The effects of the magnetic parameter M and the Prandtl number Pr ha"e on the flow field were studied. Similarity equations of the quantity and energy equations were derived by introducing the same transformations. Ambient fluid temperature: velocity component in the x-direction: velocity component in the y direction: flow coordinate.
The manipulation of natural convection of the heat transfer can be abandoned in the case of fixed Reynolds number and diminutive Grashof nwnber. Subsequently, Mori (1961) and Sparrow and Minkowycz (1962) investigated mixed convection flow in the boundary layer of a micro-polar fluid above a horizontal flange plate by means of a perturbation series in terms of the buoyancy parameter. Matching solutions are obtained in his work for the case of wall temperature inversely proportional to the square root of the distance.
The solutions of the basic equations were obtained by using the approximate momentum integral technique. In the weak magnetic fields and in the immediate vicinity of the leading edge, their results apply. Along this line, Chida and Katto (1976) investigated the conjugate problems with the help of the vertical dimensional analysis.
Finally, all references from the book can be found in the last part of the thesis.
Effects of conduction and convection hydrodynamic flow from a vertical flat plate
Governing equations of the flow
The mathematical statement of the basic conservation laws of mass, momentum and energy for the steady viscous incompressible and electrically conductive fluid is given by [Crammer and Pic (J 973). Here B = Il, Hu, JI, being the magnetic permeability of the fluid, He is the applied magnetic field and V is the vector differential operator and is defined by. and the yare the unit vector along x and y axes respectively. When the external electric field is zero and the induced electric field is negligible, the current density is related to the velocity by Olun's law as follo,vs. 2.4) Where a indicates the electrical conductivity of the liquid.
On the basis that the magnetic Reynolds number is small, the induced magnetic field is negligible compared to the applied field. F in equation (2.2) and the variation of p in the inertia tcrm are neglected. 2.7) where p", and ToO are thc density and temperature, respectively, outside the lateral boundary layer. We now consider the steady two-dimensional laminar free convection boundary layer of a viscous incompressible and electrically conducting fluid along one side of a vertical flat plate of thickness 'b' isolated on the edges with temperature To maintain on the other side in presence.
Using equations (2.4) to (2.6) with respect to our above considerations for the fundamental equations, the steady two-dimensional laminar draft convection boundary. now layer of a viscous incompressible and electrically conducting fluid ", with viscosity and also constant thermal conductivity past a vertical flat plate in the presence of a. The temperature and heat flux are required continuously at the interface of the coupled eondithiol15 and at the interface we shall have.
Transformation of the governing equations
2.12) The temperature and heat flux are required to be constant at the interface for the paired eondithiol15 and at the interface we should have.
Method of Solution
We approximate the quantities j, u, v,p in the points (t ,'I)) of the net byIj.u"j,vnJ.P"j, which we call net function. We also use the notation i'J for quantities midway between net points shown in figure (2.2) and for any net function as. The finite difference approximations according to the Box method to the three ordinary differential equations (223) to (2.25) of the first order are written for the midpoint (t,'IJ-In) of the segment P1P2 shown in the figure (2.2) and the finite difference approximations to the two first-order diTerential equations (2.26) to (2.27) are the Center for the midpoint. Here the coefficients (s.) J ami(s1G)i', which are zero in this case, are included here for the sake of generality. 2.82), which simply expresses the requirement that the boundary conditions remain during the iteration process.
The solutions or the above equations (2.20) and (2.11) together with the boundary conditions (2.23) enable us to calculate the skin friction r and the rate of heat transfer eet.
Results and discussion
2.8, the same result is observed on the surface temperature distribution due to increase in the value of the Prandll number when the value of the magnetic parameter M. From fig. 2.13, it is revealed that the velocity profile ((1/, x) decreases with the increase of the magnetic parameter M, indicating that the magnetic parameter delays the diligent motion. Fig. 2.14 shows that for increasing values of M. .the temperature profile 2.15 shows the velocity profile for different values of the Prandtl number.
1.0) while the magnetic parameter is M I . Corresponding distribution of the temperature profile ()('1,x) in the fluids is shown in Fig. shown, it can be seen that as the Prandtl number increases, the velocity of the fluid decreases, On the other hand, from Fig, 2.16 we observe that the temperature profile decreases inside the boundary layer due to increasing the Prandtl number Pro Fig and tctemperature profile e ('1,x) respectively for different values of the PIandtl number l 'r when the value of the magnetic parameter AI=0.6.
