ENTROPY GENERATION AND TEMPERATURE GRADIENT HEAT SOURCE EFFECTS ON MHD COUETTE FLOW WITH PERMEABLE BASE IN THE PRESENCE OF VISCOUS AND JOULES DISSIPATION

In this paper the entropy generation and temperature gradient heat source effects on MHD couette flow with permeable base in the presence of thermal radiation, viscous and joule's dissipation is studied. An exact solution of governing equations has been attained in closed form. The influences of several parameters on the velocity and temperature profiles and entropy generation are analyzed through graphs. Bejan number for different values have been calculated and displayed pictorially. The skin friction coefficient and Nusselt number at channel walls are derived and discussed their behaviour through tables. The entropy generation increases with intensifying magnetic field or thermal radiation or empirical constant or Brinkman number.


INTRODUCTION
The interpretation of heat transfer and fluid flow in a porous channel has established considerable responsiveness during the last numerous decades due to their significance in an extensive range of engineering and biological situations such as problems of drainage, ground water hydrology and irrigation and also in processes of absorption and filtration in chemical engineering. Rapid development in science and technology has led to the expansion of cumulative number of flow divisor that includes the influence of the fluid flow in numerous geometries. In fluid mechanics couette flow is one of the basic flows wherever the fluid motion is induced by movement of the bounding surface. Flow formation examination in couette motion as foretold by classical fluid mechanics was presented by Schlichting and Gerstein (2000).
Everywhere the irreversible process in which time symmetry is ruined. By the theory of entropy the difference between reversible and irreversible process was first presented in thermodynamics. Entropy generation minimization examinations are vigorous for confirming best thermal systems in modern industrial and technological fields like electronic cooling, geothermal systems and heat exchangers. Entropy generation is squarely connected with thermodynamic irreversibility, entropy generation is the amount of the destruction of existing work of the system, the purpose of the dynamic factors motivating the entropy generation is significant in improvement the system performances. Bejan (1979) studied the entropy generation in forced convective heat transfer. Shohel Mahmud et al. (2005) examined the characteristics of thermal and entropy generation with viscous dissipation in a porous channel. Entropy generation with heat sink in a uniformly heated micro channel was analyzed by Abbassi (2007). Sheng Chen (2009) described an entropy generation owing to internal heat generation in turbulent natural convection. Mina Shahi et al. (2011) analyzed entropy generation of a Nano fluid owing to natural convection cooling. Ram Satish Kaluri and Tanmay Basak (2011) investigated entropy generation on thermal management during natural convection in porous square cavities with distributed heat sources. Analysis of convective heat transfer and entropy generation of Al2O3-water Nano fluid flow in a tangential heat sink was described by Amir Shalchi Tabrizi and Hamid Reza Seyf (2012). Basant et al. (2012) studied free convective flow of heat generation/absorbing fluid with ramped temperature past a vertical plate. Makinde (2012) presented the applications of thermodynamics second law to magnetic boundary layer flow past a plate by similarity and shooting technique.  analyzed the entropy generation rate in an unsteady porous channel flow with convective cooling and Navier slip. An entropy generation analysis of magnetic flow with buoyancy effects over a porous pipe was presented by . In wavy wall the effects of heat transfer and entropy generation of H2O based Nano fluid was presented by Ching Chang Cho et al. (2013). Entropy generation analysis of Nano fluid flow through the circular micro/mini channel heat sink was investigated by Sohel et al. (2013). Mansour et al. (2015) analyzed magneto hydrodynamic natural convection and entropy generation of a copper water Nano fluid with wall heat sink. Gamal and Rashed (2015) studied chemical entropy generation on unsteady fluid flow embedded in a porous medium. The effects of entropy generation and Bejan number on couette flow where lower plate is at rest and upper plate is moving with a uniform velocity is studied by Sukumar and Varma (2016). Baag et al (2017) examined entropy generation for viscoelastic flow through a stretching sheet with porous medium. Shantha Kumar Dutta and Arup Kumar Biswas (2018) studied the effects of permeability and the part of non-uniform heating in increasing the temperature and thermal energy and entropy generation minimization. Sudhakar and Balamurugan (2018) examined the effects of buoyancy force and Navier slip on the entropy with suction/injection in a porous vertical channel. Arun and Jose Prakash (2018) investigated Heat transfer deterioration effects in rocket engine cooling channels. Zaidabadi Nezad and Mohammad Mehdi Keshtkar (2018) analyzed optimization of heat transfer from effective parameters on entropy generation in a square electronic package. Dharmaiah et al. (2019) studied the viscous dissipation effect and heat transfer characteristics of magneto flow through a moving inclined plate under the influence of aligned magnetic and thermal radiation. Tahar Taye (2019) investigated the entropy generation for free convection flow where the ground surface is subjected to uniform temperature and the other surface is subjected to constant temperature whereas the chimney wall is adiabatic by using finite volume method. Shyam Lal Yadav and Singh (2019) studied the effects of Darcy and viscous dissipation on free convective heat transfer couette flow in a partially filled saturated porous medium. The authors solved the equations of thermal energy corresponding to porous layer by the method of separation of variables. Hussein et al. (2019) analyzed the heat transfer and entropy generation of laminar flow. Hai-Dong Wang and, Zeng-Yuan Guo (2019) presented the physical essence of entropy in heat transfer.
The aim of this paper is to study the effects of entropy generation and temperature gradient heat source on MHD steady Couette flow bounded below by a permeable bed in the presence of thermal radiation, viscous dissipation and joules dissipation. The equations governing the flow are solved analytically and the impact of various flow parameters on velocity, temperature, entropy generation, Bejan number has been analyzed through graphs. The shear stress and rate of heat transfer coefficients at the channel walls are also derived and discussed their behavior through tables.

