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This paper presents non-Darcy mixed convective flow of an incompressible and viscous fluid in a differentially heated vertical channel filled with a porous material in the presence of a temperature dependent source/sink. The analytical solution of fourth order non-linear ordinary differential equation for temperature field, which is formed by eliminating velocity field from system of governing equations in non-dimensional form, is obtained by using new modified Adomian decomposition method (NMADM) in terms of various parameters. In order to illustrate the interactive influences of governing parameters on the temperature and velocity fields, a numerical study of the analytical solution is performed with respect to three categories of transport processes i) when forced convection is dominated, ii) when forced and natural convection are equal and iii) when natural convection is dominated. Analysis of all categories has revealed that the temperature and velocity profiles are increasing function of modified Darcy number while decreasing function of Forchheimer number.

Non-Darcy mixed convective flow and heat transfer between two vertical walls and annuli filled with porous materials have been extensively studied in the past and attracted attention of research workers due to its many engineering applications. Some of their related applications are, for example, geothermal energy extraction, nuclear waste disposal, thermal insulation, solid matrix heat exchanger, thermal energy storage in underground aquifers, etc. The analytic solutions corresponding to different physical situations in a vertical channel are presented by several authors [

The numerical calculation of the fully developed mixed convection in a heated vertical channel filled with a porous medium and imposed uniflux at the plates is performed by Chen et al. [

The main purpose of the present paper is to examine non-Darcy mixed convection in a differentially heated vertical channel filled with a porous material in the presence of temperature dependent source/sink. The analytical solution of the temperature field has been obtained using new modified Adomian decomposition method given by Wazwaz and El-Sayed [

In order to illustrate the basic idea of this method, let us consider an ordinary differential equation taken by Wazwaz [

with prescribed boundary conditions. Here u is unknown function, L is an easily invertible linear operator having highest order derivative and R is a linear differential operator of order less than L. Nu represents the non-linear terms and f is the source terms. Applying the inverse operator

Differences among different Adomian decomposition methods can be distinguish on the basis of dividing the source term

i) one component then it is known as ADM,

ii) two components then it is known as MADM,

iii) n components then it is known as NMADM, where n is the degree of

According to NMADM, we therefore express

The standard Adomian method defines the solution of u(x) by the series

where the components

Consider the non-Darcy mixed convective flow of a viscous incompressible fluid between two vertical plane walls separated by a distance 2H and filled with a porous material in the presence of a temperature dependent source/sink and under a constant pressure gradient. We employ here a Cartesian coordinate system with origin at the central line of the channel having

the governing equations of the model such as conservation of momentum and thermal energy in non-dimensional form are obtained as follows (Chen et al. [

The first term in the R.H.S. of Equation (7) is the Brinkman term, the second is the Darcy and the third is the Forchheimer term, hence the momentum transfer in the porous region is based on the Brinkman-Forchheimer-ex- tended Darcy model. The boundary conditions of the problem in non-dimensional form are obtained as

All the symbols used in the above equations are defined in the nomenclature.

Now, putting the value of u from Equation (8) into the Equation (7), we get a fourth order non-linear ordinary differential equation in of the form:

where

For the solution of Equation (10), we use new modified Adomian decomposition method, proposed by Wazwaz and El-Sayed [

Applying the inverse operator

where

Now, we can decompose θ as

and according to Equation (12) source function

As suggested by Wazwaz and El-Sayed [

According to Equation (13), the expression for temperature field can be written as

The symbols used in the above expressions as a constant are given in appendix. After obtaining the solution for temperature field in the form in Equation (16), we have derived the solution for velocity field by using Equation (8) but not presented here for the sake of brevity.

In order to measure the shear stress and the heat transfer rate at the boundaries, the expressions for the skin friction in non-dimensional form and Nusselt number on the channel walls have been obtained by using the following relations respectively:

But they are not given here for the sake of brevity.

We investigate the interactive effects of the dimensionless parameters such as modified Darcy number Da, Forchheimer number F, source/sink parameter Q and wall temperature ratio m, on heat and fluid flow between two vertical walls having differentially heating corresponding to the following categories of flow formation:

i) when forced convection is dominated (

ii) when both convections are equal (

iii) when natural convection is dominated (

In the presence of source

In all three cases, we observed that the temperature and velocity profiles are increasing function of modified Darcy number while decreasing function of Forchheimer number. The effect of Darcy number is more visible in the case of natural convective dominated flow. The fluid velocity near the wall region slowly increases with the increase of Da, but it is more significant in the region near to the walls when flow is dominated by natural convection.

We have plottedin the

Figures 5(a)-7(a), Figures 5(b)-7(b) illustrate the effects of wall temperature ratio (m) on temperature and velocity profiles corresponding to cases three types of categories. From these figures in the presence of source

parameter, it shows that when m increases, the flow near wall 1 is more and more accelerated and flow reversal is observed in the vicinity of wall 2. Meanwhile wall 1 becomes increasingly hotter relatively to wall 2 and the temperature profiles approaches a linear distribution. In the presence of sink parameter, the temperature profile slopes on both walls remain constant in accordance with the differentially heating conditions. As

A close study of the

Q | Categories | At right wall (y = 1.0) | At left wall (y = −1.0) | ||||||
---|---|---|---|---|---|---|---|---|---|

F = 0.2 m = 0.5 Da = 0.01 | F = 0.2 m = 1.0 Da = 0.01 | F = 1.0 m = 1.0 Da = 0.01 | F = 1.0 m = 0.5 Da = 0.05 | F = 0.2 m = 0.5 Da = 0.01 | F = 0.2 m = 1.0 Da = 0.01 | F = 1.0 m = 1.0 Da = 0.01 | F = 1.0 m = 0.5 Da = 0.05 | ||

