Natural Convection of MHD over a Vertical Wavy Surface in Presence of Porous Media ()
1. Introduction
This paper studies the effects of porous medium of a vertical wavy surface in presence of the magnetic field and heat transfer, and uses a different method of numerical solution to solve the problem (finite difference-fully implicit method). Heat generation effects on MHD natural convection flow along a vertical wavy surface with variable thermal conductivity were investigated by Md. Abdul Alim et al. [1].
The study of this paper has a wide application in the fields of geology and biology. Many authors presented solutions for this type of flow just like that, mass and Heat transfer in porous media was studied by Vadasz, P. (ed) [2]. Wang, C.Y. [3] investigated the free convection over a vertical stretching surface. Yao [4] studied the natural convection over a vertical complex wavy surface. The natural convection flow of Non-Darcy over a vertical wavy plate embedded in a non-Newtonian fluid-saturated porous medium was investigated by Hady, F.M., et al. [5]. Cheng, P. and Minkowycz, W. J. [6] studied the free convection along a vertical flat plate with application to transfer from a dike embedded in a porous medium. The free convection flow of boundary layer along a vertical surface in a porous medium with Newtonian heating is also investigated by Lesnic, D., Ingham, D. B., et al. [7]. The effectiveness of convection heat transfer on a flat plate in porous media for power-law fluids has been investigated by Hady, F. M. and Ibrahim, F. S. [8]. Tashtoush and Al-odat [9] investigated the effect of Magnetic field on fluid flow with a variable heat flux over a wavy surface. The natural convection flow with uniform surface temperature over a vertical curvy surface in presence of heat generation/absorption was investigated by Molla et al. [10].
The system of equations will be converted from partial differential equations into dimensionless form. The nonlinear system which is obtained is solved numerically by finite difference (fully implicit method) [11]. The effectiveness of inverse Darcy number, wavy surface amplitude, parameter of magnetic field, Prandtl number, variation of thermal conductivity and heat generation on velocity, temperature and Nusselt number will be obtained.
Formulation of Mathematic
It is considered the free convection boundary layer flow in two-dimensions is steady over vertical wavy surfaces and porous media. It is a viscous incompressible fluid and electrically conducting. There is magnetic field of strength
. The uniform temperature at the surface
and
is the temperature at the fluid, where
.
Assume that the surface geometry function
is arbitrary. Consider the equation of the wavy surface could be determined as:
(1)
whereas
is the dimensional axis along the curvy surface,
is dimensional axis perpendicular to the curvy surface,
is the distance related to the curvy surface and a is the amplitude of curvy surface. It is shown in a Figure 1 that the wavy
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Figure 1. Physical model and coordinate system
surface and the system of Cartesian coordinates. Consider that there is no Joule heat effect and neglect the heat generation, viscous dissipation and body forces of the system.
With introducing Boussinesq approximation, the governing equations are steady, laminar and in two-dimensions. Then the following equations are the continuity equation, momentum equations and energy equation are written as:
(2)
(3)
(4)
(5)
The boundary conditions
(6)
where
are the velocity components in direction of
respectively, the density
,
is the dimensional pressure of fluid,
the Laplacian operator,
the magnetic induction,
the electrical conduction,
the thermal expansion coefficient,
the kinematic viscosity,
the acceleration of gravity,
the thermal conductivity of fluid,
the specific heat at the constant pressure,
the saturated porous Medium permeability,
the outside pressure of the boundary layer of the fluid and
is the constant of heat generation.
Transformed the curvy surface into a flat surface and boundary layer approximation by Using Prandl’s transposition theorem. The dimensionless variables are:
(7)
whereas
are the dimensionless coordinates and
are the dimensionless velocity components in
directions,
the function of dimensionless temperature,
the dynamic viscosity,
the Grashof number and
the dimensionless pressure by substituting the previous dimensionless variables into the equations (2)-(5). After ignoring terms of smaller orders of magnitude of
(the Grashof number), the system of equations are obtained, then the governing equations become in form:
(8)
(9)
(10)
(11)
whereas
the parameter of thermal conductivity,
the parameter of heat generation,
the magnetic field, Prandtl number is
the and
is Darcy number.
By multiplying Equation (10) by
. The equation becomes:
(12)
In this problem, the inviscide flow field is at rest, therefore
. By adding the Equations (9), (10), thus can be eliminate
from these equations. The equation is:
(13)
The governing equations could be in form:
(14)
(15)
(16)
by using the dimensionless formulation (7) in The boundary conditions (6), it becomes in form:
(17)
the physical quantities such as
is the shearing stress concerning the coefficient of skin friction
and the rate of heat transfer with reference to Nusselt number
, their equations:
and
(18)
By using the dimensionless formulation (7), the coefficient of local skin friction
and the heat transfer rate with reference to Nusselt number Nu will be like that:
(19)
(20)
whereas
and
.
