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This paper presents an application of the spectral homotopy analysis method (SHAM) to solve a problem of darcy-forcheimer mixed convection flow in a porous medium in the presence of magnetic field, viscous dissipation and thermopherisis. A mathematical model governed the flow is analyzed in order to study the effects of chemical reaction, magnetic field, viscous dissipation and thermophoresis on mixed convection boundary layer flow of an incompressible, electrically conducting fluid past a heated vertical permeable flat plate embedded in a uniform porous medium. The similarity variable is used to transform the governing equations into a boundary valued problem of coupled ordinary differential equations which are then solved using spectral homotopy Analysis Method. The spatial domains are discretized using Chebyshev-Gauss-Lobatto points and numerical computations are carried out for the non-dimensional physical parameters. A parametric study of selected parameters is conducted and the results for the velocity, temperature and concentration are illustrated graphically and physical aspects of the problem are discussed.

Mixed convective flow and heat transfer in saturated porous media is of practical interest in engineering activities because of its wide applications as seen in porous insulation design, resin transfer modeling, nuclear waste disposal, etc. Over the years, the researchers in the field of fluid mechanics have intensified their research to unravel the importance of particles deposition technology due to its numerous engineering applications. Most of the research efforts as cited in [

The problem of Darcy Forchheimer mixed convection heat and mass transfer in fluid-saturated porous media in the presence of thermophoresis was studied by [

The migration of small particles in the direction of decreasing thermal gradient is called thermophoresis [

Thermophoresis is of practical importance in a variety of industrial and engineering applications including aerosol collection, nuclear reactor safety, corrosion of heat exchanger, and micro contamination control. The use of thermophoretic heaters has led to a reduction in chip failures. In the same vein, there is potential application of thermophoresis in removing radioactive aerosols from containment domes in the event of a nuclear reactor accident. [

Mixed convection flow occurs frequently in nature. The temperature distribution varies from layer to layer and these types of flows have wide applications in industry, agriculture and oceanography. [

The influence of magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media with Soret and Dufour effects has been carried out by [

The effects of magnetic field on a boundary layer control and on the performance of many systems using electrically conducting fluid such as MHD power generators, the cooling of nuclear reactors, plasma studies, purification of molten metals from non-metallic inclusion geothermal energy extraction etc.has been discussed extensively in some literatures. The laminar boundary layer on a moving continous flat surface in the presence of suction and magnetic field was studied by [

From the literature survey, the influence of magnetic field,viscous dissipation and thermophoresis on Darcy- Forchhemer mixed convection in fluid saturated porous media in the presence of chemical reaction have not been investigated so far. Hence,we result to study the influence of magnetic field and thermophoresis on darcy- forchhemer mixed convection in fluid saturated porous media with viscous dissipation and chemical reaction is investigated using a novel SHAM approach. The transformed governing equations is solved using the spectral homotopy analysis method.

Consider a steady mixed convection boundary layer flow over a vertical flat plate of constant temperature

The initial conditions at

The boundary conditions as

In Equation (2), the plus sign corresponds to the case where the buoyancy force has a corresponding “aiding effect” on the flow and the minus sign correspond to “opposing flow” case. The term

setting

The following relations are now introduced for u, v,

It is well known that boundary layer flows have a predominant flow direction and boundary layer thickness is small compared to a typical length in the main flow direction. Boundary layer thickness usually increases with increasing downstream distance, the basic equations are transformed in order to make the transformed boundary layer thickness a slowly varying function of y, with this objective, the governing partial differential Equations (1)-(4) are transformed by following dimensionless variables for the mixed convection regime:

Here,

where the prime denote differentiation with respect to

Also using the above stated similarity transformations, the transformed boundary conditions are:

The physical quantities of interest in this problem are Nusselt number and Sherwood number which are de- fined respectively as

Nusselt number:

Sherwood number:

In this paper, the numerical version of the homotopy analysis method called SHAM is used to solve a set of or- dinary differential equations (ODEs) that models the problem of darcy-forcheimer mixed convection flow over a vertical plate embedded in a saturated porous medium in the presence of Magnetic field and chemical reaction. SHAM is a numerical version of the homotopy analysis method (HAM) that has been widely applied to solve a wide variety of nonlinear ordinary and partial differential equation with applications in applied mathematics, physics, nonlinear mechanics, finance and engineering. A detailed systematic description of the HAM and its applications can be found in two books (and a huge list of references cited therein) ( [

In essence, the HAM works by transforming a nonlinear ODE or PDE into an infinite number of linear ODEs which should be solvable analytical. The HAM solutions are required to conform to a predefined rule of solution expression. The SHAM was introduced in ( [

