The impact of variable fluid properties on hydromagnetic boundary layer and heat transfer flows over an exponentially stretching sheet

This paper put forward an analysis of variable fluid properties and their impact on hydromagnetic boundary and thermal layers in a quiescent fluid which is developed due to the exponentially stretching sheet. The viscous incompressible fluid has been set into motion due to aforementioned sheet. We assume that the viscosity and the thermal conductivity of the Newtonian fluid are temperature dependent. The governing boundary layer equations containing continuity, momentum and energy equations are coupled and nonlinear in nature, thereby, cannot be solvable easily by using analytical methods. Since the general boundary layer equations admits a similarity solutions, thus a generalized Howarth-Dorodnitsyn transformations have been exploited for the set-up of a coupled nonlinear ODEs. These transformed ODEs are solved numerically by a shooting method and is verified from MATLAB built-in collocation solver bvp4c for different parameters appearing in the work. We show results graphically and in a tabulated form for a constant and a variable fluid properties. We find that the temperature dependent variable viscosity and a thermal conductivity influence a velocity and a temperature profiles. We show that the thermal boundary layer decreases for constant variable fluid properties and increases for variable fluid properties


Introduction
The principles of heat transfer in manufacturing industry is a chief theory behind the design and production of many household appliances and commercially used devices. The examples of heat transfer can be found in air conditioning system, refrigerators, the TV and the DVD player, to name a few. Even heat transfer flows are more important due to stretching sheet which has abundance of applications in industries, engineering, metallurgy, paper production, drawing of plastic films, hot rolling wires, elongation bubbles, extrusion processes in which the deformed materiel is pass out from die for final product, geological stretching of the tectonic plates during earthquake etc.
A Blasius type moving flow due to a stretching sheet issuing steadily from the slit has been investigated by Sakiadis [1]. The numerical and integral methods have been carried out to obtain the solution of the underlying study. He indicated that the boundary layer behavior on such surface is different than the surface of finite length. Owing to the need of definitive experiment for the boundary layer of continuous surface, the combination of experimental and analytical verifications have been considered in Tsou et al [2]. A three page article by Crane [3] extended the work of Sakiadis [1] in that he took the boundary layer flow over a stretching sheet where velocity varies linearly from the slit. The work on unsteady viscous flow has been only assumed adjacent to stagnation point by Rott [4] but far away from the plate the flow is taken as steady. The plate performed harmonic motion in its own plane i.e. along x-direction and he has shown that this problem is solvable exactly. Danberg and Fansle [5] enhanced this idea further for non-similar stretching wall where velocity is proportional to the distance x. Chakrabarti and Gupta [6] has extended the specialized case of Danberg and Fansle [5] and considered an electrically conducting fluid with a uniform transverse magnetic field. The motion in the fluid is caused by a stretching of the wall. Soundalgekar and Murty [7] tackled a heat transfer problem past a continuous semiinfinite flat plate in which temperature varies nonlinearly i.e. Ax n , where A is a constant and n is never o or 1. They observed that the Nusselt number increases with increasing the exponent n. Wang [8], on the other hand, moved one step further and presented analysis for the three dimensional flow caused by two lateral directions where wall velocities varies linearly. The list of available literature on boundary layer flows for different fluids and flows over a stretching sheet with different aspects is long. For detail the reader is referred to Dutta et al [9], Grubka and Bobba [10][11][12][13][14][15][16][17][18][19][20][21], and forthcoming cited literature in next paragraphs.
In boundary layer flow, if a temperature difference is strong then the assumption of fluid properties are constant may lead to different results and hence wrong interpretation of the post processing. The dynamic viscosity is highly dependent on a temperature and is weakly dependent on thermodynamic pressure. Takhar et al [22] was the first who has discussed variable fluid properties. Pantokratoras [23] have discussed results of variable viscosity on the flow due to a continuous moving flat plate. He assumed that the Prandtl number is variable across a boundary layer. His assumption is based on the definition of Prandtl number which depends on viscosity i.e. if viscosity is variable so do the Prandtl number. This assumption is not correct as discussed in Andersson and Aarsaeth [24] . A compact analysis on variable fluid properties for Sakiadis problem have been presented by Andersson and Aarsaeth [24]. They clarify some of the misconceptions prevalent in scientific community over a variable fluid properties. Lai and Kulacki [25] investigated variable fluid properties for convective heat transfer in a saturated porous medium since previous studies mostly dealt with constant fluid properties for water. Their work