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This paper considers the problem of hydrodynamics and thermal boundary layers of Darcy flow over horizontal surface embedded in a porous medium. The solutions of such problems for the cases of uniform surface temperature and variable surface temperature have been studied and analysed in many papers. This paper, however, attempts to find similarity solutions for the Darcy flow problem with a convective boundary condition at the plate surface. It is found that the solution is possible when the heat transfer coefficient is proportional to
*x*
^{–2/3}. The numerical solutions thus obtained are analyzed for a range of values of the parameter characterizing the hot fluid convection process. Analytical expressions are provided for local surface heat flux and total surface heat transfer rate while the flow variables are discussed graphically.

The study of natural convection flow and heat transfer in saturated porous medium from surfaces which are at a temperature different from the surrounding medium has always been a topic of great interest due to several engineering and geophysical applications. An extensive study of convection flow in porous media is provided by Nield and Bejan [

The similarity solutions for many different problems of free convection fluid flow in saturated porous medium have been obtained and analyzed by various authors. Cheng and Chang [

In all the papers cited above, the problems are solved when the flow is driven either by a prescribed surface temperature or by a prescribed heat flux. Here, in this paper, a somewhat different driving mechanism for a free convection along the surface is considered, where it is assumed that the flow is also set up by the convective heating of the surface. Aziz [

However these convection problems constitute a relatively new and active area in the field of convection in porous media. To the best of author’s knowledge, no such solution has been attempted for the Darcy flow in porous medium with a convective surface boundary condition. Hence, in the present study the problem of laminar boundary layers over flat plate in Darcy regime with a convective boundary condition has been considered. The paper demonstrates that a similarity solution is possible if the convective heat transfer of the fluid heating the plate on its lower surface is proportional to

We consider the problem of hydrodynamics and thermal boundary layer flow over a flat plate embedded in a porous medium saturated with Newtonian fluid. The momentum and thermal boundary layers exist along horizontal surface whenever wall temperature differs from that of the surrounding fluid. We take x-axis along the plate and y-axis along the normal to the plate. The flow is based on the following assumptions:

1) two-dimensional flow,

2) the flow is assumed to be slow enough to conform to the Darcy regime i.e. Darcy flow model,

3) total thermodynamic equilibrium between the surfaces,

4) negligible viscous dissipation,

5) isotropic porous medium and

6) constant thermo-physical properties, with the exception of the assumed linear relation between density and temperature in the buoyancy term (the Oberbeck-Boussinesq approximation).

Then, under boundary layer approximations, the equations governing the flow as given by Cheng and Chang [

where u and v are the velocity components in the horizontal and vertical directions respectively.

In this paper we assume that the bottom surface of the plate is heated by convection from a hot fluid at constant temperature

To determine similarity solutions to momentum and energy equations coupled with boundary conditions (4) and (5), we introduce the following dimensionless variables

where

modified local Rayleigh number.

In terms of the new variables, it can be shown that the velocity components are given by

and the governing equations become

coupled with boundary conditions

where

The primes in Equations (8)-(11) denote differentiation with respect to

For momentum and energy equations to have a similarity solution, the parameter a must be constant and not a function of x. This condition is met if the heat transfer coefficient

Therefore, we assume that

where c is a constant.

With these assumptions we have

With a as defined by Equation (12), the solutions generated are the local similarity solutions.

Equations (8) and (9) are coupled non-linear differential equations for f and θ with boundary conditions given by (10) and (11) where a is defined by (12). Numerical solutions to these coupled non-linear boundary value problem can be obtained by using shooting method technique. We first convert the boundary value problem to an initial value problem and make a systematic guessing of slopes at

The graphs for

From Equation (6), it can be seen that

equation

The values of the dimensionless velocity

reveal that for each value of a the value of

a | ||
---|---|---|

0.05 | 0.173110 | 0.041344 |

0.1 | 0.266035 | 0.073396 |

0.5 | 0.582140 | 0.208930 |

0.8 | 0.679166 | 0.256667 |

1.0 | 0.721672 | 0.278328 |

5.0 | 0.922731 | 0.386346 |

10.0 | 0.959309 | 0.406912 |

The local surface heat flux can be computed from

In terms of dimensionless temperature θ, this can be rewritten as

The total surface heat-transfer rate for a surface with a length l and a width w can be computed from

which gives

when a is constant and

when a is a function of x.

This is because when a is constant the surface temperature

It is found that under boundary layer approximations, a similarity solution for the Darcy flow over a flat plate embedded in a porous medium with a convective boundary condition at the plate surface is possible if the convective heat transfer of the fluid heating the plate on its lower surface is proportional to

Rama CharanChaudhary,Gopal NaraianPurohit,PreetiGarg, (2016) Similarity Solutions for Laminar Boundary Layer Flow of Darcian Fluid over Horizontal Plate with a Convective Boundary Condition. Applied Mathematics,07,313-319. doi: 10.4236/am.2016.73028