In applying the spectral-homotopy analysis method, it is convenient to first transform the domain of the

problem from

Substituting (15)-(16) in the governing equation and boundary conditions (11)-(14) gives

Subject to

where the primes denote differentiation with respect to

And

The initial approximation is taken to be the solution of the nonhomogeneous linear part of the governing Equa- tions (17)-(19) given by

subject to

We use the Chebyshev pseudospectral method to solve (25)-(28). The unknown function

where

Derivatives of the functions

where r is the order of differentiation and

Subject to

where

And

the last rows and columns of A and delete the first and last rows of

dary conditions (33) are imposed on the resulting first and last rows of the modified matrix A and setting the resulting first and last rows of the modified matrix

To find the SHAM solutions of (17)-(19), we begin by defining the following linear operator:

where

Differentiating (38)-(40) m times with respect to q and then setting

Subject to

where

And

Applying the Chebyshev pseudospectral transformation on (44)-(46) gives

Subject to

where

To implement the boundary conditions above we delete the first and last rows of

and delete the first and last rows and first and last columns of

Thus, starting from the initial approximation,obtain from (34), higher-order approximations

For the problem under investigation, numerical computations are carried out for different flow parameters such as inertia parameter

putation except otherwise stated, we used prandtl number Pr = 0.73, Schmidt number

Figures 1-8 illustrate the effects of the various pertinent parameters on the dimensionless velocity

parameter

Figures 2(a)-(c) illustrate the influence of Eckert number Ec, a viscous dissipation term on the velocity, temperature and concentration distribution. It is observed from these figures that Ec have quite opposite effects on the velocity and the associated thermal boundary layer thickness. That is, velocity and temperature profile of the flow increase with increase in the the viscous dissipation term. This further establishes the fact that Eckert number Ec enhances temperature distribution in a mixed convective flow. Whereas, concentration profile de- creases more quickly for an increasing value of Ec. Figures 3(a)-(c) show the influence of thermophoresis parameter

In

The influence of inertia parameter

The effects of thermophoresis, chemical reaction, Hartmann number, on heat flux and mass flux are presented in Tables 1-3 respectively.

As defined previously in the initial/boundary conditions (14), the velocity, temperature and concentration distributions decay as

Parameter | Present study | N. Kishan [ | ||
---|---|---|---|---|

0.0 | 0.44698 | 0.40591 | 0.44632 | 0.40305 |

0.05 | 0.44996 | 0.41663 | 0.44578 | 0.41264 |

0.1 | 0.44583 | 0.42201 | 0.44515 | 0.42234 |

0.5 | 0.44308 | 0.49971 | 0.44062 | 0.50022 |

1.0 | 0.43552 | 0.5906 | 0.43525 | 0.59970 |

0 | 0.43842 | 0.49973 |

0.5 | 0.41945 | 0.88663 |

1 | 0.41004 | 1.14647 |

1.5 | 0.40398 | 1.35534 |

2.0 | 0.39954 | 1.53512 |

0 | 0.39951 | 1.53512 |

0.2 | 0.42056 | 0.88687 |

0.4 | 0.41988 | 0.88673 |

0.6 | 0.41905 | 0.88655 |

0.8 | 0.41834 | 0.88632 |

1.0 | 0.41755 | 0.88611 |

In this paper, we have used the spectral homotopy analysis method (SHAM) to solve a second-order nonlinear boundary value problem that governs the two-dimensional steady darcy-forcheimer mixed convection flow in fluid saturated porous media in the presence of chemical reaction and viscous dissipation. The non-linear mo- mentum, energy and species boundary layer equations are transformed into ordinary differential equations using suitable similarity variables.The transformed boundary layer equations are solved using SHAM. We have dis- cussed the effects of magnetic field,chemical reaction,viscous dissipation and thermophoresis on the flow pro- files. The main observations are as follows:

Due to stronger magnetic field the dimensioless velocity decreases whereas, temperature and concentration distributions increase.

An increase in the ratio of the buoyancy parameter N corresponds to an increase in velocity profile

The fluid velocity and temperature rise with an increase in Eckert number Ec, a viscous dissipation term but caused a reduction the concentration profile.

The influence of thermophoresis parameter on the local heat transfer rate is very significant when compared with the sherwood number.

Using magnetic field we can control the heat and mass transfer flow characteristics.

The species concentration profiles increases significantly for the increasing chemical reaction parameter

We thank the editor and the referee for their useful comments.