Effects of thermal radiation and variable fluid viscosity on free convective flow and heat transfer past a porous stretching surface

doi:10.1016/j.ijheatmasstransfer.2007.11.038

International Journal of Heat and Mass Transfer

Volume 51, Issues 9–10, May 2008, Pages 2167-2178

https://doi.org/10.1016/j.ijheatmasstransfer.2007.11.038 Get rights and content

Abstract

Free convective boundary layer flow and heat transfer of a fluid with variable viscosity over a porous stretching vertical surface in presence of thermal radiation is considered. Fluid viscosity is assumed to vary as a linear function of temperature. The symmetry groups admitted by the corresponding boundary value problem are obtained by using a special form of Lie group transformations viz. scaling group of transformations. A third-order and a second-order coupled ordinary differential equations corresponding to the momentum and the energy equations are obtained. These equations are then solved numerically. It is found that the skin-friction decreases and heat transfer rate increases due to the suction parameter. Opposite nature is noticed in case of blowing. With the increase of temperature-dependent fluid viscosity parameter (i.e. with decreasing viscosity), the fluid velocity increases but the temperature decreases at a particular point of the sheet. Due to suction (injection) fluid velocity decreases (increases) at a particular point of the surface. Effects of increasing Prandtl number as well as radiation parameter on the velocity boundary layer is to suppress the velocity field and the temperature decreases with increasing value of Prandtl number. Due to increase in thermal radiation parameter, temperature at a point of the surface is found to decrease.

Introduction

The study of hydrodynamic flow and heat transfer over a porous stretching sheet has gained considerable attention due to its vast applications in the industry and important bearings on several technological and natural processes. The production of sheeting material arises in a number of industrial manufacturing processes and includes both metal and polymer sheets. It is well known that the flow in a boundary layer separates in the regions of adverse pressure gradient and the occurrence of separation has several undesirable effects in so far as it leads to increase in the drag on the body immersed in the flow and adversely affects the heat transfer from the surface of the body. Several methods have been developed for the purpose of artificial control of flow separation. Separation can be prevented by suction as the low-energy fluid in the boundary layer is removed [1], [2]. On the contrary, the wall shear stress and hence the friction drag is reduced by blowing.

Free convective phenomenon has been the object of extensive research. The importance of this phenomenon is increasing day by day due to the enhanced concern in science and technology about buoyancy induced motions in the atmosphere, the bodies in water and quasisolid bodies such as earth. Natural convection flows driven by temperature differences are very much interesting in case of Industrial applications. Buoyancy plays an important role where the temperature differences between land and air give rise to a complicated flow and in enclosures such as ventilated and heated rooms (Elbashbeshy and Bazid [3]).

So such type of problem, that we are dealing with, is very much useful to polymer technology and metallurgy. Cheng and Minkowycz [4] and Cheng [5] studied the free convective flow in a saturated porous medium. Wilks [6] had studied the combined forced and free convection flow along a semi-infinite plate extending vertically upwards with its leading edge horizontal. Boutros et al. [7] solved the steady free convective boundary layer flow on a non-isothermal vertical plate. Recently, any studies were made on the steady free convective boundary layer flow on moving vertical plates considering the effect of buoyancy forces on the boundary layer Chen and Strobel [8], Ramachandran et al. [9], Lee and Tsai [10].

The radiative effects have important applications in physics and engineering particularly in space technology and high temperature processes. But very little is known about the effects of radiation on the boundary layer. Thermal radiation effects may play an important role in controlling heat transfer in polymer processing industry where the quality of the final product depends on the heat controlling factors to some extent. High temperature plasmas, cooling of nuclear reactors, liquid metal fluids, power generation systems are some important applications of radiative heat transfer from a vertical wall to conductive gray fluids. The effect of radiation on heat transfer problems have studied by Hossain and Takhar [11], Takhar et al. [12], Hossain et al. [13].

In all of the above mentioned studies, fluid viscosity was assumed to be constant. However, it is known that the physical properties of fluid may change significantly with temperature. For lubricating fluids, heat generated by the internal friction and the corresponding rise in temperature affects the viscosity of the fluid and so the fluid viscosity can no longer be assumed constant. The increase of temperature leads to a local increase in the transport phenomena by reducing the viscosity across the momentum boundary layer and so the heat transfer rate at the wall is also affected. Therefore, to predict the flow behaviour accurately it is necessary to take into account the viscosity variation for incompressible fluids. Gary et al. [14] and Mehta and Sood [15] showed that, when this effect is included the flow characteristics may changed substantially compared to the constant viscosity assumption. Recently Mukhopadhyay et al. [16] investigated the MHD boundary layer flow with variable fluid viscosity over a heated stretching sheet.

