INVESTIGATION OF HEAT TRANSFER OF NON-NEWTONIAN FLUID IN THE PRESENCE OF A POROUS WALL

1 Department of Mathematics Babu Banarasi Das Northern India Institute of Technology Lucknow U.P., India *2 Department of Mathematics Babu Banarasi Das National Institute of Technology & Management Lucknow, U.P., India Abstract: This study deals the investigation of heat transfer of non-Newtonian fluid in the presence of a porous bounding wall. Perturbation method is applied for the solution of non-linear differential equation. The main focus of this paper is to investigate the effects of parameters such as Reynolds number Re, Prandtl number Pr, permeability parameter K and n in the velocity of fluid and temperature coefficient. For fulfilling the purpose Matlab software has been used. The results show that velocity of non-Newtonian increases with increase of Reynolds number Re and temperature increases with increases of Prandtl number Pr.


Introduction
The non-Newtonian fluid flow through a porous medium produced by temperature differences is one of the most considerable and contemporary subjects and found the great applications, in geothermic, geophysics, chemicals, cosmetics, pharmaceuticals and industrial technology. The practical interest in convective heat transfer in porous medium is expanded rapidly, due to the wide range of applications in engineering fields. Rivlin (1955) has studied plane strain of a net formed by inextensible cords. Christopher and Middleman (1965) have studied power-law flow through a packed tube. Blottner (1970) has studied finite-difference methods of solution of the boundary-layer equations. Chamkha (1977) has studied similarity solution for thermal boundary layer on a stretched surface of a non-Newtonian fluid. Dharmadhikari and Kale (1985) have studied flow of non-Newtonian fluids through porous media. Chen and Chen (1988) have studied free convection of non-Newtonian fluids along a vertical plate embedded in a porous medium. Wang Chaoyang and Tu Chuanjing (1989) have studied boundary-layer flow and heat transfer of non-Newtonian fluids in porous media. Hooper et al. (1993) have studied mixed convection from a vertical plate in porous media with surface injection or suction. Choi (1995) has studied enhancing thermal conductivity of fluids with nanoparticles in developments and applications of non-Newtonian flows. Das et al. (1996) have studied radiation effects on flow past an impulsively started infinite isothermal plate. Yih (1998) has studied Coupled heat and mass transfer in mixed convection over a wedge with variable wall temperature and concentration in porous media: The entire regime. Magyari et al. (1999) have studied heat and mass transfer in the boundary layers on an exponentially stretching continuous surface. Yih (2001) has studied radiation effects on mixed convection over an isothermal wedge in the porous media: The entire regime. Wang (2002) has studied flow due to a stretching boundary with partial slip: an exact solution of the navier-stokes equations. Ingham et al. (2004) have studied Emerging Technologies and Techniques in Porous Media. Kluwer, Dordrecht, Plate with time dependant temperature and concentration. Aboeldahab and A. G. El-Din (2005) have studied thermal radiation effects on MHD flow past a semi-infinite inclined plate in the presence of mass diffusion. Mahmoud and Mahmoud (2006) have studied Analytical solutions of hydro magnetic boundary layer flow of a non-Newtonian power law fluid past a continuously moving surface. Marinca and Herisanu (2008) have studied optimal homotopy asymptotic method with application to thin film flow. Cheng (2009) has studied combined heat and mass transfer in natural convection flow from a vertical wavy surface in a power-law fluid saturated porous medium with thermal and mass stratification. Suratiand  In this study perturbation Technique is applied to find the approximate solution of nonlinear differential equation for non-Newtonian fluid flow in axisymmetric channel with a porous wall for turbine cooling applications. The results are given for the velocity of non-Newtonian fluid and temperature for various values of Reynolds number Re, Prandtl number Pr, permeability parameter K and parameter n.