FORMULATION OF THE PROBLEM
Consider a steady flow of a Newtonian optically thick viscous incompressible fluid between two infinite horizontal parallel plates separated by a distance H. The upper plate moves with a uniform velocity 0 u in the fluid flow direction. A uniform magnetic field of strength B is applied perpendicular to the fluid flow direction. A Cartesian coordinate system is chosen with x -axis along the lower stationary permeable surface and the y-axis is perpendicular to the plates. T T > .

Permeable base
6. Viscous dissipation, joules dissipation and temperature gradient heat source effects are taken into account. 7. The radiative heat flux in the energy equation is assumed to follow Roseland approximation. Under the above assumptions the governing momentum and energy equations for a steady flow of viscous incompressible fluid are The boundary conditions are where µ is dynamic viscosity, k is thermal conductivity, * K is permeability, 1 T is the temperature of the lower boundary, 2 T is the temperature of the upper boundary, H is the width of the channel. B is magnetic field intensity, σ is the electric conductivity, * Q is temperature gradient heat source parameter.
The radiation heat flux r q in the energy equation is assumed to follow Roseland approximation and is given by The corresponding boundary conditions are is Radiation parameter.

Entropy Generation
The local volumetric rate of entropy generation for a viscous incompressible conducting fluid in the presence of magnetic field, thermal radiation and viscous and joules dissipation is given by Woods (1975) and Arpaci (1987).  The entropy generation number in terms of the dimensionless velocity and temperature is Where 1 Be = is the limit, at which heat transfer irreversibility dominates, 0 Be = is the limit at which fluid friction irreversibility dominates and 1/ 2 Be = implies that both of them contributes equally.

MATHEMATICAL SOLUTION OF THE PROBLEM
Solving the equations (7) and (8) The velocity and thermal regimes obtained are utilized in (11) to obtain the entropy regime.

Skin Friction
The skin friction across the channel's wall is given by

Nusselt Number
The rate of heat transfer coefficient in terms of Nusselt number across the channel wall's is given by

RESULTS AND DSICUSSION
The effects of different parameters such as Hartmann number M, Brinkman number Br, radiation parameter N, permeability parameter K, dimensionless empirical constant α, characteristic temperature ration ω and temperature gradient heat source parameter Q, are examined on the velocity, temperature, Bejan number and entropy generation. The closed form solution obtained for velocity and temperature are employed to compute the entropy generation. The findings are depicted through graphs. Figures 2 and 3 illustrate the variations of velocity u and temperature distribution θ for different values of empirical constant α and permeability parameter K. It is observed that the velocity decreases with increasing empirical constant but it improves the temperature distribution as wells as the thermal boundary layer thickness. The increasing K reduces fluid friction with channel wall and velocity enhances which is perceptible from figure 2. Physically, this means that the porous medium on the boundary layer growth is significant due to increase in the permeability of the porous medium leads to the rise in the flow of fluid through it when the holes of the porous medium become large, the resistance of the medium may be neglected. But the opposite behavior in temperature distribution. The variations in temperature θ for different values of Brinkman number Br, temperature gradient heat source parameter Q and radiation parameter N are presented in Figure 4. It is noticed that the temperature θ is increases with increasing Brinkman number Br and temperature gradient heat source parameter Q. Due to the improvement in Brinkman number results in enhanced convective transport. The result of Q is very much significant for the flow where heat transfer is given prime important whereas rising values of radiation parameter depreciate the fluid temperature.  Table 1 elucidates that the Nusselt number decreases with an increase in heat source parameter Q and Brinkman number Br at the wall y = 0 but this trend is reversed at the wall y =1. It is evident to conclude that the heat transfer rate is high at upper plate when compared with lower plate.

CONCLUSIONS
The studies of Entropy generation minimization are significant role for confirming optimal thermal methods in modern technological and industrial fields like heat exchangers, geothermal systems, electronic cooling,. Entropy generation, heat transfer, thermal radiation and temperature gradient heat source effects on a steady Couette flow bounded below by a permeable bed has been studied and analyzed. Based on the results obtained the following conclusions are made.  The Brinkman number enhances the thermal boundary layer thickness.  The fluid velocity can be enhanced and reduced temperature by increasing porosity suitably.  Temperature θ increases with increasing temperature gradient heat source parameter Q. But thermal radiation depreciates the fluid temperature.  The entropy generation number increases with increasing values of M or α or Br or N or ω . Reverse trend is observed due to K.
 By increasing temperature gradient heat source parameter Q the entropy generation falls in the lower and upper part of the channel but in the middle region the reverse effect is witnessed.  Bejan number Be increases with increasing values of M or N and decreases with increasing values of ω or Q.  Bejan number decrease in the region 0≤y≤0.8 with increasing and reverse phenomena is observed in the region 0.8≤y≤1.  Nusselt number decreases with an increase in Q and Br at the wall y = 0 but this trend is reversed at the wall y =1.  The heat transfer rate is high at upper plate when compared with lower plate.