STEADY MHD MIXED CONVECTION NEWTONIAN FLUID FLOW ALONG A VERTICAL STRETCHING CYLINDER EMBEDDED IN POROUS MEDIUM

A viscous electrically conducting luid is considered and its steady mixed convective low along a vertical stretching cylinder is investigated. It is assumed that the cylinder is embedded in a porous medium and, external magnetic ield, heat source/sink are also taken into account. Suitable similarity transformations are used to reduce the governing equations and associated boundary conditions into a system of nonlinear ordinary differential equations. This system along with the boundary conditions is solved by fourth order RungeKutta method with shooting technique. Variations in luid velocity and temperature due to various physical parameters such as heat source/sink parameter, mixed convection parameter, magnetic parameter are presented through graphs. Effect of these parameters on dimensionless shear stress and rate of heat transfer are discussed numerically through tables.


INTRODUCTION
MHD mixed convective low along a stretching vertical cylinder has many applications in areas such as heat exchangers, geothermal power generation, nuclear reactors, drilling operations, insulation system and plastic products formation, polymer processing units, etc.
Mixed convection with heat source/sink along a vertical stretching cylinder has been investigated by a few researchers compared to the horizontal stretching cylinder. Crane Crane (1970) was the irst who considered low past a stretching plate. Chen and Mucoglu Chen and Mucoglu (1975) investigated buoyancy effects on forced convection along a vertical cylinder. Cheng Cheng (1977) took a horizontal lat plate in a saturated porous medium and analyzed the mixed convective low along it. Wang Wang (1984) extended the study of Crane Crane (1970) and obtained an exact Steady mhd mixed convection newtonian luid low along a vertical stretching cylinder embedded in porous medium similarity solution for three-dimensional luid low caused by the stretching boundary. Grubka and Bobba Grubka and Bobba (1985) considered a linearly stretching continuous surface and discussed the heat transfer with power law temperature distribution. Vajravelu Vajravelu (1994) numerically analyzed the convective low and heat transfer of viscous luid along a vertical stretching surface with suction or blowing. Ganesan and Rani Ganesan et al. (2000) studied the applied magnetic ield effects on unsteady mixed convective low past a vertical cylinder. Steady linear and nonlinear convection in a micropolar luid was discussed by Siddheshwar and Srikrishna Siddheshwar et al. (2003). Chang Chang (2008) considered micropolar luid low along a vertical slender cylinder and investigated free convective heat transfer. Ishak and Nazar Ishak et al. (2009) discussed axisymmetric boundary layer low of a viscous incompressible luid along a continuously stretching cylinder. Aydin and Kaya Aydin and Kaya (2011) analyzed the heat transfer characteristic in steady laminar mixed convective low past a vertical slender cylinder. Wang Wang (2012) extended his own study Wang (1984) by considering a vertical stretching cylinder. He discussed natural convection and obtained both analytic and numerical solutions of the problem. Patil et al. Patil et al. (2012) considered a permeable nonlinear stretching vertical slender cylinder and inspected the unsteady mixed convective low with buoyancy force and thermal diffusion. Proceeding in the same order Mishra and Singh Mishra et al. (2014) used momentum and thermal slip and investigated the low and heat transfer over a permeable shrinking cylinder. Simultaneously, effects of heat source and porous medium on MHD free convective low over a horizontal stretching cylinder were analyzed by Yadav and Sharma Yadav et al. (2014). Hayat et al. Hayat et al. (2015) studied mixed convective low of viscoelastic luid due to a vertical stretching cylinder. In their study they considered temperature dependent thermal conductivity. MHD boundary layer slip low over a permeable stretching cylinder was explained by Reddy and Das Reddy and Das (2016). In their research they also included chemical reaction and investigated by arti icial neural network. Following this Sohut et al. Sohut et al. (2017) investigated the chemical reaction and heat source effects on boundary layer over a vertical stretching cylinder. In our previous paper Sharma et al. Sharma et al. (2018), we have investigated heat generation/absorption on MHD mixed convective stagnation point low in the presence of external magnetic ield. Afterwards Vinita and Poply Vinita (2019) discussed steady MHD nano luid slip low past a stretching cylinder with heat generation. They explained thermophoresis and Brownian motion effects on the low and heat, mass transfer. Motivated by the above works here we have taken a viscous incompressible electrically conducting Newtonian luid and attempting to investigate the steady mixed convection low over a vertical stretching cylinder. Moreover, it is assumed that the cylinder is embedded in porous medium in the presence of magnetic ield and heat source/sink. For the validation of the results of the present study, these are compared with the earlier published results and found to be in good agreement.

MATHEMATICAL FORMULATION
Consider steady laminar boundary layer low of a viscous incompressible electrically conducting Newtonian luid past a stretching vertical cylinder. Here it is assumed that the radius of the cylinder is u w (x) = U x/l, where l where is the characteristic length and U is the reference velocity. The coordinate system is taken such that r-axis is measured in the radial direction and axis of the cylinder represents the -axis. An external magnetic ield B is applied along the r -axis. The induced magnetic ield is neglected as the magnetic Reynolds number is considered very small. In view of the above assumptions, the governing equations of conservation of mass, momentum and energy are given as where the velocity components along x and r -axis are u and v, respectively. ρ is density of the luid, σ is electrical conductivity, β is the coef icient of thermal expansion, g is acceleration due to gravity, ν is the kinematic viscosity, K p is the permeability of the porous medium, C p is the speci ic heat at constant pressure, Q is the volumetric rate of heat source/sink, α is the thermal diffusivity, T and T ∞ are temperature within boundary layer and in free stream, respectively.
The associated boundary conditions are where b is a constant. To solve the governing equations along with boundary conditions we have used the following parameters and similarity variables: where η is the similarity variable and ψ is the stream function? Using equation (5) into momentum and energy equations (2) and (3) respectively, we get where the prime is differentiation with respect to η ,λ is the mixed con- is the heat generation/absorption parameter. Here, continuity equation (1) is identically satis ied. Using equation (5), the boundary conditions (4) are reduced to Equations (6) and (7) are nonlinear coupled differential equations with boundary conditions (8). Here we have used fourth order Runge-Kutta method with shooting technique to solve these equations.

