Functional Differential and Difference Equations with Applications 2013View this Special Issue
Research Article | Open Access
Heat Transfer Analysis on the Hiemenz Flow of a Non-Newtonian Fluid: A Homotopy Method Solution
The mathematical model for the incompressible two-dimensional/axisymmetric non-Newtonian fluid flows and heat transfer analysis in the region of stagnation point over a stretching/shrinking sheet and axisymmetric shrinking sheet is presented. The governing equations are transformed into dimensionless nonlinear ordinary differential equations by similarity transformation. Analytical technique, namely, the homotopy perturbation method (HPM) with general form of linear operator is used to solve dimensionless nonlinear ordinary differential equations. The series solution is obtained without using the diagonal Padé approximants to handle the boundary condition at infinity which can be considered as a clear advantage of homotopy perturbation technique over the decomposition method. The effects of the pertinent parameters on the velocity and temperature field are discussed through graphs. To the best of authors’ knowledge, HPM solution with general form of linear operator for two-dimensional/axisymmetric non-Newtonian fluid flows and heat transfer analysis in the region of stagnation point is presented for the first time in the literature.
Stagnation point flow is of great importance in the prediction of skin friction as well as heat/mass transfer near stagnation regions of bodies in high speed flows and also in the design of thrust bearings and radial diffusers, drag reduction, transpiration cooling, and thermal oil. In 1911, Hiemenz  revealed that stagnation point flow can be examined by the Navier-Stokes (NS) equations. He used the similarity of the solution to reduce number of variables by means of a coordinate transformation. Later Howann  discovered the stagnation point flow in case of axisymmetric situation. Recently, a number of researchers studied the stagnation point flow considering different fluids models, geometries, and assumptions that were proposed in the literature. The literature on the topic is quite extensive and hence cannot be described here in detail. However some most recent works of eminent researchers regarding the analytical/numerical solution of stagnation point for different geometries may be mentioned in [3–5]. Attia , Massoudi and Ramezan , and Garg  extended the stagnation point flow for heat transfer.
The main aim of this paper is to extend the HPM [9–17] for solving non-Newtonian fluid flow and heat transfer analysis in the region of stagnation point flow towards a stretching/shrinking and axisymmetric shrinking sheet. Also the main motivation is to perform such analysis  (shrinking/axisymmetric shrinking sheet) for a non-Newtonian fluid in the presence of heat transfer. Heat transfer plays very important role in nuclear energy because nuclear chain reaction creates heat, and it is used to boil water, produce steam, and drive a steam turbine. The steady Navier-Stokes equations are reduced to the nonlinear ordinary differential equations by using similarity solutions. Graphical results explicitly reveal the complete reliability and efficiency of the suggested algorithm.
2. Governing Equations
The flow and heat characteristics are governed by the following equations :
The similarity transformations for two-dimensional stagnation flow case are as follows :
The steady Navier-Stokes equations yield a system of nonlinear ordinary differential equations in the form and corresponding boundary conditions take the form
The similarity transformations for axisymmetric stagnation flow towards an axisymmetric shrinking surface are as follows :
3. Analytical Solution
For the HPM  solution, we select as initial approximations of , , and . We further choose the following auxiliary linear operators:
On solving (14) in any software like Mathematica, Maple or MATLAB we can get any order of approximation.
4. Concluding Remarks
In this paper, we have studied non-Newtonian Stagnation point flow in the presence of heat transfer by using HPM. The HPM is used in a direct way without using linearization, discretization, or restrictive assumption. The variations of various emerging parameters on the velocities () and temperature field () are discussed through Figures 1, 2, 3, 4, 5, and 6. The main results of the present analysis are as follows: (i)for two-dimensional case, the velocity decreases for shrinking parameter while for axisymmetric shrinking surface, the velocity shows opposite behavior for ;(ii)for two dimensional case and axisymmetric shrinking surface, the velocity profiles and increase with increasing value of ;(iii)the effects of Prandtl number Pr are same on the temperature field for both cases.
|:||Density of fluid|
|:||Second grade parameter|
|:||Stretching and shrinking parameter|
|and :||The temperatures at and far away from the plate|
|:||Dimensionless second grade parameter|
|:||Dimensionless velocity profiles|
|:||Dimensionless temperature profile|
|:||Velocity component in direction|
|:||Velocity component in direction|
|:||Velocity component in direction|
|:||Independent dimensionless parameter.|
This work is supported by the Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology, Czech Republic.
