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## Latin American applied research

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*versión impresa* ISSN 0327-0793

### Lat. Am. appl. res. vol.39 no.4 Bahía Blanca oct. 2009

**ARTICLES**

**Heat and mass transfer in MHD free convection along a vertical wavy plate with variable surface heat and mass flux**

**A. Mahdy ^{1,3}, R.A. Mohamed^{2} and F.M. Hady^{2}**

^{1} *Math. Department, Science, South Valley University, Qena, EGYPT.*

^{2} *Math. Department, Science, Assiut University, Assiut, EGYPT*

^{3} *sci_eng@yahoo.com*

** Abstract - **The problem of combined heat and mass transfer in buoyancy-induced MHD natural convection flow of an electrically conducting fluid along a vertical wavy plate with power-law variation of both heat and mass flux was investigated. The resulting transformed governing equations are solved numerically by an implicit finite-difference scheme. The results are presented for the major parameters including the wave amplitude

*a*, the magnetic parameter

*Mn*, the buoyancy ratio between species and thermal diffusion

*Br*, the Lewis number

*Le*, and the power-law parameter

*λ*. A systematic study on the effects of the various parameters on flow, heat and mass transfer characteristics is carried out.

* Keywords* - Wavy Plate; Heat Mass Flux; Magnetic Field; Porous Media.

**I. INTRODUCTION**

Investigation of magneto-hydrodynamic flow for an electrically conducting fluid past a heated surface has attracted the interest of many researchers in view of its important applications in many engineering problems such as plasma studies, petroleum industries, MHD power generators, cooling of nuclear reactors, the boundary layer control in aerodynamics, and crystal growth. This study has been largely concerned with the flow and heat transfer characteristics in various physical situations. For example, Watanabe and Pop (1994) investigated the heat transfer in thermal boundary layers of magneto-hydrodynamic flow over a flat plate. Kumari *et al.* (1990) treated convection in porous media near a horizontal uniform heat flux surface. Elbashbeshy (1997) studied heat and mass transfer along a vertical plate in the presence of a magnetic field. Chamkha (1997) considered the problem of MHD natural convection from an isothermal inclined plate embedded in a thermally stratified porous medium. Chamkha and Khaled (2001) investigated the problem of coupled heat and mass transfer by hydromagnetic free convection from an inclined plate in the presence of internal heat generation or absorption, and similarity solutions were presented. Chen (1998) studied the mixed convection heat transfer from a horizontal plate with variable surface heat flux in a porous medium. However, the monograph by Gebhart *et al.* (1998) and Nield and Bejan (1999) provided an overview of the early studies concerning the natural convection boundary-layer flow due to simultaneous heat and mass transfer over heated surfaces with various geometries.

In the other hand, few studies have been carried out to examine the effect of geometric complexity, such as irregular surfaces, on the convection heat transfer. That is because complicated boundary conditions or external flow fields are difficult to deal with. However, the prediction of heat transfer from an irregular surface is of fundamental importance, and is encountered in several heat transfer devices, such as flat-plate solar collectors and flat-plate condensers in refrigerators. Moreover, surfaces are sometimes intentionally roughened to enhance heat transfer for the presence of rough surfaces disturbs the flow and alters the heat transfer rate. However, all of the previous studies considered only the case of a flat plate or simple two-dimensional bodies, and few have been done on wavy surfaces.

Cheng (2000) studied the phenomenon of natural convection heat and mass transfer near a vertical wavy surface with constant wall temperature and concentration in a porous medium. Rees and Pop (1994a) investigated the effects of transverse surface waves on the free convective boundary layer induced by a uniform heat flux vertical surface embedded in a porous medium. Rees and Pop (1994 b,c and 1995) done many of related papers. The magnetohydrodynamic (MHD) flow and heat transfer from a horizontal wavy surface with variable heat flux due to the effect of magnetic fields on flow control and performance is considered by Bourhan and Al-Odat(2004). Wang and Chen (2005) reported the effect of magnetic field on forced heat and fluid flow over a wavy surface subjected to a heat flux proportional to (where n is constant). Hossain and Pop (1996) and Hossain *et al.* (1997) investigated the problem of magnetohydrodynamic boundary layer flow and heat transfer on a continuous moving wavy surface and the problem of magnetohydrodynamic free convection along a vertical wavy surface.

In the formulation of our problem it can be demonstrated that the system of momentum, heat and mass conservation equations can be reduced to a five parameter problem by introducing suitable transformation variables. The major problem parameters include the amplitude of the wavy surface *a*, the magnetic parameter *Mn*, the Lewis number *Le*, the buoyancy ratio *Br*, the heat and mass flux exponent *λ*. The resulting transformed governing equations are then solved using the method of Runge-Kutta. The results allow us to predict the different behavior for the velocity, temperature, and concentration distributions that can be observed when the relevant parameters are varied.

