**4. Numerical results and discussion**

The steady velocity components *u* and *v* are plotted against independent variable *z* for different values of Ekman number *E*, Hartmann number *H* and material parameter and results are compared for two types of fluids: the Newtonian fluid, for which 0 ( 1,2,3) *<sup>i</sup> i* , and the non-Newtonian fluid, in which we choose 1. Fig. 1 shows the effect of Hartmann number *H* on the velocity components *u* and *v*. We fixed *E* 1 and varied H= 0,1,3,5. It is observed that an increase in the Hartmann number reduces the velocity components *u* and *v* due to the effects of the magnetic force against the flow direction. Figs. 1a and 1b show that with an increase of Hartmann number *H* , the curvature of the velocity component *u* profile increases for both a Newtonian fluid and non-Newtonian fluid. Quite contrary, increasing Hartmann number *H* causes the velocity component *v* profile to become less parabolic, see Figs. 1c and 1d. It is also noted that decrease in *u* and *v* in the Newtonian fluid is larger as compared with non-Newtonian fluid. Furthermore, the boundary layer thickness is drastically decreased by increasing . *H* It means that the magnetic field provides some mechanism to control the boundary layer thickness.

The dependence of the velocity components *u* and *v* on the Ekman number is shown in Fig. 2. In Fig. 2 we fixed *H* 1 and varied *E* 0.1, 0.2, 0.3. It is observed that velocity component *u* increases with an increase in Ekman number *E* for the Newtonian fluid while it remains almost unaffected for non-Newtonian fluid (Figs. 2a, 2b). Moreover, a 202 Topics in Magnetohydrodynamics

2 ( ) 2() ( ) ( ) ( ). *uz h uz uz h u z h h*

This converts the given boundary value problem into a linear system of equations involving values of the function *u* at *aa ha h* , , 2 , ,b. For higher accuracy, one should choose *h* small. However, this increases the number of equations in the system which in turn

Depending upon the size of this resulting system of linear equations, it can either be solved

In the present problem the governing differential equations (36) and (37) are highly nonlinear which cannot be solved analytically. These equations are discretized using second order central finite difference approximations defined in Eqs. (41) and (42). The resulting system of algebraic equations is solved using successive under relaxation scheme. The difference equations are linearized employing a procedure known as lagging the coefficients [59]. The iterative procedure is repeated until convergence is obtained according to the

( 1) ( ) max , *n n u u*

The steady velocity components *u* and *v* are plotted against independent variable *z* for

results are compared for two types of fluids: the Newtonian fluid, for which 0 ( 1,2,3) *<sup>i</sup>*

number *H* on the velocity components *u* and *v*. We fixed *E* 1 and varied H= 0,1,3,5. It is observed that an increase in the Hartmann number reduces the velocity components *u* and *v* due to the effects of the magnetic force against the flow direction. Figs. 1a and 1b show that with an increase of Hartmann number *H* , the curvature of the velocity component *u* profile increases for both a Newtonian fluid and non-Newtonian fluid. Quite contrary, increasing Hartmann number *H* causes the velocity component *v* profile to become less parabolic, see Figs. 1c and 1d. It is also noted that decrease in *u* and *v* in the Newtonian fluid is larger as compared with non-Newtonian fluid. Furthermore, the boundary layer thickness is drastically decreased by increasing . *H* It means that the magnetic field provides some mechanism to

The dependence of the velocity components *u* and *v* on the Ekman number is shown in Fig. 2. In Fig. 2 we fixed *H* 1 and varied *E* 0.1, 0.2, 0.3. It is observed that velocity component *u* increases with an increase in Ekman number *E* for the Newtonian fluid while it remains almost unaffected for non-Newtonian fluid (Figs. 2a, 2b). Moreover, a

different values of Ekman number *E*, Hartmann number *H* and material parameter

where superscript ' ' *n* represents the number of iteration and ' '

is taken as <sup>8</sup> 10 .

is the order of accuracy. In

1. Fig. 1 shows the effect of Hartmann

and

*i* ,

increases the computational time.

following criterion

the present case

control the boundary layer thickness.

