Abstract

The present paper studies the heat transfer flow of a third grade fluid between two heated parallel plates for the two models: constant viscosity model and Reynold's model. In each case the nonlinear momentum equation and the energy equation have been solved using HPM and PEM. Graphs for the velocity and temperature profiles are presented and discussed for the various values of parameters entering the problem. The dominant effect is governed by whether or not the fluid is non-Newtonian, the temperature effects being relegated to a less dominant role.

Keywords:

Constant viscosity model, homotopy perturbation mathod, momentum equation, PEM, Reynold's model, third grade fluid,

References

1. Abbasbandi, S., 2006. Application of He's homotopy perturbation method for laplace transform. Chaos Soliton. Fract., 30(5): 1206-1212.
   CrossRef    
2. He, J.H., 1998a. An approximation solution technique depending upon an artificial parameter: A special example. Commun. Nonlinear Sci., 3(2): 92-97.
   CrossRef    
3. He, J.H., 1998b. Newton-like iteration method for solving algebraic equations. Commun. Nonlinear Sci., 3(2): 106-109.
   CrossRef    
4. He, J.H., 1999. Homotopy perturbatin technique. Comput. Methods Appl. Mech. Eng., 178(3-4): 257-262.
   CrossRef    
5. He, J.H., 2000. A coupling method of a homotopy technique and pertubation technique for non-linear problems. Int. J. Nonlinear Mech., 35(1): 37-43.
   CrossRef    
6. He, J.H., 2003. Homotopy perturbation method: A new nonlinear analytical technique. Appl. Math. Comput., 135(1): 73-79.
   CrossRef    
7. He, J.H., 2004a. Asymptotology by homotopy perturbation method. Appl. Math. Comput., 15(3): 591-596.
   CrossRef    
8. He, J.H., 2004b. The homotopy perturbation method for nonlinear oscillators and discontinuities. Appl. Math. Comput., 151(1): 287-292.
   CrossRef    
9. He, J.H., 2005a. Application of homotopy perturbation method to nonlinear wave equations. Chaos Soliton. Fract., 26(3): 695-700.
   CrossRef    
10. He, J.H., 2005b. Homotopy perturbation method for bifurcation of nonlinear problems. Int. J. Nonlinear Sci., 6(2): 207.
    CrossRef    
11. Massoudi, M. and I. Christie, 1995. Effects of varibale Dissipation on the flow of a third grade fluid in a pipe. Int. J. Nonlinear Mech., 30(5): 687-699.
    CrossRef    
12. Shakil, M., T. Khan, H.A. Wahab and S. Bhatti, 2013. A comparison of Adomian Decomposition Method (ADM) and Homotopy Perturbation Method (HPM) for nonlinear problems. IMPACT: Int. J. Res. Appl. Nat. Soc. Sci., 1(3): 37-48.
    
13. Siddiqui, A.M., M. Ahmed and Q.K. Ghori, 2006a. Couette and Poiseuille flows for non-newtonian fluids. Int. J. Nonlinear Sci., 7(1): 15.
    CrossRef    
14. Siddiqui, A.M., R. Mahmood and Q.K. Ghori, 2006b. Homotopy perturbation method for thin film flow of a fourth grade fluid down a vertical cylinder. Phys. Lett., 352(4-5): 404-410.
    CrossRef    
15. Siddiqui, A.M., M. Ahmed and Q.K. Ghori, 2007. Thin film flow of non-Newtonian fluids on a moving belt. Choas Soliton. Fract., 33(3): 1006-1016.
    CrossRef    
16. Siddiqui, A.M., R. Mahmood and Q.K. Ghori, 2008. Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane. Choas Soliton. Fract., 35(1): 140-147.
    CrossRef    
17. Szeri, A.Z. and K.R. Rajagopal, 1985. Low of non-Newtonian fluid between heated parallel plates. Int. J. Nonlinear Mech., 20(2): 91-101.
    CrossRef    
18. Truesdell, C. and W. Noll, 1965. The Non-linear Field Theories of Mechanics. 2nd Edn., In: Flugge (Ed.), Handbuch der Physik (German). Springer-Verlag, Berlin.
    CrossRef    
19. Tsai, C.Y, M. Novack and G. Roffle, 1988. Rheological and heat transfer characteristics of flowing coal-water mixtures. Final Report, DOE/ MC/23255-2763.
    
20. Wahab, H.A., M. Shakil, T. Khan, S. Bhatti and M. Naeem, 2013a. A comparative study of a system of Lotka-Voltera type of PDEs through perturbation methods. Computat. Ecol. Software, 3(4): 110-125.
    
21. Wahab, H.A., T. Khan, M. Shakil, S. Bhatti and M. Naeem, 2013b. Analytical treatment of system of KdV equations by Homotopy Perturbation Method (HPM) and Homotopy Analysis Method (HAM). Comput. Ecol. Software, 4(1): 63-81.
    
22. Wahab, H.A., S. Bhatti, M. Naeem, M.T. Qureshi and M. Afzal, 2013c. A mathematical model for the rods with heat generation and convective cooling. J. Basic Appl. Scientific Res., ISSN: 2090-4304 (Print), ISSN: 2090-424x (Online), (Accepted).
    

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The authors have no competing interests.

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