Effects of Variable Viscosity and Thermal Conductivity on Micropolar Fluid Flow Due to a Stretching Cylinder in Presence of Magnetic Field

In the presence of magnetic field, steady flow of a micropolar fluid due to a stretching cylinder is studied. Viscosity and thermal conductivity are assumed to be inverse linear functions of temperature. The governing partial differential equations are converted into ordinary differential equations using suitable similarity transformations and then solved by fourth order Runge-Kutta shooting method and developing Matlab programme. The graphs show the effects of different parameters and the skin friction coefficient and Nusselt numbers are shown in tabular form. KeywordsMicropolar fluid, Stretching cylinder, Variable viscosity, Thermal conductivity.


Introduction
From the very beginning of the introduction of micropolar fluid theory by Eringen (1964Eringen ( , 1966Eringen ( , 1972, it becomes an attractive part of so many researchers. This theory is theoretical, for which attracts Mathematicians as well as practical attracting Physicists and Engineers. The micropolar fluid theory is the most appropriate theory among various theories to study the behaviour of non-Newtonian fluids. There are so many fluids which are industrially important and showing non-Newtonian behaviours. Biological fluids, multiphase mixtures, food products are some such fluids. Boundary layer theory for micropolar fluid was studied by Peddieson and McNitt (1970). Guram and Smith (1980) investigated the stagnation flow of micropolar fluids with strong and weak interactions. Time dependent slip flow of a micropolar fluid between two parallel plates through a state space approach was examined by Slayi et al. (2016). Sheri and Shamshuddin (2015) studied heat and mass transfer on the MHD (magnetohydrodynamic) flow of micropolar fluid in the presence of viscous dissipation and chemical reaction. MHD unsteady flow and heat transfer of micropolar fluid through the porous channel with expanding or contracting walls were investigated by Asia et al. (2016). example, in the extrusion of a polymer sheet, the properties of the final product considerably depend on the rate of cooling. MHD flow and heat transfer of a micropolar fluid over a stretching surface with heat generation (absorption) and slip velocity was studied by Mahmoud and Waheed (2012). Hazarika and Phukan (2016) investigated the effects of variable viscosity and thermal conductivity on magnetohydrodynamic free convection flow of a micropolar fluid past a stretching plate through a porous medium with radiation, heat generation, and Joule dissipation. Effects of variable viscosity and thermal conductivity on magnetohydrodynamics mixed convective flow over a stretching surface with radiation have been studied by Hazarika and Hazarika (2015). Ishak et al. (2008) examined heat transfer over a stretching surface with variable heat flux in micropolar fluids. Convective heat transfer in an electrically conducting fluid at a stretching surface in the uniform free stream was studied by Vajravelu and Hadjinicalaou (1997).
Effects of variable viscosity and thermal conductivity on unsteady micropolar fluid about a permeable cylinder under moving boundaries have been studied by Baruah and Hazarika (2017). Hayat et al. (2016) investigated magnetohydrodynamics flow by a stretching cylinder with Newtonian heating and homogeneous heterogeneous reactions.
In this study, we have extended the work of Nazir and Shafique (2015) and analyse the effects of variable viscosity and thermal conductivity on micropolar fluid flow due to a stretching cylinder in the presence of magnetic field. In most of the earlier studies, physical properties were assumed to be constant. Here we consider the viscosity and thermal conductivity as inverse linear functions of temperature. The governing partial differential equations are reduced to ordinary differential equations using similarity transformations and then solved numerically by fourth order Runge-Kutta shooting method under prescribed boundary conditions.

Mathematical Formulation
We consider the steady viscous incompressible micropolar fluid flow due to a stretching cylinder in the presence of magnetic field. The cylindrical coordinate system ( , , ) is considered and the cylinder is described with radius = . Here, the material constants of the micropolar fluid are assumed to be independent of position and neglect body force and body couple. Due to axial symmetry in the prescribed cylindrical coordinate system only two components are there. The velocity components are given by ( ( , ), 0, ( , )) and micro rotation components are given by(0, ( , ), 0). Under these conditions the governing equations in the cylindrical coordinate system are as below: where is the co-efficient of dynamic viscosity, is the vortex viscosity, is the density, 0 is the acceleration due to gravity, is the angular velocity, is the fluid temperature, is the electrical conductivity, 0 is the magnetic field intensity, is the micro rotation viscosity, is the micro rotation density, λ is the thermal conductivity, is the specific heat at constant pressure, is the pressure. represents the constant temperature at the surface of the cylinder and ∞ represent the temperature far away from the surface where > ∞ .
According to Lai and Kulacki (1990) where is transformed reference temperature corresponding to viscosity parameter, is the viscosity, ∞ is the viscosity of the ambient fluid, and ∞ are constants whose values are depend upon the reference state and thermal property of the fluid.
According to Khound and Hazarika (2000) where ∞ is the thermal conductivity of the ambient fluid, and are constants depending on the reference state and thermal properties of the fluid.
We introduce the following similarity transformations and parameters: Using the above transformations continuity equation (1) is identically satisfied and equations (2) -(4) becomes where and are the dimensionless reference temperatures corresponding to viscosity and thermal conductivity parameter respectively.
The boundary conditions (6) become The dimensionless parameters are defined here as: where is the heat transfer from the surface given by After some simple steps we get (1).

Results and Discussion
The system of differential equations (10) and . It is observed that velocity increase for the increase of micropolar parameters 1 and 2 , viscosity parameter and thermal conductivity parameter .    Angular velocity profile for various parameters is shown from the Figures 9 to 12. It is observed that angular velocity increases with micropolar parameter 3 while it decreases for the increase of the micropolar parameter 2 . Angular velocity decrease first for the increase of the viscosity parameter and thermal conductivity parameter but after some time it behave opposite.

Conclusion
From the above discussions, we can conclude that the variable viscosity and thermal conductivity along with other parameters have a significant effect on velocity, temperature and micro-rotation.
The following conclusions can be made:  Fluid velocity increase with micropolar parameter 1 and 2 , viscosity parameter and thermal conductivity parameter .
 The temperature of the fluid enhance with magnetic parameter and micropolar parameter 1 while there is a loss of temperature with the increase of viscosity parameter and thermal conductivity parameter .  The angular velocity of the fluid increase with the increase of the micropolar parameter 3 and decrease with the increase of the micropolar parameter 2 .  Angular velocity at first decrease and then increase with the increase of and .
This study will be helpful to study the properties of fluid with more parameters.

Conflict of Interest
The author confirms that this article contents have no conflict of interest.