Numerical Study of Convective Heat Transfer on the Power Law Fluid over a Vertical Exponentially Stretching Cylinder
M. Naseer^{1, *}, M. Y. Malik^{1}, Abdul Rehman^{2}
^{1}Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
^{2}Department of Mathematics, University of Balochistan, Quetta, Pakistan
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To cite this article:
M. Naseer, M. Y. Malik, Abdul Rehman. Numerical Study of Convective Heat Transfer on the Power Law Fluid over a Vertical Exponentially Stretching Cylinder. Applied and Computational Mathematics. Vol. 4, No. 5, 2015, pp. 346-350. doi: 10.11648/j.acm.20150405.13
Abstract: The present paper is the study of boundary layer flow and heat transfer of Power law fluid flowing over a vertical exponentially stretching cylinder along its axial direction. The governing partial differential equations and the associated boundary conditions are reduced to nonlinear ordinary differential equations after using the boundary layer approximation and similarity transformations. The obtained system of nonlinear ordinary differential equations subject to the boundary conditions is solved numerically with the help of Fehlberg method. The effects of Power law index, Reynolds number, Prandtl number, the natural convection parameter and local Reynolds number are presented through graphs. The skin friction coefficient and Nusselt number are presented through tables for different parameters.
Keywords: Boundary Layer Flow, Exponential Stretching, Vertical Cylinder, Power Law Fluid, Natural Convection Heat Transfer, Fehlberg Method
1. Introduction
Large amount of work has been done on laminar boundary layer flow over stretching sheet. For example in extursion processes such as polymer extursion from a dye and wire driling, drawing, tunning and annealing of copper wires, the cooling of a metalic plate in a cooling bath and so on. Crane (1970) was the first who studied the stretching sheet problem. After Crane many researchers have extended this work (D. R. Jeng at al.,1986; F. Labropulu at al.,2010; E. Magyari and B. Keller, 1999; R.Ellahi, 2009; M. Y. Malik at al., 2013). The above mentioned studies are about linear stretching but in many practical situations involves non linear stretching such as exponential stretching. Many authors vary velocity of sheet exponentially with distance from slit. Elbasbesh (2001) was the first who studied the exponentially stretching sheet problem. He take a perforated sheet and notice the effect of wall mass suction on the flow and heat transfer over an exponentially stretching surface using similarity transformation.
Later on, Sanjayan and Khan (2005,2006) extended the work on exponential stretching. They studied a similar kind of problem considering viscoelastic fluid model under viscousdissipation effects. The non-Newtonian fluids are very useful in industrial and engineering applications. Schowalter (1960) studied the applications of boundary layer using power law fluid. Similarity solutions for non
Newtonian power law fluids were obtained by Kapur and Srivastave (1963) and Lee et al. (1966). The power law fluids over a continous moving flat plate with constant surface velocity and temperature distribution was given by Fox et al. (1969). Anderson and Dandapat (1991) extended the pioneery work of Crane (1970) for a non-Newtonian power law fluids. Later on Hassnain (1998) extended the work for heat transfer analysis. Abel et al. (2009) studied the power law fluid over a vertical stretching sheet with variable thermal conductivity and non uniform heat source. Few relavent intresting works concerning the stretching flowes are cited in (S. Nadeem et al., 2009; Abdul Rehman et al.,2013;M.Naseer et al.,2014; C.Y.Wang and Z Angew, 1989; A. Ishak et al., 2008; I.A. Hassanien et al., 1998; S. Nadeem and Anwar Hussain, 2010; A. Ishak et al., 2011; C. Y. Wang, 2012; Abdul Rehman et al.,2013;). In this paper we have studied the flow and heat transfer of a power law fluid over a vertical exponentially stretching cylinder.