The present work deals with free convective flow and radiative heat transfer of viscous incompressible fluid having variable viscosity over a stretching porous vertical plate. The system remains invariant due to some relations among the parameters of the scaling group of transformations. Using these invariants, a third-order and a second-order coupled ordinary differential equations corresponding to the momentum and the energy equations are derived. These equations are solved numerically using shooting method. The effects of the temperature-dependent fluid viscosity parameter, suction/injection parameter, the influence of Prandtl number and radiation parameter on velocity and temperature fields of the fluid are investigated and analysed with the help of their graphical representations.

Section snippets

Equations of motion

We consider a free convective, laminar boundary layer flow and heat transfer of viscous incompressible fluid over a porous stretching sheet emerging out of a slit at origin (x = 0, y = 0) and moving with non-uniform velocity U(x) in presence of thermal radiation (Fig. 1).

The governing equations of such type of flow are, in the usual notations, $\frac{\partial u}{\partial x} + \frac{\partial υ}{\partial y} = 0,$ $u \frac{\partial u}{\partial x} + υ \frac{\partial u}{\partial y} = \frac{1}{ρ} \frac{\partial μ}{\partial T} \frac{\partial T}{\partial y} \frac{\partial u}{\partial y} + \frac{μ}{ρ} \frac{\partial^{2} u}{\partial y^{2}} + g β (T - T_{\infty}),$ $u \frac{\partial T}{\partial x} + υ \frac{\partial T}{\partial y} = \frac{κ}{ρ c_{p}} \frac{\partial^{2} T}{\partial y^{2}} - \frac{1}{ρ c_{p}} \frac{\partial q_{r}}{\partial y},$ when the viscous dissipation term in the energy equation is neglected

Numerical method for solution

The above Eqs. (24), (25) along with boundary conditions are solved by converting them to an initial value problem. We set $f^{'} = z, z^{'} = p, θ^{'} = q,$ $p^{'} = (2 z^{2} - 3 fp - 4 A θ + 4 Apq) / (4 (a + A - A θ)), q^{'} = - \frac{3}{4} \Pr fq / (1 + \frac{4}{3 N}) .$ with the boundary conditions $f (0) = S, f^{'} (0) = 1, θ (0) = 1 .$ To integrate Eqs. (28), (29) as an initial value problem we require a value for p(0), i.e. f″(0) and q(0), i.e. θ′(0) but no such values are given in the boundary. The suitable guess values for f″(0) and θ′(0)are chosen and then integration is carried out. We

Results and discussion

To analyse the results, numerical computation has been carried out using the method described in the previous section for various values of the temperature-dependent viscosity parameter (A), suction/injection parameter (S), Prandtl number (Pr) and radiation parameter (N). For illustrations of the results, numerical values are plotted in the Fig. 1a, Fig. 1b, Fig. 2a, Fig. 2b, Fig. 3a, Fig. 3b, Fig. 4a, Fig. 4b, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10. In all cases we take a = 1.

Fig. 1a,

Conclusion

The present study gives the similarity solutions for steady free convective boundary layer flow and heat transfer over a porous stretching surface with power-law velocity distribution in presence of thermal radiation and temperature-dependent fluid viscosity. The effect of increasing temperature-dependent fluid viscosity parameter on a viscous incompressible fluid is to increase the flow velocity which in turn, causes the temperature to decrease. The results pertaining to the present study

Acknowledgement

The authors are thankful to the honourable reviewers for their constructive suggestions.

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Effects of thermal radiation and variable fluid viscosity on free convective flow and heat transfer past a porous stretching surface

Abstract

Introduction

Section snippets

Equations of motion

Numerical method for solution

Results and discussion

Conclusion

Acknowledgement

Int. J. Eng. Sci.

Int. J. Heat Mass Transfer

Appl. Math. Comp.

Int. J. Heat Mass Transfer

Int. J. Heat Mass Transfer

Int. J. Heat Mass Transfer

Int. J. Eng. Sci.

Int. J. Heat Mass Transfer

Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a disk

J. Geophys. Res.