5 | i) | 3.82514 | 3.89299 | 4.1029 | 4.08650 | 1.97408 | 3.89299 | 3.96088 | 1.84139 |

ii) | 1.29565 | 1.61654 | 1.6658 | 1.32525 | 0.323148 | 0.616548 | 0.624665 | 0.1652 | |

iii) | 0.42969 | 0.440571 | 0.8364 | 0.488220 | 0.24551 | 0.44057 | 0.007820 | 0.00477 | |

0 | i) | 3.00068 | 3.19296 | 3.6410 | 3.24082 | 2.255497 | 3.18796 | 3.22421 | 2.81004 |

ii) | 1.49129 | 1.79016 | 2.1718 | 1.61191 | 1.71632 | 1.79266 | 2.38322 | 1.82967 | |

iii) | 0.97701 | 1.77142 | 1.8718 | 1.05870 | 1.42852 | 1.776423 | 2.09665 | 1.49581 | |

−5 | i) | 4.19511 | 4.84815 | 4.8692 | 3.90279 | 4.08168 | 4.73043 | 5.58713 | 5.55570 |

ii) | 3.40051 | 3.83252 | 3.9902 | 3.00829 | 3.80597 | 3.86243 | 5.48712 | 5.48357 | |

iii) | 3.13667 | 3.72504 | 3.8168 | 2.8008 | 3.71639 | 3.72070 | 5.45826 | 5.43080 |

Q | Categories | At right wall (y = 1.0) | At left wall (y = −1.0) | ||||||
---|---|---|---|---|---|---|---|---|---|

F = 0.2 m = 0.5 Da = 0.01 | F = 0.2 m = 1.0 Da = 0.01 | F = 1.0 m = 1.0 Da = 0.01 | F = 1.0 m = 0.5 Da = 0.05 | F = 0.2 m = 0.5 Da = 0.01 | F = 0.2 m = 1.0 Da = 0.01 | F = 1.0 m = 1.0 Da = 0.01 | F = 1.0 m = 0.5 Da = 0.05 | ||

5 | i) | 0.241903 | 0.208009 | 0.27780 | 0.29610 | −0.20800 | −0.60056 | −0.23450 | −0.15620 |

ii) | 0.13017 | 0.09215 | 0.10850 | 0.16350 | −0.90782 | −0.09279 | −0.06770 | −0.16060 | |

iii) | 0.26350 | 0.08780 | 0.22730 | 0.29890 | −0.18240 | −0.08570 | −0.16520 | −0.26460 | |

0 | i) | 0.77320 | 0.46930 | 0.64540 | 0.75000 | −0.31510 | −0.46380 | −0.18430 | −0.29520 |

ii) | 0.96470 | 0.74550 | 0.83420 | 0.95000 | −0.44070 | −0.74560 | −0.31730 | −0.43870 | |

iii) | 1.02970 | 0.83350 | 0.89320 | 1.02024 | −0.48500 | −0.83360 | −0.36102 | −0.48094 | |

−5 | i) | 1.54960 | 1.22200 | 1.22210 | 1.41892 | −0.64670 | −1.22320 | −0.26430 | −0.43160 |

ii) | 1.56722 | 1.29240 | 1.34240 | 1.43290 | −0.62317 | −1.29298 | −0.24740 | −0.41050 | |

iii) | 1.5600 | 1.31870 | 1.44144 | 1.44580 | −0.62600 | −1.31183 | −0.24917 | −0.4740 |

However, the Nusselt number at the right wall (y = 1) in each category positive, which indicates physically that heat flows from the walls into the porous region. We observed that Nusselt number for each category decreases with wall temperature ratio m and increases with Forchheimer number F. On comparing the Nusselt number for each category of the wall at y = 1.0 and y = −1.0, it is observed that it increases with Da when

New modified Adomian decomposition method has been used to find the analytical solution of the governing equations describing the non-Darcy mixed convective flow of an incompressible and viscous fluid in a differentially heated vertical channel filled with a porous material in the presence of a temperature dependent source/ sink. The effect of all physical parameters appearing in the governing equations is more visible when natural convection is dominated in comparison with fully forced convection in the presence of source parameter. The impact of all categories is more effective in the presence of source than sink. Also, it has been observed that the numerical values of all skin friction components increase with the increase of wall temperature ratio (m) and Forchheimer number (F). Increasing value of Gr from P i.e. forced convection to natural convection induces flow acceleration near both walls and consequently flow deceleration in the central line of the channel. Eventually this tendency can lead to flow reversal at the centerline but the corresponding value of Gr either greater, equal or less than P is of questionable physical signification (see discussion by Barletta and Zanchini [

A. K.Tiwari,PremlataSingh, (2015) Non-Darcy Mixed Convection between Differentially Heated Vertical Walls Filled with a Porous Material: Application of New Modified Adomian Decomposition Method. Open Journal of Fluid Dynamics,05,380-390. doi: 10.4236/ojfd.2015.54037

Da Modified Darcy number, defined in Equation (1)

F Forchheimer number, defined in Equation (1)

g Acceleration due to gravity

Gr Grashof number or free convection parameter, defined in Equation (1)

m wall temperature ratio, defined in Equation (1)

P constant pressure gradient,

p Dimensionless fluid pressure defined in Equation (1)

Q dimensionless source/sink parameter defined in Equation (1)

u Dimensionless velocity component

x, y Dimensionless Cartesian coordinates