Conclusion
Effects of conduction and convection on magneto hydrodynamic flow with viscous dissipation from a
Governing equations ofthe flow
We consider a steady two-dimensional two-dimensional boundary layer flow without laminar convection of a viscous incompressible and electrically conductive fluid along the side of a vertical flat plate of thickness 'b', insulated at the edges with temperature Tt>, maintained on the other side in the presence of a uniformly distributed transverse magnetic field. Temperature and heat flux are required to be constant at the interface for coupled conditions and at the interface we must have. where k and kfarc are the thermal conductivity of the solid and liquid, respectively. the temperature TM in the solid as given by A. 3.6) where T(.x,O) is the unknown temperature at the boundary determined from solutions or Eqs. We note that the equations together with the boundary conditions are nonlinear partial differential equations.
Transformation of the governing equations
To solve the equations subject to the boundary conditions (3.11), the following transformations were introduced for the catchment starting from upstream to downstream.
3,3. Method of Solution
- Conclusion
3.8, it is revealed that the velocity profile f'(fI,x) increases slowly with the increase of the viscous dissipation parameter N indicating that viscow; dissipation increases fluid movement. A small increase is shown in Figure 3.9 on the temperature profile δ(fI,x) for increasing values of N. From Figure 3.12 we see that the velocity profiles decrease monotonically with the increase of the magnetic parameter M when the values of the Prandtl number Pr and the dissipation parameter N are 0.73 and 0.5, respectively. The opposite result is shoVvTI in Figure 3.13 for the temperature distribution for the same values of the magnetic parameter M.
Figure 3.10 shows that as the Prandtl number increases, the velocity of the fluid decreases. On the other hand, we see in Figure 3.11 that the temperature profile within the boundary layer decreases due to the increase in the Prandtl numberPro. The effect of viscous dissipation and magnetic parameters Nand M for a small Prandtl number Pr, respectively, on the magnetohydrodynamic (MHD) natural convection of the lower layer current with viscoll dissipation of a vertical natural plate has been studied by introducing a new class of tral15fOmlations.
The translated non-homogeneous boundary layer equations governing the flow together with the boundary conditions based on conduction and convection shear were solved numerically using the very efficient implicit I1nitic difference method together with Keller box scheme. The coupled effect of natural convection and conduction required that the temperature and the heat flux be continuous at the interface. The skin friction coefficient and the velocity distribution increase to increase the value of the viscous dissipation parameter N.
2, Increased value of the viscous dissipation parameter N leads to increase of the surface temperature distribution as well as the temperature distribution. 1t has been observed that the coefficient of skin friction, the surface temperature distribution, the temperature distribution over the entire boundary layer, and the velocity distribution decrease with the increase of Prandtl nwnbcr Pro. The skin friction coefficient, the surface temperature distribution and the velocity profile decrease while the temperature profile increases for the increased values of the magnetic parameter 114, .
Conclusions'
From chapter 2
The skin friction coefficient and the velocity distribution decrease for increasing value ofthc magnetic parameter M
- Increased value of thc magnetic parameter M leads to increase III the surface temperature distribution as well as the temperature distribution
From chapter 3
The skin friction coefficient and the velocity distribution increase lor increasing value of the viscous dissipation parameter i'/
- Increased value of the viscous dissipation parameter N leads to increase in the surface temperature distribution as wcll as the temperature distribution
- It has been observcd that the skin friction coefficient, the sUlface temperature distribution, the temperature distribution over the whole boundary layer and the velocity
- lbe skin friction coetTieient, the surface temperalure distrihution and the velocity profile decrease while the temperature profile increases for the increased values of the
1961) '.stcady and transient free convection or an electrically conductive fluid of a vertical plate in the presence of a magnetic field.'. Hassanein, LA Combined forced and free convection in boundary layer. 1991), Viscous ami joule heating effects on MHO free flow with variable plate temperature Non-Darc} forced convective boundary layer flow over a flat plate embedded in a saturated porous medium.
Gosse (1980), Une analyse simplifiée du couplage conduction-convection pour l'écoulement laminaire de la couche limite sur une plaque plate, Rev. Padet (1985), Une étude théorique de l'assemblage de conduction convective en convection libre laminaire sur plaque verticale, lnt. KallO (\976), stable en transfert thermique conjugué par analyse dimensionnelle vectorielle, InU. Transfert de masse thermique.
Combined forced and free convection flow on a vertical surface, Int. Natural Convection of an Electrically Conducting Fluid in the Presence of a Magnetic Field.". Nakamura (1980): Effect of Axial Thermal Conductivity in a Vertical Flat Plate on Free Convection Heat Transfer, Int. .