2. Method of Solution
The Equations (14)-(16) are nonlinear partial differential and boundary conditions (17) can be solved numerically by finite difference fully (implicit method). By using the implicit method The derivatives w.r.t x1 and x2 are approximated by using central difference.
For example the momentum equation of q1:
(15)
In the above,
is a weighting factor. Take the value of
.
This equation will be written in form:
where
3. Results
The numerical results are represented by graphs for the velocity q1 and temperature θ and Nusselt number Nu*. Use the different values of parameters to illustrate their effective on the velocity, temperature and Nusselt number. Figure 2 reveals the influence of magnetic parameter MÄ to velocity profiles q1, this figure shows that the increasingof magnetic parameter tends to increase the velocity q1 at the other parameters are constant as prÅ = 0.7, q* = 0.1, a = 0.2, DaÅ−1 = 0.1 and γ* = 2. Figure 3 indicates to the effective of increase the inverse Darcy number tends to increase velocity q1, at prÅ = 0.7, q* = 0.1, a = 0.2, MÄ = 0.2 and γ* = 2. Figure 4 shows that the heat generation q* increase tends to the velocity q1 increase at prÅ = 0.7, DaÅ−1 = 0.1, a = 0.2, MÄ = 0.2 and γ* = 2. Figure 5 illustrated that the thermal conductivity variation increase γ tends to the velocity q1 increase at prÅ = 0.7, DaÅ−1 = 0.1, a = 0.2, MÄ = 0.2 and q* = 0.1. Figure 6 indicates
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Figure 2. Velocity distribution q1 for different magnetic parameter MÄ.
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Figure 3. Velocity distribution q1 with various inverse Darcy number.
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Figure 4. Velocity distribution q1 for different heat generation parameter q*.
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Figure 5. Velocity distribution q1 for different thermal conductivity variationγ*.
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Figure 6. The temperature θÄ with various magnetic.
to the influence of increase the magnetic field tends to decrease the temperature θÄ, at prÅ = 0.7, q* = 0.1, a = 0.2, DaÅ−1 = 0.1 and γ* = 2. Figure 7 and Figure 8 present the temperature θ* decrease with increase both the inverse of Darcy number and prandtle number prÅ at q* = 0.1, a = 0.2, MÄ = 0.2 and γ* = 2. Figure 9 and Figure 10 indicate to influence of prandtle number and magnetic field on Nusselt number, the increase of prandtle number and magnetic field parameter tends to increase the Nusselt number Nuat q* = 0.1, a = 0.2, DaÅ−1 = 0.1 and γ* = 2. Figure 11 and Figure 12 present increase the Nusselt number cause increase both of inverse Darcy number and heat generation parameter q* at prÅ = 0.7, a = 0.2, MÄ = 0.2 and γ* = 2. Figure 13 shows that the increase in thermal conductivity variation parameter γ tends to increase the Nusselt number Nu* at MÄ = 0.2, DaÅ−1 = 0.1, a = 0.2, q* = 0.1, and prÅ = 0.7.
4. Conclusion
This paper presents the influence of porous medium wavy vertical surface of
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Figure 7. The temperature θÄ with various inverse Darcy number DaÅ−1.
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Figure 8. The temperature θÄ for different prandtle number PrÅ.
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Figure 9. Nusselt number with various prandtle number.
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Figure 10. Nusselt number with various magnetic parameter.
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Figure 11. Nusselt number with various inverse Darcy number.
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Figure 12. Nusselt number with various heat generation.
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Figure 13. Nusselt number with various thermal conductivity.
natural convection flow with thermal conductivity variable. The system of dimensional partial differential equations is turned into equations of partial differential whose non-linear, they are solved numerically by using finite difference (fully implicit method). This paper has a new result of influence of inverse Darcy number with the velocity, temperature and Nusselt number. The results have been obtained to velocity, temperature and Nusselt number with various parameters, such as magnetic field, Prandtl number, amplitude of wavelength, heat generation parameter and thermal conductivity variation parameter.
Nomenclature
a the dimensionless amplitude of the wavy surface
magnetic induction
magnetic field parameter
characteristic reference wave-length
local Nusselt number
the dimensional pressure of fluid
Prandtl number
Grashof number
electrical conductivity
gravitational acceleration
temperature
axial and normal dimensionless velocity Components, respectively
the velocity components along
(
)
dimensionless coordinates
dimensional coordinates
Darcy number
the dimensionless pressure.
heat generation parameter
heat generation constant
specific heat at constant pressure
permeability of the saturated permeable Medium
thermal conductivity
Greek Symbol
kinematics viscosity
density
surface geometry function
dimensionless temperature
the dynamic viscosity
coefficient of thermal expansion
thermal conductivity variation parameter
the shearing stress
Superscripts
dimensional quantity
Subscripts
w wall surface
free stream
derivative with respect to x