2. Formulation
Consider the problem of natural convection boundary layer flow of a power law fluid flowing over a vertical circular cylinder of radius . The cylinder is assumed to be stretched exponentially along the axial direction with velocity The temperature at the surface of the cylinder is assumed to be and the uniform ambient temperature is taken as such that the quantity in case of the assisting flow, while in case of the opposing flow, respectively. Under these assumptions the boundary layer equations of motion and heat transfer are
(1)
(2)
(3)
where the velocity components along the axes are , is fluid density, is the consistency coefficient, is pressure, is the gravitational acceleration along the direction, is the coefficient of thermal expansion, is the temperature, is the thermal diffusibility. The corresponding boundary conditions for the problem are
(4)
(5)
where is the fluid velocity at the surface of the cylinder.
3. Solution of the Problem
Introduce the following similarity transformations:
(6)
(7)
Where the characteristic temperature difference is calculated from the relations With the help of transformations and , to take the form
(8)
(9)
In which is the natural convection parameter, is the Prandtl number, is the local Reynolds number and is the Reynolds number. The boundary conditions in non dimensional form become
(10)
(11)
The important physical quantities such as the shear stress at the surface the skin friction coefficient the heat flux at the surface of the cylinder and the local Nusselt number are
(12)
(13)
The solution of the present problem is obtained by using Fehlberg Method.
4. Results and Discussion
The problem of natural convection boundary layer flow of a Power law fluid over an exponentially stretched cylinder is studied in this paper. The cylinder is assumed to be stretched exponentially along its axial direction. The exponential stretching velocity at the surface of the cylinder is assumed to be The solution of the problem is obtained numerically with the help of Fehlberg Method. The effect of the various parameters such as the Reynolds number the local Reynolds number , the power law index , the Prandtl number and the natural convection parameter over the non dimensional velocity and temperature profiles are presented graphically and in the form of tables. shows the effects of natural convection parameter on the velocity profile when. From it is observed that by increasing the values of natural convection parameter the velocity profile increases. Shows the influence of local Reynolds number over the velocity profile when . From it is clear that by increasing the values of local Reynolds number the velocity profile decreases. and shows the effects of Prandtl number and Reynolds number on temperature profile when . Similar characteristics are observed for Prandtl number and Reynolds number in and , by increasing the values of these numbers temperature profile decreases. shows the effects of natural convection parameter on the velocity profile when . The velocity profile decreases by increasing the values of natural convection parameter. shows opposite behavior of velocity profile when, the velocity profile increases by increasing local Reynolds number . In and temperature profiles are presented for . The temperature profiles behave just like for . Table shows the boundary derivatives for the velocity profile at the surface of the cylinder that corresponds to the skin friction coefficient at the surface tabulated for different values of and . From the Table 1 it is observed that the magnitude of the boundary derivative increases with increase in both and . Table shows the values for local Nusselt numbers calculated for different values of and .From entries in the Table it is noticed that with increase in both and , the Local Nusselt number decreases.
λ \ Rea | 0 | 0.1 | 0.2 | 0.3 | 0.4 |
1 | 0.9859 | 0.9903 | 0.9953 | 1.0011 | 1.0078 |
3 | 1.2212 | 1.2366 | 1.2544 | 1.2754 | 1.3012 |
5 | 1.4494 | 1.4755 | 1.5065 | 1.5452 | 1.5972 |
10 | 1.9274 | 1.9809 | 2.0499 | 2.1505 | 2.3941 |
15 | 2.3145 | 2.3968 | 2.5121 | 2.7246 | 2.9537 |
Pr \ Re | 0 | 0.1 | 0.2 | 0.3 | 0.4 |
1 | 1.1971 | 1.1967 | 1.1962 | 1.1957 | 1.1952 |
7 | 1.7912 | 1.7890 | 1.7866 | 1.7838 | 1.7808 |
10 | 3.5901 | 3.5808 | 3.5699 | 3.5566 | 3.5396 |
15 | 5.5503 | 5.5360 | 5.5182 | 5.4944 | 5.4580 |
25 | 6.6652 | 6.6491 | 6.6285 | 6.5999 | 6.5508 |
References