**II. MATHEMATICAL FORMULATION**

Consider the steady-state with magnetohydrodynamic combined heat and mass transfer by natural convection flow aligned parallel to a permeable wavy plate. Both the heat and mass flux are assumed to vary with the distance from the leading edge along the plate according to a power-law model. The fluid is assumed to be electrically conducting. With measured along the plate, a magnetic field of strength *B* is applied in the -direction that is normal to the streamwise direction. The coordinate system is chosen such that measures the distance along the wavy surface and measures the distance normal outward see Fig. (1). Here we consider a wavy surface that is described by

(1) |

**Fig. 1**. Schematic of the problem

Under the foregoing assumptions with introducing Boussinesq approximations, the equations governing the steady-state, conservation of mass, momentum, energy, and constituent for MHD Darcy flow through a homogeneous porous medium can be written in two-dimensional Cartesian coordinates , in usual notation as,

(2) | |

(3) | |

(4) | |

(5) | |

(6) |

The associated boundary conditions are:

(7) |

In the above equations, *B* is the magnetic field strength, *C* is the concentration, *D* is the mass diffusivity coefficient, β is the thermal expansion coefficient, β^{*} is the concentration coefficient of volumetric expansion, and σ is the electrical conductivity. *T* is the fluid temperature, *K* is the permeability of the porous medium, g is the acceleration due to gravity, the properties, ρ are the density, and thermal diffusivity coefficients of the saturated porous medium, and *n*, *k* are the unit vector normal to the wavy surface, effective thermal conductivity, *m _{0}*,

*q*are the reference mass and heat flux, respectively. , and are the velocity components along the and axes respectively.

_{0}To facilitate the analysis, the governing differential equations are to be made Non-dimensional with suitable transformation variables; let us suggest the following dimensionless variables:

(8) |

Substituting the above transformation into the governing equations and eliminating the pressure from Eqs. (3), and (4) we get the following form of non-dimensional governing equations

(9) | |

(10) | |

(11) |

where is the Darcy-Rayleigh number, *Le* = *α⁄D* is the Lewis number, is the buoyancy ratio, is the magnetic field parameter.

Equations (9), (10), and (11) may be transformed by introducing suitable variables, in order to subtract out the effect of the wavy surface from the boundary conditions. The new transformation variables are given by:

(12) |

Using Eq. (12), then Eqs. (9), (10), and (11) turn into

(13) | |

(14) | |

(15) |

where

Again, let us consider the following boundary layer variables:

(16) |

On introducing Eqs. (13), (14), and (15) and formally letting (*Ra* → ∞), and neglecting small order terms of *Ra*, we get the following boundary layer equations

(γ^{2} + Mn)f'' = | (17) |

(18) | |

(19) |

where

The corresponding boundary conditions turn into

(20) |

where the prime denotes differentiation with respect to *η*.

**III. RESULTS AND DISCUSSION**

Equations (17), (18) and (19) are coupled non-linear parabolic differential equations, thus, they must be solved numerically. The system of non-linear partial differential Eqs. (17)-(19) with the relevant initial boundary conditions Eq. (20) have been solved numerically using the Runge-Kutta fourth-order. The resulting system of non-linear algebraic equations is solved using the Newton-Raphson iteration scheme.

The effect of various parameters (five parameters) on the MHD Darcy free convection with heat and mass transfer along a vertical wavy plate is examined and discussed in this section. The parameters that the solution are affected are the amplitude of the wavy surface *a*, the magnetic field parameter *Mn*, the buoyancy ratio *Br*, the Lewis number *Le*, and the power law index, λ

Figure 2 displays the effect of the wave amplitude *a* = 0.0 (i.e. flat plate), 0.1, 0.2, 0.3, 0.4, and, 0.5 while *Mn* = 1.0, *Br* = 1, *Le* = 2.0, *x* = 1.0, and λ = 0.2 on the velocity, *f'*(1, η), temperature θ(1, η), and concentration, , profiles versus η and it is clearly that as the wave amplitude increases the velocity profile decreases while the temperature and concentration profile increase.

**Fig. 2.** Variation of respectively, as a function of *η* for different values of amplitude *a.*

Figure 3 shows the effect of the magnetic field parameter *Mn* = 0.0, 1.0, 2.0, 3.0, 4.0, and, 5.0 with *a* = 0.3, *Br* = 1, *Le* = 2.0 and λ = 0.2 on *f'*(1, η), *θ*(1, η) and and as we see that as *Mn* increases, both θ, increase while the velocity decreases. The effect of the power law index λ = 0.0, 0.2, 0.5, 0.7, and, 1.0, on the velocity, temperature and concentration profiles is plotted in Figures (4 a, b, c) with *a* = 0.3, *Mn* = 2.0, *Br* = 1, *Le* = 2.0 and as the power law index increases both of the velocity *f'*(1, η), temperature θ(1, η) and the concentration, , decreases.

