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The dynamics of steady, two-dimensional magnetohydrodynamics (MHD) free convective flow of micropolar fluid along a vertical porous surface embedded in a thermally stratified medium is investigated. The ratio of pressure drop caused by liquid-solid interactions to that of pressure drop caused by viscous resistance are equal; hence, the non-Darcy effect is properly accounted for in the momentum equation. The temperature at the wall and at the free stream which best accounts for thermal stratification are adopted. Similarity transformations are used to convert the nonlinear partial differential equation to a system of coupled non-linear ordinary differential equation and also to parameterize the governing equations. The approximate analytical solution of the corresponding BVP are obtained using Homotopy Analysis Method (HAM). The effects of stratification parameter, thermal radiation and other pertinent parameters on velocity, angular velocity and temperature profiles are shown graphically. It is observed that increase in the stratification parameter leads to decrease in both velocity and temperature distribution and also makes the microrotation distribution to increase near the plate and decrease away from the plate. The influence of both thermal stratification and exponential space dependent internal heat source on velocity, micro-rotation and temperature profiles are presented. The comparison of the solutions obtained using analytical techniques (HAM) and MATLAB package (bvp4c) is shown and a good agreement is observed.

The study and structure of a fluid as it flows over a surface results in vertical density variations which is of great importance in industry due to its vast application in industry. Dake and Harleman [

Micropolar fluids are fluids with internal structures or micro-structures which belong to a class of fluids with nonsymmetric stress tensor that can be called polar fluids. The theory of micropolar fluids introduced by Eringen [

Porous medium is a very important aspect in Science and Engineering which is described as a medium or material that contains pores or spaces between solid materials or solid matrix through which liquids or gases can pass. Common examples of naturally occurring porous medium include sand, soil, sandstone, sponges, ceramics and foams. Fluid flow in porous media is an important dimension in many areas of reservoir engineering, such as petroleum, environmental and groundwater hydrology. A number of studies have been reported in the literature focusing on the problem of combined heat and mass transfer in porous media and the analysis of convective transport in a porous medium with the inclusion of non-Darcian effects has also been a matter of study in recent years. Non-Darcy behavior is important for describing fluid flow in porous media in situations where high velocity occurs. Hence, due to its important applications in many fields, a full understanding of heat transfer by non-Darcy natural convection from a heated vertical surface embedded in fluid saturated porous medium is meaningful. Mohammed et al. [

In the literatures above, little attention has been given to investigate free convective micropolar fluid flow along a vertical surface embedded in non-Darcian thermally medium. In addition, no attempt has been made to investigate the behaviour of micro- polar fluid in the presence of exponential space dependent and temperature dependent internal heat source along a vertical surface embedded in non-Darcian thermally stratified porous medium using Homotopy Analysis Method. In view of this, it is imperative to highlight that, the present study will offer helpful information to scientists and engineers in industry.

We consider steady two-dimensional free convective boundary layer flow of an incompressible, electrically conducting micropolar fluid along a vertical surface embedded in non-Darcian thermally stratified porous medium. Keeping the origin fixed, the sheet is then stretched with a velocity

Continuity Equation

Momentum Equation

Angular Momentum Equation

Energy Equation

Subject to boundary conditions

In this study, wall temperature and free stream temperature are defined as

where u and v are components of velocity in x and y directions respectively,

stream temperature,

diffusivity,

The Micropolar parameter or material parameter is

fluid and

where

and then neglecting higher order terms beyond the first degree in

In view of the Equations (8) and (10), Equation (4) becomes;

The continuity Equation (1) is satisfied by introducing a stream function

The momentum, angular momentum and energy equations can be transformed into the corresponding ordinary differential equations by the following transformation

where

subject to the boundary conditions (5) and (6) which become;

In the above equations, primes denote differentiation with respect to

and

terest are the skin friction coefficient

where the wall shear stress

where

Nonlinear differential equations are usually arising from mathematical modeling of many physical systems. Some of them are solved using numerical methods and some are solved using the analytic methods such as perturbation techniques, Adomian Decomposition and d-expansion method. Generally speaking, it is still difficult to obtain analytical solutions of nonlinear problems. In this research, Homotopy analysis method is considered as a method of solution because of its efficiency as an approximate solution of linear and nonlinear differential equations and also; HAM is valid for strongly nonlinear problems even if a given nonlinear problem does not contain any small/large parameters. Animasaun et al. [

where N is a nonlinear operator,

Instead of using the traditional Homotopy

we considered a nonzero auxilary parameter

When

Next step is to find solution of

Equation (26) is the solution of

When

Consider the solution of

Equating to Equation (22)

Algebrically,

In many cases, by mean of analyzing the physical background and the initial/ boundary conditions of the nonlinear differential problem, we might know what kinds of base functions are proper to represent the solution, even without solving the given nonlinear problem. In view of the boundary conditions (17) and (18),

