Applied mathematical model for a heat transfer

It is implicit for some time ago, that the radiative magneto hydrodynamic stands for an interacting between a radiation field and magneto hydrodynamic (MHD) field that is concerned with interacted electrical conducting fluids and electromagnetic field. An accurate solution to the heat transfer equation is obtained by calculating the radiant heat emission in the boundary layer MHD because of the eccentric rotations of the pore disk and the fluid for long time. It is found that the asymptotic solution is present in both the cases of suction and blowing states, whereas no such solution exists for the blowing state in the absence of radiative emission of heat. The heat transfer rate at the disk has been determined and the condition is gotten for the heat to flow from the liquid to the disk.

sections significance of the problem, section three problem description, section four the mathematical analysis and solution of the problem, exact solution has been established and description of parameters and discussion followed by conclusion in section 5.

Significance of the problem
This research we expect is important in applied mathematics and useful for fieldwork in engineering.

Problem Formulation
We consider a system of the Cartesian coordinate in which the z-axis has been taken normally to the plan z=0 of the pore disc. It is assumed that the disc and the fluid have been rotating with uniform angular velocity Ω regarding the axes with distance L in the plane x=0 between them, so that the boundary conditions are Where the components of the fluid velocity are w,v,u . The fluid is assumed electrically conducting with a transverse applied magnetic field 0 in the direction of z-axis. Reynolds number magnetic has been assumed so small that the induced magnetic field is negligible as compared to the applied magnetic field. Following the velocity profits in the magnetohydrodynamic (MHD) boundary layer caused by the rotation of disc are caused by the rotation of disc given by: Here, S: is suction parameter N: is magnetic parameter The heat transfer equation to be solved is of the form.
where stands for the specific heat at a constant pressure T denotes absolute temperature K denotes a coefficient of thermal condition denotes the flux of radiative heat V denotes the kinematic coefficient of viscosity.

Solution of the problem
Based on optically thin approximation, the mean free path of radiation is much larger than the characteristic length of flow field. The situation is sometimes expressed as that of radiation with negligible self-absorption so that the plank mean absorption coefficient = 1 is very small, where is the characteristic length of mean free path of radiation. Each element of the fluid exchanges radiation straightforwardly with the bounding surfaces and so there is no rad-dative interaction between various fluid elements. Sparrow, and Cess [24] have shown that the equation of radiative heat transfer for a non-grey gas near equilibrium under optically thin limit assumes a simple from: Where ∶ the Stefan -Boltzmann constant : is the Planck mean absorption coefficient For the case of extremely optically thin limit → 0 as → ∞ and thus the flux of radiative heat will be a constant as has been shown.
Since the temperature 0 at the pore disc is the highest and the temperature ∞ of the liquid for long time is the lowest, it can be assumed that the liquid at infinity is the lowest, it can be assumed that the liquid at infinity attains an isothermal state of equilibrium whereas the temperature difference in the whole flow filed is small. Under such condition it is possible to linearize the equation (4), and for the present case the best-fit linear approximation of (4) can be expressed in the form: The liquid at infinity is in an isothermal state of equilibrium. The boundary conditions to be satisfied by the temperature profile of (4) are. The equation (11) and (12) show that the solution is valid only for the suction case (s>0) and no asymptotic solution exists for the blowing case (s<0). At the disc the rate of heat transfer is given by The heat will flow from the disc to the liquid, if the following condition is satisfied: or otherwise, there will be flow of heat from the liquid to the disc.

Conclusion and recommendation
1-This research is about model in applied mathematics for an interact with fluid flow, heat transfer and radiative optical thin limit, it includes a solution of heat transfer equation and radiative heat emission in magneto hydrodynamic (MHD) boundary layer with eccentric rotations of pores and disk and fluid for long time.
2-It is found that the asymptotic solution exists in both the cases of blowing and suction whereas no such solution exists for the case of blowing in the absence of radiative heat emission. 3-The heat transfer rate at the disc is determined, and the condition is obtained for the flowing of heat from the liquid to the disc. 4-The equations (11) and (12) show that the solution is valid only for the suction case of (s>0), and the asymptotic solution does not exist for the blowing case, (s < 0). 5-This research is an important and wide subject especially for engineering students in undergraduate and postgraduate in most of the Universities. Also the important of the researches from the above material and its equations (ordinary differential and partial differential) and their solutions especially for the faculty interest in applied mathematics