
178.5 |

0.79 |
61.9 |
15.9 |
201.1 |

1.60 |
71.4 |
17.9 |
221.3 |

2.40 |
80.9 |
19.9 |
236 |

3.9 |
100 |

It is likely that the model parameters are strongly dependent on the shear rate range covered by the rheological data. Compare the values of the model parameters by considering the following shear rate intervals:

1.5 The following shear stress-shear rate data demonstrate the effect of (b) temperature on the power-law constants for a c ncentrated orange juice containing 5.7% fruit pulp.

T = |
254.3 K |
267.7 K |
282.6 K |
302.3 K | |||

P (s-1) |
t (Pa) |
P (s-1) |
T (Pa) |
P (s-1) |
t (Pa) |
P (s-1) |
t (Pa) |

0.5 |
14.4 |
0.6 |
4.3 |
1.1 |
2.6 |
8 |
3.6 |

1 |
24.3 |
1 |
6.5 |
8 |
10.3 |
20 |
7.6 |

10 |
142 |
10 |
38.4 |
15 |
17 |
40 |
13.1 |

20 |
240.4 |
20 |
65.4 |
30 |
29.5 |
60 |
17.5 |

30 |
327 |
30 |
89 |
60 |
50.3 |
120 |
31.2 |

T D |
= 254.3 K |
267.7 K |
282.6 K |
302.3 K | |||

Y (s" |
1 ) r (Pa) |
Y (s"1 ) |
r (Pa) |
Y (s"1) |
r (Pa) |
Y (s"1) |
r (Pa) |

40 |
408 |
40 |
111 |
90 |
69 |
240 |
54.5 |

50 |
484 |
50 |
132 |
150 |
103 |
480 |
94.4 |

60 |
556 |
60 |
152 |
250 |
154 |
800 |
142 |

70 |
635 |
70 |
171.3 |
350 |
200 |
1000 |
170 |

80 |
693 |
80 |
189.4 |
450 |
243 |
1100 |
183 |

150 |
1120 |
150 |
309 | ||||

300 |
527 |

How do the values of the power-law flow behaviour and consistency indices depend upon temperature? Estimate the activation energy of viscous flow (E) from these data by fitting them to the equation m = m0exp(E/rT) where m0 and E are constants and r is the universal gas constant.

1.6 The following shear stress-shear rate data are available for an aqueous (b) carbopol solution at 293 K.

Y (s"1 ) |
r (Pa) |
Y(s"1) |
r (Pa) |

0.171 |
53.14 |
1.382 |
78.18 |

0.316 |
57.86 |
1.92 |
84.37 |

0.421 |
61.59 |
2.63 |
90.23 |

0.603 |
66 |
3.67 |
98.26 |

0.812 |
70 |
5.07 |
106.76 |

1.124 |
75.47 |

By plotting these data on linear and logarithmic scales, ascertain the type of fluid behaviour exhibited by this solution. Suggest a suitable viscosity model and evaluate the parameters for this solution. Does the fluid appear to have a yield stress? Using the vane method (Q.D. Nguyen and D.V. Boger, Annu. Rev. Fluid Mech., 24 (1992) 47), the yield stress was found to be 46.5 Pa. How does this value compare with that obtained by the extrapolation of data to p = 0, and that obtained by fitting Bingham, Casson and Herschel-Bulkley fluid models?

1.7 The following rheological data have been reported for a 100 ppm (b) polyacrylamide solution in 96% (by weight) aqueous wheat syrup solution at 294 K.

