**3. Numerical procedure**

Consider a simplest boundary value problem

$$F(\mu'', \mu', \mu, z) = 0,\tag{39}$$

$$
\mu(a) = A \text{ and } \mu(b) = B. \tag{40}
$$

To solve the boundary value problem the derivatives *u* and *u* involved in the problem are approximated by finite differences of appropriate order. If we employ second order central difference formulation, then we can write

$$
\mu'(z) = \frac{\mu(z+h) - \mu(z-h)}{2h} + \mathcal{O}(h^2),
\tag{41}
$$

0


(c) (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

Fig. 1. Variation of velocity components *u* and *v* with z for 0,1,3,5; *H* 1, 0 *E*




**v**



0

0 0.5 1 **z**

0 0.5 1

non-Newtonian fluid

**z**

E = 1 β = 1

in (a),

H = 0 H = 1 H = 3 H = 5

H = 0 H = 1 H = 3 H = 5

non-Newtonian fluid

E = 1 β = 1

0.2

0.4

0.6

**u**

(a) (b)

0.8

1

1.2

0


(c); *E* 1, 1 




**v**



0

0 0.5 1 **z**

0 0.5 1

Newtonian fluid

**z**

in (b), (d).

as compared with non-Newtonian fluid.

E = 1

H = 0 H = 1 H = 3 H = 5

H = 0 H = 1 H = 3 H = 5

Newtonian fluid

E = 1

0.2

0.4

0.6

**u**

0.8

1

1.2

$$
\mu''(z) = \frac{\mu(z+h) - 2\mu(z) + \mu(z-h)}{h^2} + O(h^2). \tag{42}
$$

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 increases the computational time.

Depending upon the size of this resulting system of linear equations, it can either be solved by exact methods or approximate methods.

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 following criterion

$$\max \left| \mu^{(n+1)} - \mu^{(n)} \right| < \varepsilon\_n$$

where superscript ' ' *n* represents the number of iteration and ' ' is the order of accuracy. In the present case is taken as <sup>8</sup> 10 .