- K. Hiemenz, “Die Grenzschict neinem in den gleichformigen flussigkeitsstrom eingetauchten geraden Kreiszylinder,” Dingler's Polytechnic Journal, vol. 326, pp. 321–410, 1911.
- F. Howann, “Der Einfluss grosser Zahigkeit bei der Stromung um den Zylinder und um die Kugel,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 16, pp. 153–164, 1936.
- C. Y. Wang, “Stagnation flow towards a shrinking sheet,” International Journal of Non-Linear Mechanics, vol. 43, no. 5, pp. 377–382, 2008.
- C. Y. Wang, “Similarity stagnation point solutions of the Navier-Stokes equations—review and extension,” European Journal of Mechanics, B/Fluids, vol. 27, no. 6, pp. 678–683, 2008.
- A. Yildirim and S. A. Sezer, “Analytical solution of MHD stagnationpoint flowin porous media by means of the homotopy perturbation method,” Journal of Porous Media, vol. 15, no. 1, pp. 83–94, 2012.
- H. A. Attia, “Hydromagnetic stagnation point flow with heat transfer over a permeable surface,” Arabian Journal for Science and Engineering, vol. 28, no. 1, pp. 107–112, 2003.
- M. Massoudi and M. Ramezan, “Boundary layers heat transfer analysis of a viscoelastic fluid at a stagnation point,” Journal of Heat Transfer, vol. 130, pp. 81–86, 1990.
- V. K. Garg, “Heat transfer due to stagnation point flow of a non-Newtonian fluid,” Acta Mechanica, vol. 104, no. 3-4, pp. 159–171, 1994.
- J. H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters A, vol. 350, no. 1-2, pp. 87–88, 2006.
- Y. Khan, N. Faraz, A. Yildirim, and Q. Wu, “A series solution of the long porous slider,” Tribology Transactions, vol. 54, no. 2, pp. 187–191, 2011.
- L. Xu, “He's homotopy perturbation method for a boundary layer equation in unbounded domain,” Computers and Mathematics with Applications, vol. 54, no. 7-8, pp. 1067–1070, 2007.
- P. D. Ariel, “Homotopy perturbation method and the natural convection flow of a third grade fluid through a circular tube,” Nonlinear Science Letters A, vol. 1, no. 2, pp. 43–52, 2010.
- Y. Khan and Q. Wu, “Homotopy perturbation transform method for nonlinear equations using He's polynomials,” Computers and Mathematics with Applications, vol. 61, no. 8, pp. 1963–1967, 2011.
- Y. Khan, “A method for solving nonlinear time-dependent drainage model,” Neural Computing and Applications, vol. 23, no. 2, pp. 411–415, 2013.
- Y. Khan, Q. Wu, N. Faraz, and A. Yildirim, “The effects of variable viscosity and thermal conductivity on a thin film flow over a shrinking/stretching sheet,” Computers and Mathematics with Applications, vol. 61, no. 11, pp. 3391–3399, 2011.
- J. H. He, “Asymptotic methods for solitary solutions and compactons,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012.
- M. Madani, Y. Khan, Gh. Mahmodi, N. Faraz, A. Yildirim, and B. Nasernejad, “Application of homotopy perturbation and numerical methods to the circular porous slider,” International Journal of Numerical Methods for Heat and Fluid Flow, vol. 22, no. 6, pp. 705–717, 2012.
Copyright © 2013 Yasir Khan and Zdeněk Šmarda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.