Y (s"1 ) |
r (Pa) N1 (Pa) |
Y (s"1 ) |
r (Pa) |
N (Pa) |

0.025 |
0.70 - |
0.315 |
8.79 |
18.1 |

0.0315 |
0.89 - |
0.396 |
11.01 |
27.7 |

0.0396 |
1.12 - |
0.500 |
13.87 |
42.0 |

P (s-1) |
t (Pa) |
N (Pa) |
P(s-1) |
T (Pa) |
N (Pa) |

0.050 |
1.42 |
0.0892 |
0.628 |
17.46 |
65.4 |

0.0628 |
1.78 |
0.158 |
0.790 |
21.80 |
97.5 |

0.0791 |
2.25 |
0.890 |
0.995 |
27.16 |
141.0 |

0.0995 |
2.83 |
1.26 |
1.25 |
33.75 |
196.0 |

0.125 |
3.56 |
2.47 |
1.58 |
42.34 |
283.0 |

0.158 |
4.49 |
3.64 |
1.99 |
53.53 |
440.0 |

0.199 |
5.65 |
6.71 |
2.50 |
67.03 |
661.0 |

0.250 |
7.08 |
11.60 |
3.15 |
84.11 |
863.0 |

(i) Is this solution shear-thinning?

(ii) Can this solution be treated as a Newtonian fluid? If not, why?

(iii) Is it a visco-elastic fluid? Estimate the value of the Maxwellian relaxation time for this solution.

(i) Is this solution shear-thinning?

(ii) Can this solution be treated as a Newtonian fluid? If not, why?

(iii) Is it a visco-elastic fluid? Estimate the value of the Maxwellian relaxation time for this solution.

880 kgm-3 |
905 kgm-3 |
937 kgm-3 |
965 kgm-3 |
995 kgm-3 | |||||

p |
T |
p |
T |
p |
T |
p |
T |
p |
T |

(s-1) |
(Pa) |
(s-1) |
(Pa) |
(s-1) |
(Pa) |
(s-1) |
(Pa) |
(s-1) |
(Pa) |

3.9 |
3.46 |
1.9 |
3.79 |
1.1 |
5.20 |
1.9 |
6.25 |
0.9 |
7.55 |

5.3 |
3.54 |
2.6 |
3.90 |
2.1 |
5.40 |
3.4 |
6.46 |
1.8 |
8.01 |

5.9 |
3.61 |
3.8 |
4.02 |
3.3 |
5.62 |
4.7 |
6.67 |
2.8 |
8.33 |

7.0 |
3.68 |
4.9 |
4.12 |
5.6 |
5.90 |
7.2 |
6.92 |
4.9 |
8.75 |

8.2 |
3.74 |
5.9 |
4.19 |
7.9 |
6.12 |
9.1 |
7.16 |
8.0 |
9.25 |

9.2 |
3.80 |
7.0 |
4.30 |
10.8 |
6.32 |
10.7 |
7.27 |
9.9 |
9.43 |

10.3 |
3.86 |
8.0 |
4.38 |
12.4 |
6.48 |
12.0 |
7.38 |
11.6 |
9.64 |

11.3 |
3.91 |
9.0 |
4.44 |
13.9 |
6.58 |
13.1 |
7.48 |
13.0 |
9.88 |

12.2 |
3.96 |
12.2 |
4.64 |
15.2 |
6.68 |
14.2 |
7.55 |
14.3 |
9.95 |

14.9 |
4.10 |
13.5 |
4.70 |
16.4 |
6.78 |
15.1 |
7.70 |
15.5 |
10.10 |

16.2 |
4.17 |
14.9 |
4.79 |
17.6 |
6.88 |
15.9 |
7.74 |
16.6 |
10.27 |

19.3 |
4.30 |
18.2 |
5.00 |
23.1 |
7.34 |
17.7 |
7.98 |
19.4 |
10.53 |

22.5 |
4.43 |
21.4 |
5.15 |
25.7 |
7.41 |
19.4 |
8.10 |
22.0 |
10.80 |

25.5 |
4.48 |
24.7 |
5.30 |
28.3 |
7.59 |
24.2 |
8.47 |
24.5 |
11.10 |

28.6 |
4.70 |
28.0 |
5.40 |
33.2 |
7.91 |
27.0 |
8.62 |
26.9 |
11.25 |

34.6 |
4.98 |
34.4 |
5.82 |
30.0 |
8.87 |
31.8 |
11.60 | ||

40.5 |
5.15 |
40.7 |
5.97 |
37.1 |
9.30 |
36.5 |
11.91 |

(i) Plot these data on linear and logarithmic scales and fit the Herschel-Bulkley viscosity model to represent the behaviour of these suspensions.