is also concerned on heat transfer analysis for gases too. Kameswaran et al [26] studied the effect of radiation on the MHD Newtonian fluid flow due to an exponentially stretching sheet when considering the effects of viscous dissipation and frictional heating on the heat transport. Hayat et al [27] have deliberated axisymmetric hydromagnetic flow of a third grade fluid. The idea was to observe characteristics of flow over a stretching cylinder. They reported that the velocity and momentum boundary layer thickness is dependent on the curvature parameter. They also mentioned that velocity profile is higher for third grade fluid than the Newtonian and second grade fluid with and without MHD. Very recently Babu et al [28] discussed MHD dissipative flow across slendering stretching sheet with temperature dependent variable viscosity. Study of viscoelastic boundary layer flow and heat transfer over an exponentially stretching sheet was examined by Khan and Sanjayanand [29]. Pop et al [30] have examined the influence of variable viscosity on laminar boundary layer flow. They assumed the fluid viscosity varies inversely with temperature. Ali [31] considered heat transfer characteristics over a nonlinearly stretching sheet. Prasad et al [32] similar to Ali [31] have studied the effect of variable viscosity and thermal conductivity over a nonlinearly stretching sheet. Magyari and Keller [33] considered mass and heat transfer in the boundary layers on acontinuous surface which is stretched exponentially. The flow of a viscoelastic fluid over a stretching sheet with transverse magnetic field is assumed by Andersson [34]. He showed that the MHD has the same effect on the flow as viscoelasticity. In a similar work, a power-law fluid over a stretching sheet was investigated by Andersson et al [35]. They have shown that the magnetic field make the boundary layer thinner for the underlying case. Nadeem et al [36] analyzed the heat transfer characteristic while presenting two cases, Prescribed exponential order surface temperature (PEST) and prescribed exponential order heat flux (PEHF). They studied Jeffrey fluid over an exponentially stretching surface. Although, viscous dissipation is a key term appearing in energy equation but considered by very few scientists. Pavithra et al [37] took this task to include viscous dissipation in dusty fluid over an exponentially stretching sheet and also discussed two cases for heat transfer analysis: Prescribed exponential order surface temperature (PEST) and prescribed exponential order heat flux (PEHF). Mabood et al [38] did analysis on viscous incompressible flow along with radiation effect while taking exponentially stretching sheet. They obtained the solution by using homotopy analysis method (HAM). Mukhopadhyay [39] studied MHD boundary layer flow and heat transfer towards an exponentially stretching sheet embedded in a thermally stratified permeable medium. Singh and Agarwal [40] investigated the effects of variable fluid properties of Maxwell fluid over an exponentially stretching sheet. They applied Keller-Box method to find a numerical solution. A variable thermal conductivity has been accounted with Cattaneo-Christov heat flux formulation in Hayat et al [13].
All studies of the past have considered variable fluid properties with many different fluids over a different type of stretching sheets. Not much work has been done on variable fluid properties, specifically temperature dependent viscosity and thermal conductivity, over an exponentially stretching sheet with MHD effect. We fill these gaps and present some interesting results on this topic.
The present paper has been organized as follows. In section 2, we present a mathematical model for the flow and heat transfer analysis. The three distinct cases have been discussed in section 3. The computational procedure has been explained in section 4. In section 5, we present the graphs, tables and their discussion. The conclusion has been drawn in section 6.

Problem formulation
Consider a steady, two dimensional, incompressible flow of an electrically conducting fluid over a sheet that has been stretched exponentially. The x-axis is taken along the sheet and y-axis is normal to it. B o is the strength of uniform magnetic field which is applied normal to the sheet. The induced magnetic field is neglected because the value of a magnetic Reynolds number is less than unity in an electrically conducting fluids. T w is a temperature of the sheet and T o is the temperature of the ambient fluid. The geometrical configuration of the problem can be seen in the figure 1 for better understanding and visualization. The governing equations with these assumptions are given by Andersson and Aarseth [4] u v a 0, 1 with boundary conditions where u is a xcomponent and v is a y-component of a fluidʼs velocity. Fluid density is represented by ρ, B 0 is the strength of an applied magnetic field, μ is the dynamic viscosity, specific heat is denoted by C p , fluidʼs temperature is symbolized by T and the factor k appearing in energy equation is commonly known as a thermal conductivity. U w represents the velocity of the sheet, wall temperature is denoted by T w (x).
Since governing equations are written in general set-up, we cannot apply usual similarity transformation. But we take the following Howarth-Dorodnitsyn transformations Dorodnitsyn [41], Howarth [42]: here the stream function is denoted by ψ and its relation with u and v have been given as Andersson and Aarseth [4].