NUMERICAL SOLUTION
According to Runge-Kutta method equations (6) and (7) with boundary conditions (8) are decomposed into system of equations of order one. By denoting f, f ′ , f ′′ , θand θ ′ by f 1 , f 2 , f 3 , f 4 and f 5 the following system of equations is obtained: and associated boundary conditions are In process to solve the system numerically, we require ive conditions at η = 0 , but here only three conditions are available. Apart from these three conditions two conditions are given at η → ∞ . In pursuance of shooting technique, we assume that values of f 3 and f 5 at η = 0 are s 1 ands 2 . To check the authenticity of these assumed values, we compute f 2 and f 4 at η → ∞ with the help of given and assumed initial conditions and match it with the values given in (14). The process is repeated until the result of the desired accuracy is obtained.
In the study physical quantities of interest are dimensionless shear stress and rate of heat transfer at the wall. Expression for these is given below

Skin friction Coef icient
The shearing stress at the surface is given by where µ is the coef icient of viscosity?
The skin friction coef icient at the surface is given by  Figure 1 illustrates the effect of magnetic parameter on luid velocity. It is seen from the igure that luid velocity decreases with increasing values of magnetic parameter. The reason behind this is larger magnetic parameter corresponds to stronger Lorentz force which resist the motion of the luid. This decreasing velocity causes lesser skin friction as noticed from Table 2 . Effect of curvature parameter on luid velocity is depicted in Figure 2 . The curvature parameter is inversely proportional to the radius of the cylinder, so for higher curvature parameter there will be lesser interaction between the luid particles and the surface, as a result velocity goes up. It is observed from Figure 3 that luid temperature increases with larger magnetic parameter. As explain earlier that Lorentz force works as a resistance and retards the luid motion, due to this resistance some amount of energy transforms into heat and increases the temperature. This increasing temperature reduces the temperature difference between the surface and the luid, and because of this Nusselt number decrease, as noted from Table 2 . Figure 4 illustrates that increasing permeability parameter causes a drop in temperature of the luid. Larger permeability parameter shows more assistance to the luid to low through the porous medium, due to this orderly mannered low, temperature goes down. The effect of curvature parameter on luid temperature is shown through Figure 5 . It is quoted in the explanation of Figure 2 that luid velocity increases with increasing curvature parameter. Due to this increased velocity there will be more kinetic energy and, the temperature is de ined by the average kinetic energy therefore temperature increases. Impact of Prandtl number on luid temperature is examined in Figure 6 . For larger values of Prandtl number, kinematic viscosity dominates the thermal diffusivity and this lesser thermal diffusivity results in temperature drop near the surface. This decreasing temperature creates more temperature difference between the surface and the luid and as an outcome Nusselt number increase. It is noted from Figure 7 that luid temperature increases with increasing heat generation parameter. This increasing temperature corresponds to low temperature difference and Nusselt number. Figure 8 is plotted to see the impact of mixed convection parameter on luid temperature. As the mixed convection parameter increases, Grashof number dominates the Reynolds number and, in outcome more buoyancy force works on lowing luid. This increased buoyancy force gives a hike to the temperature near the surface.     Table 1 is prepared to compare the numerical value of Nusselt number in the present study with previously published results by Ishak and Nazar Ishak et al. (2009). The comparison is performed for various values of Prandtl number and it can be noted that the results are in a good agreement with data provided by Ishak et al. (2009). It is observed from Table 2 that shear stress at the surface decreases with increasing Prandtl number, while heat generation/absorption parameter and permeability parameter are having reverse effect on it. The dimensionless rate of heat transfer at the surface increases with increasing Prandtl number, curvature parameter, permeability parameter and it decreases for enhancing heat generation/absorption parameter, magnetic parameter and mixed convection parameter.  Steady mhd mixed convection newtonian luid low along a vertical stretching cylinder embedded in porous medium Table 2 continued 7 0.5 0.5 0.5 2.0 0.2 -1.982703 2.009901 7 0.5 0.5 0.5 0.5 0.5 -1.517001 2.211902 7 0.5 0.5 0.5 0.5 0.8 -1.517001 2.081971

CONCLUSIONS
In this article a viscous incompressible Newtonian luid is considered and its steady low past a linearly stretching vertical cylinder is investigated. In addition, the low ield is manifested with heat source/sink, external magnetic ield and porous medium. The conclusions of the study are itemized as follows: 1. To enhance the luid velocity, curvature of the cylinder should be taken large 2. Temperature of the luid can be decreased by increasing permeability parameter or Prandtl number 3. In order to achieve the goal of lesser skin friction, Prandtl number, curvature parameter or magnetic parameter must be increased.
4. For higher rate of heat transfer, larger values of Prandtl number, curvature parameter or permeability parameter can be used.