(ii) How do the model parameters depend upon the concentration?

(i) Plot these data on linear and logarithmic scales and fit the Herschel-Bulkley viscosity model to represent the behaviour of these suspensions.

(ii) How do the model parameters depend upon the concentration?

1.9 The following rheological data have been obtained for a 0.244% Poly- (b) isobutylene/92.78% Hyvis Polybutene/6.98% kerosene (by weight) solution at 293 K.

y (s-1) |
t (Pa) N1 (Pa) |
P (s-1) |
t (Pa) |
N (Pa) |

0.05 |
0.165 - |
0.792 |
2.55 |
- |

0.0629 |
0.202 - |
0.998 |
3.18 |
0.30 |

0.0792 |
0.260 - |
1.26 |
3.98 |
0.57 |

0.0998 |
0.330 - |
1.58 |
4.58 |
1.20 |

0.126 |
0.413 - |
1.99 |
6.24 |
2.21 |

0.158 |
0.518 - |
2.51 |
7.72 |
3.29 |

0.199 |
0.663 - |
3.15 |
9.61 |
5.09 |

0.251 |
0.823 - |
3.97 |
12 |
8.10 |

0.315 |
1.03 - |
5.00 |
15 |
12.10 |

0.397 |
1.29 - |
6.29 |
18.6 |
18.20 |

0.50 |
1.61 - |
7.92 |
23.2 |
28.00 |

0.629 |
2.03 - |
9.97 |
29.2 |
46.90 |

(i) Does this fluid exhibit Newtonian, or shear-thinning fluid behaviour?

(ii) Is the liquid visco-elastic? Show the variation of the Maxwellian relaxation time with shear rate.

2.1 The following volumetric flow rate - pressure gradient data have been (c) obtained using a capillary viscometer (D = 10 mm and L = 0.5 m) for a viscous material. Obtain the true shear stress-shear rate data for this substance and suggest a suitable viscosity fluid model.

Q (mm3/s) 1.3 15 140 1450 14500 1.5 x 105 1.4 x 106 —Ap (Pa) 0.5 1.5 5 15 50 160 500

2.2 A polymer solution was tested in a cone-and-plate viscometer (cone (b) angle 0.1 rad and cone radius 25 mm) at various rotational speeds. Use the following torque - speed data to infer shear stress-shear rate behaviour and suggest an appropriate fluid model to describe the fluid behaviour.

fl (rad/s) 10-4 10-3 0.01 0.1 1 10 100 T (Nm) 0.003 0.033 0.26 1 2.2 3.3 66

2.3 The following data has been obtained with a capillary viscometer for (c) an aqueous polymer solution, density 1000 kg/m3 at 293 K.

Capillary data |
Ap (kPa) |
mass flow rate (kg/s) |

L = 200 mm |
224.3 |
1.15 x 10-3 |

D = 2.11 mm |
431 |
2.07 x 10-3 |

596.3 |
2.75 x 10-3 | |

720.3 |
3.88 x 10-3 |

Capillary data |
Ap (kPa) |
mass flow rate (kg/s) |

L = 50 mm |
87.5 |
1.01 x 10-3 |

D = 2.11 mm |
148.1 |
1.71 x 10-3 |

361.7 |
4.29 x 10-3 | |

609.8 |
6.69 x 10-3 | |

L = 50.2 mm |
43.40 |
3.95 x 10-3 |

D = 4.14mm |
79.22 |
8.18 x 10-3 |

117.12 |
1.092 x 10-2 | |

160.53 |
1.48 x 10-2 |

3.1 A low molecular weight polymer melt, which can be modelled as a (a) power-law fluid with m = 5 kPa-s" and n = 0.25, is pumped through a 13 mm inside diameter tube over a distance of 10 m under laminar flow conditions. Another pipe is needed to pump the same material over a distance of 20 m at the same flow rate and with the same frictional pressure loss. Calculate the required diameter of the new pipe.

