Heat Transfer in a Micropolar Fluid over a Stretching Sheet with Newtonian Heating

This article looks at the steady flow of Micropolar fluid over a stretching surface with heat transfer in the presence of Newtonian heating. The relevant partial differential equations have been reduced to ordinary differential equations. The reduced ordinary differential equation system has been numerically solved by Runge-Kutta-Fehlberg fourth-fifth order method. Influence of different involved parameters on dimensionless velocity, microrotation and temperature is examined. An excellent agreement is found between the present and previous limiting results.


Introduction
Understanding the flow of non-Newtonian fluids is a problem of great interest of researchers and practical importance. There are several natural and industrial applications of such fluids, for instance volcanic lava, molten polymers, drilling mud, oils, certain paints, poly crystal melts, fluid suspensions, cosmetic and food products and many others. The flow dynamics of non-Newtonian fluids can be described by non-linear relationships between the shear stress and shear rate. Further these fluids have shear dependent viscosity. In literature there exist many mathematical models with different constitutive equations involving different set of empirical parameters. The micropolar fluid model is adequate for exocitic lubricants, animal blood, liquid crystals with rigid molecules, certain biological fluids and colloidal or suspensions solutions. The micromotion of fluid elements, spin inertia and the effects of the couple stresses are very important in micropolar fluids [1,2]. The fluid motion of the micropolar fluid is characterized by the concentration laws of mass, momentum and constitutive relationships describing the effect of couple stress, spin-inertia and micromotion. Hence the flow equation of micropolar fluid involves a micro -rotation vector in addition to classical velocity vector. In micropolar fluids, rigid particles in a small volume element can rotate about the centroid of the volume element. The micropolar fluids in fact can predict behavior at microscale and rotation is independently explained by a microrotation vector. More interesting aspects of the theory and application of micropolar fluids can be found in the books of Eringen [3] and Lukazewicz [4] and in some studies of Peddieson and McNitt [5] Willson [6] Siddheshwar and Pranesh [7,8], Siddheshwar and Manjunath [9].
During the past few decades, several researchers have concentrated on the boundary layer flows over a continuously stretching surface. This is because of their in several processes including thermal and moisture treatments of materials in metallurgy, in the manufacture of glass sheets, in textile industries in polymer processing of chemical engineering plants. Further the stretching flow with heat transfer is quite important in polymer extrusion, cable coating etc. The boundary layer flow of a viscous fluid over a stretching sheet was initially studied by Crane [10], then followed by many investigators for the effect of heat transfer, rotation, MHD, suction/injection, non-Newtonian fluids, chemical reaction etc. It is well known that in many industrial processes, heat transfer is an integral part of the flow mechanism. Now there is an abundant literature available on the flow induced by a stretching sheet with heat transfer [11][12][13][14][15][16][17][18][19][20]. Heat transfer characteristics are dependent on the thermal boundary conditions. In general, there are four common heating processes representing the wall-toambient temperature distribution, prescribed surface heat flux distribution, and conjugate conditions, where heat transfer through a bounding surface of finite thickness and finite heat capacity is specified. The interface temperature is not known a priori but depends on the intrinsic properties of the system, namely, the thermal conductivities of the fluid and solid. In Newtonian heating, the rate of heat transfer from the bouncing surface with a finite heating capacity is proportional to the local temperature surface which is usually termed as conjugate convective flow (see Merkin [21], Lesnic et al. [22], Chaudhary and Jain [23], Salleh et al. [24], Makinde [25]). Salleh et al. [24] numerically investigated the boundary layer flow of viscous fluid over a stretched surface in the regime of Newtonian heating. Numerical solution of the differential system is obtained by Keller box method. Desseaux and Kelson [26] investigated the flow of a micropolar fluid over a stretching sheet. In another attempt, Kelson and Desseaux [27] have investigated the effects of surface conditions on the flow of a micropolar fluid over a stretching sheet. The presented the closed form solution using the perturbation method and made a comparison between the analytical solution with numerical solution obtained by shooting method with fourthorder-Runge-Kutta algorithm. Bhargava et al. [28] studied mixed convection flow of a Micropolar fluid over a porous stretching sheet by implementing finite element method. The stagnation point flow of a micropolar fluid over a stretching surface has been discussed by Nazar et al. [29]. The steady MHD mixed convection flow towards a vertical stretching surface immersed in an incompressible micropolar fluid was investigated by Ishak et al. [30].
To the best of authors' knowledge, the flow of micropolar fluid over a stretching sheet with heat transfer in the presence of Newtonian heating has not been addressed so far. The resulting problems are solved numerically and solutions obtained are compared with the existing results. It is found that the present results are in a very good agreement. Variations of several pertinent physical parameters are also analyzed in detail by plotting graphs.

Basic Equations
We consider the steady boundary layer flow of an incompressible Micropolar fluid induced by a stretching surface. The sheet is stretched with a velocity u w (x)~cx (where c is a real number). The heat transfer in the presence of Newtonian heating is considered. The governing equations of the boundary layer flow in the present study are where u and v are the velocity components parallel to the x{ and y{axes, respectively, r the fluid density, n the kinematic viscosity, T is temperature, N the microrotation or angular velocity, c p the specific heat, k the thermal conductivity of the fluid, j~n=c ð Þ is microinertia per unit mass, c Ã~( mzk=2)j and k are the spin gradient viscosity and vortex viscosity, respectively. Here k~0 corresponds to situation of viscous fluid and the boundary parameter n varies in the range 0ƒnƒ1: Here n~0 corresponds to the situation when microelements at the stretching sheet are unable to rotate and denotes weak concentrations of the microelements at sheet. The case n~1=2 corresponds to the vanishing of anti-symmetric part of the stress tensor and it shows weak concentration of microelements and the case n~1 is for turbulent boundary layer flows.
The boundary conditions of the present problem are [24] u~u w (x)~cx,v~0,N~{n Lu Ly , LT Ly~h s T (NH) at y~0, u~0, N?0, T~T ? as y??: Here h s is heat transfer coefficient and T ? is the ambient temperature. In terms of similarity variables, we write Introducing above Eqs. (6) into Eqs. (1)-(5) one has 1z where Pr is the Prandtl number, c is the conjugate parameter for Newtonian heating and micropolar parameter K. These quantities are given by The skin friction coefficient C f and local Nusselt number Nu x are in which the wall skin friction t w and the heat transfer q w from the plate can be expressed as follows: Now using Eqs. (6) and (14) into Eq. (13) we have where Re x~c x 2 =n À Á is the local Reynolds number.

Results and Discussion
This section the effects of different parameters on the velocity, microrotation and temperature profiles (Fig. 1, 2, 3, 4, 5, and 6). Skin friction coefficient and the local Nusselt number are also Computed (see Tables 1, 2    obtained when K~0: Figs. 1 and 2 represent the velocity profiles for various values of vortex viscosity parameter K when n~0:5 and n~0:0 respectively. It is seen that results here are similar in both cases but change in Fig. 1 is slightly smaller when compared with Fig. 2. From Figs. 3 and 4 we can also observed that the microrotation profile for n~0 is different than n~0:5.  Tables 1, 2, 3, and  4 with already existing results in ½24. All the solutions are found in good harmony. In Table 1