The solutions

In which

Linear operators

The operators

In which

Subject to boundary conditions

where the nonlinear operators are defined as

when

With the property

Subject to

when

Based on the fact that

Equating Equations (54) to (56) with Equations (22), we have

Subject to

Expanding

The auxiliary parameters are so properly chosen that the series (62), (63) and (64) converge at

For the mth order deformation, differentiate (38) to (40) m times with respect to q, divide by

Subject to

where

And

The general solutions of equations are given by

Here,

In order to gain an insight into the behavior of the fluid as it flows, analytic approximate solution of the dimensionless governing equation described in the previous section has been carried out using various values of stratification parameter

In addition, it is also noticed from the table that the magnitude of the local heat transfer rate increases with an increase in

Pertinent to inquire the effects of variation of

Figures 1-3 represent the velocity, micro-rotation and temperature profiles with variation in the magnitude of thermal stratification parameter

HAM | bvp4c | |||||
---|---|---|---|---|---|---|

0.4 | −1.07323092325528 | 0.29572619181767834 | −0.1817220296067319 | −1.0728 | 0.2953 | −0.1800 |

0.6 | −1.14391400312143 | 0.31350600136745343 | −0.1402800127453434 | −1.1436 | 0.3132 | − 0.1397 |

0.8 | −1.21490100151245 | 0.33164300216441233 | −0.0961960011275332 | −1.2140 | 0.3315 | −0.0960 |

Order of approximation | |||
---|---|---|---|

1 | −1.102850698204918 | 0.3142916584602799 | −0.298855474342964 |

2 | −1.091353120619502 | 0.3070703970974996 | −0.256685615712014 |

3 | −1.095907204113479 | 0.3074890117246342 | −0.217902469545101 |

4 | −1.079781088675545 | 0.309348721194765 | −0.208535910987546 |

5 | −1.080611230231204 | 0.3023404211947201 | −0.194456011581959 |

6 | −1.076262532448428 | 0.300770604344728 | −0.192129902182097 |

7 | −1.075622898014302 | 0.2981949587200132 | −0.186197762941896 |

8 | −1.074421204429870 | 0.2970219453865645 | −0.208444518149192 |

9 | −1.073223644756899 | 0.2952554670367616 | −0.181690035630036 |

10 | −1.073230923255284 | 0.2957261918176783 | −0.181722029606731 |

this can be traced to the fact that, as

and this results in reduction in the velocity distribution and likewise micro-rotation distribution and this also corresponds to reduction in the thickness of velocity boundary layer.

The variation of the dimensionless velocity, micro-rotation, temperature profiles for different values of magnetic parameter M is illustrated in Figures 8-10 respectively. It is observed from

the velocity of the fluid and angular velocity of micro-rotation in the boundary layer and to increase its temperature. It is observed from

Figures 11-13 depicts the effect of space-dependent and temperature-dependent heat source parameters A and B. It is shown that increase in A and B leads to an increase in

velocity, micro-rotation and temperature profiles respectively. The presence of the exponential term in the space-dependent heat source is to produce additional heat energy across the fluid region, leading to increase in velocity and temperature of the fluid and also the thickness of the velocity and thermal boundary layer increase. It is revealed in

the surface. In other words, we can still say that when heat source parameters A and B have increasing positive values, it is observed that substantial heat source will be generated within the fluid domain and hence influence the thermal boundary layer.

Convergence of the Homotopy SolutionIt is obvious that the series Equations (65)-(67) consists of the non-zero auxiliary parameters

set of admissible values of

approximate optimal values of

The study describes free convective boundary layer flow of a conducting micropolar fluid in the presence of exponential space and temperature dependent heat source is analyzed, the set of non-linear ordinary differential equations are then solved by an analytic approximate techniques (Homotopy Analysis Method) and the behaviours of embedded parameters are investigated. The following conclusions are drawn from the

analysis:

1) Velocity profiles and micro-rotation profiles are strongly influenced by the magnetic field in the boundary layer, which decreases with increase in the Magnetic parameter M.

2) Increase in the stratification parameter

3) Increasing the value of Micropolar parameter results in increase in micro-rotation profile.

4) Micro-rotation profile has a parabolic distribution when micro-gyration para- meter

5) Micropolar fluids reduce the shear stresses and enhance couple stress as compared to Newtonian fluids.

6) Variation of stratification parameter result in decrease in the local skin friction coefficient and increase the couples stress and local heat transfer rate respectively.

Koriko, O.K., Ore- yeni, T., Omowaye, A.J. and Animasaun, I.L. (2016) Homotopy Analysis of MHD Free Convective Micropolar Fluid Flow along a Vertical Surface Embedded in Non- Darcian Thermally-Stratified Medium. Open Journal of Fluid Dynamics, 6, 198-221. http://dx.doi.org/10.4236/ojfd.2016.63016