**Fig. 3.** Variation of respectively, as a function of *η* for different values of magnetic field *Mn*

**Fig. 4.** Variation of respectively, as a function of *η* for different values of power law index λ

Figure 5 depicts the influence of the buoyancy ratio *Br* = 0, 1, 2, 3, and, 4, on the velocity, temperature and concentration profiles. It is clearly that as the buoyancy ratio increases both of the temperature and concentration decreases, while the velocity profile increases.

**Fig. 5**. Variation of respectively, as a function of *η* for different values of Buoyancy ratio *Br*

Figure 6 illustrates the effect of Lewis number *Le* = 1, 5, 10, 20, 50, and, 100, on *f'*(1, η), θ(1, η) and from these Figures we see that both of velocity and concentration decrease with increases of Lewis number, but the temperature increases. All the above figures we plotted *f'*, θ and as a function of η.

**Fig. 6**. Variation of respectively, as a function of η for different values of Lewis number *Le.*

Now we shall study the effect of all pervious parameters on *f'*, θ and as a function of *x*. The influence of the waviness-amplitude *a*, on velocity *f'*(*x*, 0), temperature θ(*x*, 0), and concentration (*x*, 0) is shown in Figure (7 a, b, c) and as it is shown that both of velocity, temperature and concentration vary periodically with the waviness-amplitude *a*. Moreover, increasing the amplitude-wave-length tends to increase the amplitude of the *f'*, θ and profiles.

**Fig. 7.** Variation of respectively, as a function of *x* for different values of amplitude *a *

Figure 8 displays the influence of the magnetic field parameter *Mn*, on velocity *f'*(*x*, 0), temperature θ(*x*, 0), and concentration (*x*, 0), and from these Figures we observe that both of velocity, temperature and concentration vary periodically with the *Mn*. Moreover, increasing the magnetic field parameter tends to increase the amplitude of the *f'*, θ and profiles.

**Fig. 8**. Variation of respectively, as a function of *x* for different values of magnetic field *Mn.*

Figure 9 depicts the effect of the power law index λ = 0.0, 0.2, 0.5, 0.7, and, 1.0, on velocity *f'*(*x*, 0), temperature θ(*x*, 0), and concentration (*x*, 0), and from these Figures it's observed that both of velocity, temperature and concentration vary periodically with the λ. Moreover, increasing the power law index tends to decrease the rate of *f'*, θ and profiles.

**Fig. 9.** Variation of respectively, as a function of *x* for different values of power law index λ.

Figure 10 displays the influence of the buoyancy ratio *Br* = 0, 1, 2, 3, and, 4, on velocity *f'*(*x*, 0), temperature θ(*x*, 0), and concentration (*x*, 0). These Figures observe that the velocity, temperature and concentration vary periodically with the *Br*. Moreover, increasing the buoyancy ratio parameter tends to increase the amplitude of the *f'*, while decrease the amplitude of both θ and profiles.

**Fig. 10**. Variation of respectively, as a function of *x* for different values of buoyancy ratio *Br.*

The influence of Lewis number *Le* = 1, 5, 10, 20, 50, and, 100 is plotted in Figures (11 a, b, c). From these figures we observe that the velocity, temperature and concentration vary periodically with the *Le*. Moreover, increasing Lewis number tends to decrease the amplitude and the rate of , *f'* while increases the rate of θ profiles.

**Fig. 11**. Variation of respectively, as a function of *x* for different values of Lewis number *Le.*

**REFERENCES**

1. Bourhan T., and M. Al-Odat, "Magnetic field effect on heat and fluid flow over a wavy surface with a variable heat flux," *Journal of Magnetism amd Magnetic Materials*, **268**, 357-363 (2004). [ Links ]

2. Chamkha, A.J., "Hydromagnetic natural convection from an isothermal inclined surface adjacent to a thermally stratified porous medium,"* Int. J. Eng. Sci.*, **34**, 975-986 (1997). [ Links ]

3. Chamkha, A.J., and A.R. Khaled, "Similarity solutions for hydromagnetic simultaneous heat and mass transfer by natural convection from an inclined plate with internal heat generation or absorption. *Heat Mass Transfer*," **37**, 117-123 (2001). [ Links ]

4. Chen, C.H., "Mixed convection heat transfer from a horizontal plate with variable surface heat flux in a porous medium," *Heat Mass Transfer*, **34**, 1-7 (1998). [ Links ]

5. Cheng, C.-Y., "Natural convection heat and mass transfer near a vertical wavy surface with constant wall temprature and concentration in a porous medium," *Int. Comm. Heat Mas Transfer*, **27**, 1143-1154 (2000). [ Links ]

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**Received: September 25, 2007 Accepted: November 9, 2008 Recommended by Subject Editor: Eduardo Dvorkin**