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Comment on “Heat Generation/Absorption and Viscous Dissipation Effects on MHD Flow of a Micropolar Fluid Over a Moving Permeable Surface Embedded in a Non-Darcian Porous Medium, by Mostafa A. A. Mahmoud, Journal of the Korean Physical Society, Vol. 54, No. 4, April 2009, pp. 1526-1531”
Asterios Pantokratoras
School of Engineering, Democritus University of Thrace,
67100 Xanthi – Greece
e-mail:
In the above paper the flow and heat transfer of an electrically conducting micropolar fluid on a continuously moving plate embedded in a porous medium in the presence of a transverse magnetic field and heat generation or absorption has been studied numerically. The transformed ordinary differential equations are solved by employing the Runge-Kutta scheme coupled with the shooting technique. The effects of various parameters on velocity, temperature and microrotation are shown graphically.
The problem is described by the following partial differential equations
(1)
(2)
(3)
(4)
where u and v are the velocity components in x and y direction, υ is the fluid kinematic viscosity, k1 is the coupling constant, σ is the microrotation, G1 is a microrotation constant, k is the porous medium permeability, φ is the porosity of the porous medium, c is a constant, σ0 is the fluid electrical conductivity, B0 is the strength of the magnetic field, ρ is the fluid density, T is the fluid temperature, K is the fluid thermal conductivity, Cp is the specific heat and Q is the coefficient of heat generation.
Let us consider the case of a normal fluid (not micropolar) without porous medium in a magnetic field. In this case the momentum equation (2) takes the form
(5)
Equation (5) expresses the flow along a plate moving with constant velocity inside a constant magnetic field. The momentum equation (2) has been transformed by Mahmoud (2009) as follows
(6)
where prime denotes differentiation with respect to similarity variable
(7)
and the stream function is
(8)
M is the magnetic parameter
(9)
and L and Da are the microrotation and Darcy parameters. For the case of a normal fluid (not micropolar) without porous medium in a magnetic field the transformed equation (6) takes the form
(10)
The above equation (10) corresponds to equation (5).
This special case of equation (5) is included in the work of Kumari and Nath (2001) which concerns the boundary layer flow of a non-Newtonian fluid over a continuously moving plate within a parallel free stream. The transformed momentum equation in the work of Kumari and Nath (2001, equation 6) is the following
(11)
For a Newtonian fluid along a plate moving with constant velocity in a calm fluid (without free stream) the quantities N and λ are N=1 and λ=1 and the equation (11) takes the form
(12)
where M* is the magnetic parameter defined as
(13)
where L* is a length scale. The above equation (12) corresponds again to equation (5). The transformed equation (10) by Mahmoud (2009) and the transformed equation (12) by Kumari and Nath (2001) are equivalent and both refer to the same momentum equation (5). However, it is clear that equations (10) and (12) are quite different although both correspond to the same momentum equation. Τhe similarity variable η and the stream function f are defined with the same way in the two papers except of a factor 2 and the quantity M*ξ () in equation (12) is the same with magnetic parameter M in equation (10). The main difference is the two terms in the right hand side of equation (12) which are missing from equation (10) by Mahmoud (2009). It is reminded here that the present problem is non-similar, that is, the velocity, temperature and the other physical quantities change along the plate and are functions of the distance x (Kumari and Nath, 2001, Pantokratoras, 2011, Pantokratoras and Fang, 2011). Kumari and Nath (2001) considered the problem as non-similar in contrast to Mahmoud (2009) who made an approximation and considered the problem as locally similar, although this is not true.
Taking into account all the above the credibility of the results presented by Mahmoud (2009) is doubtful.
References
1. Mostafa A. A. Mahmoud (2009). Heat Generation/Absorption and Viscous Dissipation Effects on MHD Flow of a Micropolar Fluid Over a Moving Permeable Surface Embedded in a Non-Darcian Porous Medium, Journal of the Korean Physical Society, Vol. 54, No. 4, pp. 1526-1531.
2. Kumari, M., Nath., (2001). MHD boundary-layer flow of a non-Newtonian fluid over a continuously moving surface with a parallel free stream, Acta Mechanica, Vol. 146, pp. 139-150.
3. A. Pantokratoras (2011). Further results on hydromagnetic boundary-layer flow of a non-Newtonian power-law fluid over a continuously moving surface with suction, Chemical Engineering Communications, Vol. 198, pp. 1405-1414.
4. A. Pantokratoras and T. Fang (2011). A note on the Blasius and Sakiadis flow of a non-Newtonian power-law fluid in a constant transverse magnetic field, Acta Mechanica, Vol. 218, pp. 187-194.