**4. Numerical results and discussion** 

and the non-Newtonian fluid, in which we choose

by exact methods or approximate methods.

2

(42)

Fig. 1. Variation of velocity components *u* and *v* with z for 0,1,3,5; *H* 1, 0 *E* in (a), (c); *E* 1, 1 in (b), (d).

backflow is observed near the boundary 0 *z* for 0.1. *E* On the contrary, the magnitude of velocity component *v* decreases with an increase in Ekman number *E* for the both types of the fluids (Figs. 2c, 2d). This velocity component has larger magnitude in Newtonian fluid as compared with non-Newtonian fluid.

0

0


c d

**v**

0 0.5 1 **z**

0 0.5 1

H = 5 E = 1

Effect of material parameter

**z**

fixing 1, 5 *E H* .

β = 1 β = 3 β = 5

0.2

0.4

0.6

**u**

a b

0.8

1

β = 1 β = 3 β = 5

Effect of material parameter

H = 5 E = 1

1.2

0


**v**

0 0.5 1 **z**

0 0.5 1

Effect of material parameter

**z**

H = 1 E = 1

Fig. 3. Variation of velocity components *u* and *v* with *z* for 1, 3, 5;

β = 1 β = 3 β = 5

β = 1 β = 3 β = 5

Effect of material parameter

H = 1 E = 1

0.2

0.4

0.6

**u**

0.8

1

1.2

Fig. 2. Variation of velocity components *u* and *v* with *z* for 0.1, 0.2, 0.3; *E* 1, 0 *H* in (a), (c); 1, 1 *H* in (b), (d).

Figs. 3,4 depict the variation of the velocity components *u* and *v* with *z* for various values of material parameter fixing *E* 1 and taking *H* 1 in Figs. 3a and 3c, while 5 *H* in 3b, 3d, and in Fig. 4. It is observed from Fig. 3b and 4a that when the material parameter increases from 1 to a large value of 20, the velocity component *u* tend to approach the

204 Topics in Magnetohydrodynamics



fixing *E* 1 and taking *H* 1 in Figs. 3a and 3c, while 5 *H* in 3b,

to a large value of 20, the velocity component *u* tend to approach the

(c) (d) Fig. 2. Variation of velocity components *u* and *v* with *z* for 0.1, 0.2, 0.3; *E* 1, 0 *H*

Figs. 3,4 depict the variation of the velocity components *u* and *v* with *z* for various values

3d, and in Fig. 4. It is observed from Fig. 3b and 4a that when the material parameter




**v**



0

0.05

(a) (b)

0

0 0.5 1

E = 0.1 E = 0.2 E = 0.3

non-Newtonian fluid

H = 1 β = 1

**z**

0 0.5 1

non-Newtonian fluid

**z**

H = 1 β = 1

E = 0.1 E = 0.2 E = 0.3

0.2

0.4

0.6

**u**

0.8

1

1.2



in (a), (c); 1, 1 *H*

of material parameter

increases from 1





**v**



0

0

0 0.5 1

E = 0.1 E = 0.2 E = 0.3

Newtonian fluid

H = 1

**z**

0 0.5 1

Newtonian fluid

**z**

in (b), (d).

H = 1

E = 0.1 E = 0.2 E = 0.3

0.2

0.4

0.6

**u**

0.8

1

1.2

Fig. 3. Variation of velocity components *u* and *v* with *z* for 1, 3, 5; fixing 1, 5 *E H* .

[1] R.S. Rivlin, J.L. Ericksen, Stress deformation relations for isotropic material, J. Rational

[2] C. Truesdell and W. Noll, The Nonlinear Field Theories of Mechanics*,* 2nd ed. Springer-

[3] J. G. Oldroyd, On the formulation of rheological equations of state, Proc. Roy. Soc. Lond.

