24 S. IQBAL ET AL.

(22) for the fluid velocity with same values of

and P.

From these figures we observe that the agreement of the

numerical solution with the co rresponding exact solution

is very good. This agreement is further highlighted by

sufficiently small size of the absolute error shown

in Table 2 for u

E

=0.25

and . The

fluid velocity magnitude again appears to show an

increasing trend with increasing values of the pressure

gradient P. Thus we conclude that the fluid velocity

magnitude increases with increasing values of the non-

Newtonian parameter

0.80.4, 0.6,P

and the pressure gradient P,

when one of these is fixed.

Figure 3(a) illustrates variation of the numerical

solution for the fluid temperature obtained by the finite

element method with = 0.25,= 5

0.9 and various

values of . The corresponding exact

solution is presented in Figure 3(b) for same values of

the parameters

0.1,0.6 ,P

and P. From these figures we observe

that the numerical sol uti on for the fluid tem pe r ature e x hibits

good agreement with the corresponding exact solution.

This agreement of the numerical solution with exact

solution is further illustrated in Table 3. This table presents

variation of the absolute error =

EM Exact

E for

=0.25, =5

P and . It may be

observed from this table that the size of the error

0.1,0.6 ,0.9

E

is

small enough to illustrate accuracy of the numerical

solution. Moreover, we observe that the fluid temperature

increases with increasing values of P for fixed

and

are fixed.

Then in Figure 4(a) we present variation of the FEM

numerical solution for =0.25, =0.5P

and

1.0,3.0,5.0

6. Conclusions

One dimensional flow of a constant property pseudo

plastic fluid with heat transfer between two infinite

horizontal parallel plates is considered. Numerical

solutions for the dimensionless fluid velocity and the

fluid temperature are obtained by applying the finite

element method. The corresponding exact solutions are

also derived for the sake of comparison. It is found that

the numerical technique based on the finite element

method produces highly accurate solution both for the

fluid velocity and the fluid temperature that agrees nicely

with the corresponding exact solution. Moreover, the

fluid velocity along with the fluid temperature increases

with increasing values of the pressure gradient P, the

non-Newtonian parameter

and/or the Brinkman

number

.

7. References

[1] G. Astarita and G. Marrucci, “Principles of Non-

Newtonian Fluid Mechanics,” McGraw Hill, London,

1974.

[2] R. B. Bird, R. C. Armstrong and O. Hassager, “Dynamics

of Polymeric Liquids, Fluid Mechanics,” Wiley, New

York, 1987.

[3] M. Moradi, “Laminar Flow Heat Transfer of a

Pseudoplastic Fluid through a Double Pipe Heat

Exchanger,” Iranian Journal of Chemical Engineering,

Vol. 3, No. 2, 2006, pp. 13-19.

[4] M. Massoudi and I. Christie, “Effects of Variable

Viscosity and Viscous Dissipa tion on the Flow of a Third

Grade Fluid in a Pipe,” International Journal of

Non-Linear Mechanics, Vol. 30, No. 5, 1995, pp.

687-699.

doi:10.1016/0020-7462(95)00031-I

. The corresponding exact solution for

the same values of the parameters ,

and P is

presented in Figure 4(b) for the sake of comparison.

From these figures we again observe that the agreement

of the numerical solution for the fluid temperature with

the corresponding exact solution is very good. We

present the absolute error E

for = 0.25,

and in Table 4 and observe that the

size of is satisfactorily small to confirm agreement

of the numerical solution with the corresponding exact

solution. Moreover, as we see from Figure 4, again the

fluid temperature shows an increasing trend with

increasing values of

=0.5P

1.0,3.0,

E

[5] A. M. Siddiqui, A. Zeb, Q. K. Ghori and A. M.

Benharbit, “Homotopy Perturbation Method for Heat

Transfer Flow of a Third Grade Fluid between Parallel

Plates,” Chaos Solitons and Fractals, Vol. 36, No. 1,

2008, pp. 182-192.

5.0

, when P and

are held fixed.

Further, we report without presenting that the numerical

solution for the fluid temperature and the

corresponding absolute error shows similar

behavior with varying

E

and fixed P and

. Therefore,

we conclude that the fluid temperature increases with

increasing values of the non-Newtonian parameter

,

the Brinkman number

and the pressure gradient P,

when any two of these parameters are held fixed.

[6] T. Hayat, K. Maqbool and M. Khan, “Hall and Heat

Transfer Effects on the Steady Flow of a Generalized

Burger's Fluid Induced by Sudden Pull of Eccentric

Rotating Disks,” Nonlinear Dynamics, Vol. 51, No. 1-2,

2008, pp. 267-276. doi:10.1007/s11071-007-9209-2

[7] F. L. Stasa, “Applied Finite Element Analysis for

Engineers, ” CBS Publishing, 1985.

[8] O. C. Zienkiewicz, “The Finite Element Method,”

McGraw Hill, London, 1977.

[9] S. Iqbal, A. M. Mirza and I. A. Tirmizi, “Galerkin’s

Finite Element Formulation of the Second-Order

Boundary-Value Problems,” International Journal of

Computer Mathematics, Vol. 87, No. 9, 2010, pp.

2032-2042.

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