Mathematical Formulation
In this paper we study the simultaneous development of fluid flow and heat transfer of non-Newtonian fluid flow is investigated, it is the applicable on the turbine disc for the cooling purposes. The problem is shown schematically in Fig. 1. In this Figure y-axis is perpendicular to the heated disc surface, which is shown the x-axis. The porous wall is parallel to heated disc axis. The perforated disc of the channel is located at y = +L. The non-Newtonian fluid is injected from the other porous wall uniformly in order to cool the heated wall that coincides with the x-axis.
Here we observed in Figure 1 the cooling problem of the disk can be considered as a inactivity point flow with injection. For a steady, axisymmetric non-Newtonian fluid flow for the following equations in the cylindrical coordinates.
For the solution of this problem depicted in the figure 1 in the case of axially symmetric flow it is convenient to define a stream function so that the continuity equation is satisfied: Where λ=z/L and the velocity components can be derived as: Using equation (12) and (13), in equation (2) and (3) and then pressure term eliminated, we get the differential equation Where K_1=φ_2/(ρL^2 ),K_2=φ_3/(ρL^2 ) is injection Reynolds number, For K_2=0, the equation turned as Where K_1=K, The boundary conditions are: The energy equation for this problem with viscous dissipation in non-Newtonian form is given by Where U_r,U_z are the velocity components in the r and z directions and V is the injection velocity: ρ,P,T,C,k ̅ are the density, pressure, temperature, specific heat and heat conduction coefficient of the fluid, respectively, ϕ is the dissipation function. Assuming the blade wall (z=0) temperature distribution be And assuming the fluid temperature to have the form of [1] Where T_0 is the temperature of the incoming coolant (z = L) and neglecting dissipation effect the following equations and boundary conditions are obtained: The above differential equation (15) and (21) solved by perturbation technique, given bellow.

Method of Solution
In this paper we have applied the perturbation technique for the solution of non-linear differential equation with suitable boundary conditions, which is given above. We take the series Putting these values of equation (23) in equation (15), we get Taking the coefficient of 0 , 1 , we get Putting these values of equation (24) in equation (21), we get Taking the coefficient of Now solving the above equation (26), (27), (29), (30) and (31) using the boundary condition Taking the coefficient of 0 , 1 , 2 , we get Now putting the value of equation (32), (33), (34), (35), (36) in equation (23) and (24), we get the solution of non-linear differential equation (15) The Graphs have been plotted with described set of parameters and discussed in detail in the next section.

Results and Discussion
The objective of the present study is to find out the solution of heat transfer of non-Newtonian fluid in the presence of a porous bounding wall, for the purpose we have used the perturbation technique. In this paper the graph of non-Newtonian fluid velocity and Temperature coefficient against distance have been plotted using constant Reynolds number, Prandtl number and other parameter.   Fig 8a & 8b is graph between dimensionless variable λ and radial velocity of non-Newtonian fluid f'(λ) at constant Reynolds number Re = 2, increase of dimensionless variable λ in the range (0≤λ≤1) radial velocity of fluid increases sharply then decreases sharply. This curve is like as half sine nature; it is seen that increase of permeability parameter K the sharpness of radial velocity of fluid increases. Fig 9a represents the graph between dimensionless variable λ and radial velocity of fluid f^' (λ) at constant parameter K = 0.5 in the range(0≤λ≤1), increase of dimensionless variable λ radial velocity of fluid f^' (λ) slowly increases then decreases, it is flat type curve; it is seen that the increase of Reynolds number Re radial velocity of fluid f^' (λ) increases sharply then decreases sharply, it is peaked type of curve. Fig 9b represents the graph between dimensionless variable λ and radial velocity of fluid f^' (λ) at constant variable K = 0.5 in the range (0≤λ≤2.5), increase of dimensionless variable λ radial velocity of fluid f^' (λ) slowly increases then decreases slowly in the range (0≤λ≤1.5), and increases sharply in the range (1.5≤λ≤2.5); it is seen that the increase of Reynolds number Re radial velocity of fluid f^' (λ) increases sharply then decreases sharply.  increases sharply: it is seen that the increase of Parameter n temperature coefficient of fluid qn(λ) increases sharply. The reciprocal effect has shown in the Fig. 14b. It is seen that increase of parameter n temperature coefficient of fluid qn(λ) is decreases sharply at constant parameter k = 0.5, Pr = 3.0, Re = 2.0 in the range(0≤λ≤1).

Conclusions
In this paper the main objective is to investigate the effect of Reynolds number Re, Prandtl number Pr, permeability parameter k and parameter n. The results show that the increment in the Reynolds number has the similar effects on velocity components and heat component. It increases sharply. At higher Reynolds number the maximum velocity of fluid is shift to the solid wall where shear stress becomes larger as the Reynolds number grows. The increase of permeability parameter k velocity of fluid increases whereas the heat coefficient of fluid is also increased. Increase of Prandtl number Pr heat flow of fluid increases sharply. Increase of parameter n heat flow of fluid decreases sharply. It is investigated that Reynolds numbers sharply affect the velocity of fluid and heat flow of fluid. This study have practical applications in nuclear engineering control, plasma aerodynamics, mechanical engineering manufacturing processes, astrophysical fluid dynamics, filtration technology, geothermal energy and precipitation.