[4] K.R. Rajagopal, Mechanics of non-Newtonian fluids in Recent Developments in

[5] J.E. Dunn, K.R. Rajagopal, Fluids of differential type: critical review and thermodynamic

[6] R.R. Huilgol, Continuum mechanics of viscoelastic liquids, Hindushan Publishing

[8] M.A. Rana, A.M. Siddiqui, Rashid Qamar, Hall effects on hydromagnetic flow of an

[9] R. Bandelli, K. R. Rajagopal and G.P. Galdi, On some unsteady motions of fluids of

[10] R.L. Fosdick, K.R. Rajagopal, Anomalous features in the model of "second order

[11] R.L. Fosdick, K.R. Rajagopal, Thermaodynamics and stability of fluids of third grade,

[12] J. Malek, K.R. Rajagopal, and M. Ruzicka, Existence and regularity of solutions and

[13] Hartmann Hydrodynamics. 1. theory of the lamina flow of an electrically conductive

[14] G.W. Sutton and A. Shermann, Engineering magneto-hydrodynamics, McGraw-Hill

[15] W.F. Hughes and Y.J. Young, The electromagnetinodynamics of fluids, John Wiley,

[16] T.G. Cowling, Magnetohydrodynamics, Interscience Publishing, Inc., New York, 1957.

[18] K. R. Rajagopal, T.Y. Na, On Stokes problem for non-Newtonian fluid. Acta Mech. 48

[19] F. Mollica, K.R. Rajagopal, Secondary flows due to axial shearing of a third grade fluid between two eccentrically placed cylinders. Int. J. Engng. Sci. 37 (1999) 411 - 429. [20] A.M. Siddiqui, P.N. Kaloni, Plane steady flows of a third grade fluid. Int. J. Engng. Sci.

[17] S.I. Pai, Magneto-gas dynamics and plasma dynamics, Springer-Verlag, 1962.

stability of the rest state for the fluids with shear dependent viscosity, Math.

liquid in a homogeneous magnetic field. Kgi Danske Videnslab, Sleskuh, Mat. Fys.

Theoretical Fluid Mechanics, Pitman Res. Notes Math. Ser. 291, eds. G.P. Galdi and

Oldroyd 6-constant fluid between concentric cylinders, Chaos, Solitons and Fractals

It is noted that the flow behaviour depends strongly on the choice of the parameters.

**6. References** 

Mech. Anal. 4 (1955) 323-425.

Verlog, Brelin, 1992.

A 200 (1950) 523-541.

Corporation, 1975.

39 (2009) 204–213.

Medd. 15 (1937).

New York, 1966.

(1983) 233 -239.

25 (2) (1987) 171 -188.

Co. Inc. 1965.

J. Necas (Springer), 1993.

analysis, Int. J. Engng. Sci. 33 (1995) 689-729.

second grade. Arch. Mech., 47 (1995) 661 -676.

fluids", Arch. Ration. Mech. Anal. 70 (1979) 145-152.

Models Methods Appl. Sci. 5 (1995), no. 6, 789-812.

Proc. Roy. Soc. London Ser. A 369 (1980), no. 1738, 351-377.

[7] W.R. Schowalter, Mechanics of non-Newtonian fluids, Pergamon, 1978.

linear distribution; thus, the shearing can unattenuately extend to the whole flow domain from the boundaries, corresponding to a shear-thickening phenomenon. A further increase of will not effect this velocity component further. The magnitude of velocity component *v* decreases when increases and the curvature of the velocity profile decreases with an increase in material parameter (see Figs. 3c, 3d and 4b). It is also found that the flow behaviour depends strongly on the choice of the parameters, for example, for large *H* , *u* increases with an increase of material parameter , whereas this velocity component is independent of for small . *H* On the contrary, the magnitude of velocity component *v* decreases with for both small and large values of . *H*

Fig. 4. Variation of velocity components *u* and *v* with *z* for large values of 1,5, 10, 15,20; fixing 1, 5 *E H* .