Inserting equations
ODEs where where f ¢ denotes dimensionless velocity and θ denotes dimensionless temperature.
The skin friction coefficient C f and local Nusselt number Nu x are defined as follows: where τ w is a shear stress and q w regarded as a heat flux, and these are defined as: Using above equations (8) and (9)we get where Re denotes local Reynolds number.
It is important to note that all the fluid properties considered here are constant except the viscosity and thermal conductivity which are temperature dependent.

Special cases
The boundary conditions given in equation (7) remains the same.

Case B: variable fluid properties
For this case, we assume viscosity and thermal conductivity as variable that depends on a temperature when keeping the other physical properties as constant. For this case the momentum boundary layer equation equation (6a) becomes Lai and Kulachi [5], Ling and Dybbs [31] and Pop et al [10] suggested the following relation between viscosity and temperature: where δ is property of the fluid that depends on the reference temperature T ref .
If T ref ≈T 0 , the above formula becomes ). By using equation (14) in equation (12), we get the following momentum equation The thermal conductivity is defined as Subhas et al [36] k T k 1 , using the above relation (16) in equation (13) we get the following energy equation.

Numerical procedure
Here we find the numerical solution of nonlinear (ODEs) for each Cases A, B and C with the boundary conditions as given in equation (12). We apply shooting technique to obtain numerical results.

Results and discussions
In this part, numerical results of velocity and temperature gradients are discussed. Results are shown in tabular and graphical form. Numerical solutions for f 0 -( ) (coefficient of skin friction) and 0 q -¢( ) (temperature gradient) for various values of physical parameters that are Prandtl number, magnetic parameter and the parameter ò numerical results of have been shown from numerical results of tables 1 to 4. From tables 1-3 one can observe that the skin friction coefficient increases whereas a reduction in wall temperature have been seen as magnetic parameter arises. The Prandtl number increases wall temperature for all the three cases but skin friction changes slightly. It can also be seen that the parameter ò reduces both the skin friction coefficient and wall temperature for the Cases B and C. In table 4 numerical results for skin friction coefficient and heat transfer rate are computed for all the cases by increasing the Prandtl number. The value of skin friction coefficient increases for two Cases B and C but for case A it shows a decreasing behaviour. The wall temperature shows increasing behavior for all the three cases. In table 5 we compare our results with the previously published data.  Shooting method cpu time(bvp4c) The effect of viscosity and thermal conductivity for all the three cases have been studied. Temperature of ambient fluid is T 0 =278 K while temperature of surface is taken as T w =358 K. In figures 2-3 velocity and temperature profiles are presented for all Cases A, B and C. In comparison with Case A and C velocity profile for Table 2. Values of skin friction and wall temperature gradient for different physical parameters for Case B. bvp4c Shooting method  Table 3. Values of skin friction and wall temperature gradient for different physical parameters for Case C.

Shooting method Cases
M

Conclusions
In this paper, MHD flow and transfer of heat for viscous fluid with changeable fluid properties over an exponentially stretching surface has been discussed. The problem has following governing parameters: Magnetic parameter M, Prandtl number Pr and parameter ò. Their effect on MHD flow and transfer of heat characteristics have been discussed. Main focus of our study has been to describe viscosity and thermal conductivity as functions of temperature. The boundary layer equations together with the boundary conditions have been reduced to nonlinear ordinary differential equations by using similarity variables. The resulting differential equations are then solved numerically by shooting method and verified by bvp4c and from the literature. The results are summarized as follows: • It is observed that skin friction and thermal boundary layer both increases with increment in magnetic parameter while velocity profile and wall temperature decreases.
• The Prandtl number causes a slight change in momentum boundary layer and skin friction whereas wall temperature and momentum boundary layer thickens for the case of variable viscosity. Thermal boundary layer reduces as Prandtl number rises .
• The parameter ò reduces both the skin friction coefficient and Nusselt number whereas it enhances thermal boundary layer thickness.