3.2 The flow behaviour of a tomato sauce follows the power-law model, (b) with n = 0.50 and m = 12Pa-sn. Calculate the pressure drop per metre length of pipe if it is pumped at the rate of 1000 kg/h through a 25 mm diameter pipe. The sauce has a density of 1130 kg/m3. For a pump efficiency of 50%, estimate the required power for a 50 m long pipe.

How will the pressure gradient change if

(a) the flow rate is increased by 50%,

(b) the flow behaviour consistency coefficient increases to 14.75 Pa-sn without altering the value of n, due to changes in the composition of the sauce,

(c) the pipe diameter is doubled,

(d) the pipe diameter is halved.

Is the flow still streamline in this pipe?

3.3 A vertical tube whose lower end is sealed by a movable plate is filled (b) with a viscoplastic material having a yield stress of 20 Pa and density

1100 kg/m3. Estimate the minimum tube diameter for this material to flow under its own weight when the plate is removed. Does the depth of the material in the tube have any influence on the initiation of flow?

3.4 A power-law fluid (m = 5Pa-sn and n = 0.5) of density 1200 kg/m3 (b) flows down an inclined plane at 30° to the horizontal. Calculate the volumetric flow rate per unit width if the fluid film is 6 mm thick. Assume laminar flow conditions.

3.5 A Bingham plastic material is flowing under streamline conditions (c) in a circular pipe. What are the conditions for one third of the total flow to be within the central plug region across which the velocity profile is flat? The shear stress acting within the fluid r varies with the velocity gradient dVx/dr according to the relation:

r = ro + MB (where rB and mB are respectively the Bingham yield stress and the plastic viscosity of the material.

3.6 Tomato puree of density 1100 kg/m3 is pumped through a 50 mm (c) diameter pipeline at a flow rate of 1 m3/hr. It is suggested that, in order to double production,

(a) a similar line with pump should be put in parallel with the existing one, or

(b) a larger pump should be used to force the material through the present line, or

(c) the cross-sectional area of the pipe should be doubled.

The flow behaviour of the tomato puree can be described by the Casson equation (1.18), i.e.,

where r£, the Casson yield stress, is equal to 20 Pa and ^c, the Casson plastic viscosity has a value of 5 Pa-s.

Evaluate the pressure gradient for the three cases. Also, evaluate the viscosity of a hypothetical Newtonian fluid for which the pressure gradient would be the same. Assume streamline flow under all conditions.

3.7 A polymer solution is to be pumped at a rate of 11 kg/min through (b) a 25 mm inside diameter pipe. The solution behaves as a power-law fluid with n = 0.5 and an apparent viscosity of 63 m Pa-s at a shear rate of 10s-1, and a density of 950 kg/m3.

(a) What is the pressure gradient in the pipe line?

(b) Estimate the shear rate and the apparent viscosity of the solution at the pipe wall?

(c) If the fluid were Newtonian, with a viscosity equal to the apparent viscosity at the wall as calculated in (b) above, what would be the pressure gradient?

(d) Calculate the Reynolds numbers for the polymer solution and for the hypothetical Newtonian fluid

3.8 A concentrated coal slurry (density 1043 kg/m3) is to be pumped through a 25 mm inside diameter pipe over a distance of 50 m. The flow characteristics of this slurry are not fully known, but the following preliminary information is available on its flow through a smaller tube, 4 mm in diameter and 1 m long. At a flow rate of 0.0018 m3/h, the pressure drop across the tube is 6.9 kPa, and at a flow rate of 0.018 m3/h it is 10.35 kPa. Evaluate the power-law constants from the data for the small diameter tube. Estimate the pressure drop in the 25 mm diameter pipe for a flow rate of 0.45 m3/h.

3.9 A straight vertical tube of diameter D and length L is attached to the bottom of a large cylindrical vessel of diameter DT(^D). Derive an expression for the time required for the liquid height in the large vessel to decrease from its initial value of H0(—L) to H(—L) as shown in the following sketch.

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