Viscous dissipation effect on Williamson nanofluid over stretching/shrinking wedge with thermal radiation and chemical reaction

This paper scrutinizes the effect of viscous dissipation on unsteady two-dimensional boundary layer flow of Williamson nanofluid over a stretching/Shrinking wedge. To express the boundary condition in concentration problem the passive control concept used. The governing PDEs are converted to ODEs by means of a similarity transformation before being solved numerically by finite difference scheme called Keller-Box method. The equations were numerically solved by using Matlab software 2013a. The characteristics of parameters such as wedge angle, unsteadiness, Williamson, slip, Brownian motion, thermophoresis, chemical reaction parameters, Prandtl number, Biot-number, Eckert number and Lewis number on velocity, concentration and temperature profiles and skin friction coefficient, Nusselt number and Sherwood number are presented in graphs and tables. The result of the study designates that the velocity profiles increased with an upsurge of wedge angle, unsteady parameter and suction parameter while it is diminished with an increase of Williamson and injection parameter. The temperature profiles upsurges with the distended Williamson parameter, Biot number and injection parameter, while it is declined for large values of wedge angle, unsteady and suction parameter. With an increase of Williamson, unsteady and suction parameter the concentration profiles upsurges, while it is decreased with an increase of wedge angle and injection parameter. The numerical results are compared with available literature and obtained a good agreement.


Abstract
This paper scrutinizes the effect of viscous dissipation on unsteady two-dimensional boundary layer flow of Williamson nanofluid over a stretching/Shrinking wedge. To express the boundary condition in concentration problem the passive control concept used. The governing PDEs are converted to ODEs by means of a similarity transformation before being solved numerically by finite difference scheme called Keller-Box method. The equations were numerically solved by using Matlab software 2013a. The characteristics of parameters such as wedge angle, unsteadiness, Williamson, slip, Brownian motion, thermophoresis, chemical reaction parameters, Prandtl number, Biot-number, Eckert number and Lewis number on velocity, concentration and temperature profiles and skin friction coefficient, Nusselt number and Sherwood number are presented in graphs and tables. The result of the study designates that the velocity profiles increased with an upsurge of wedge angle, unsteady parameter and suction parameter while it is diminished with an increase of Williamson and injection parameter. The temperature profiles upsurges with the distended Williamson parameter, Biot number and injection parameter, while it is declined for large values of wedge angle, unsteady and suction parameter. With an increase of Williamson, unsteady and suction parameter the concentration profiles upsurges, while it is decreased with an increase of wedge angle and injection parameter. The numerical results are compared with available literature and obtained a good agreement.

Introduction
Due to its applications the flow of fluid in a boundary adjacent to the wedge has great attention to the researchers and Engineers. Some of its applications are geothermal systems, polymer processes, crude oil extraction, cooling or heating of films/sheets, etc. Firstly the steady laminar flow above a wedge was investigated by Fankner and systems, polymer processes, crude oil extraction, cooling or heating of films/sheet. The novel govrning partial differential equation of momentum, energy and concentration are reduced in to non linear ordinary differential equation and then solved by implicit finite difference method known as Keller box. Inaddtion graphically the impact of different physical parametres such as Suction/Injection, Streching/shrinking, Biot number, Williamson parameter, wedge angle parameter, unsteady parameter, Prandtl number, Lewis number, Thermophoresis parameter, Brownian motion paremeter, Eckret number, radiation parameter and chemical reaction parameter on velocity, temperature and concentration profiles scrutinized. Moreover, in tabular the numerical values of volume friction, mass and heat transfer rate were examined.

Mathematical formulation
Unsteady two-dimensional incompressible laminar boundary flow of non-Newtonian Williamson nanofliuds over stretching/shirnking wedge is considered. In this study we consider u w (x) the velocity of stretching/ shirnking wedge, u e (x) free steam velocity, T ∞ ambient temperature, T w stretching/shirnking wedge tempearture, C w nanoparticle at stretching/shirnking wedge and C ∞ ambient nanopaticle. The physical coordinate system is chosen with x along the surface of the wedge and is measured from the origin and y is the coordinate normal to the surface (see figure 1). It is noted that the wedge is stretching when u w (x) > 0 whereas shirnking when u w (x) > 0. For an incompressible Williamson fluid the continuity and momentum equation are given by where ρ is the density, V is the velocity vector, S is the Cauchy stress tensor, b represents the specific body force vector and d/dt represents the material time derivative. The essential equations for Williamson fluid model are specified as follows: where p is pressure, I is identity vector, τ is extra stress tensor, μ 0 and μ ∞ are the limiting viscosities at zero and at infinite shear rate respectively, Γ > 0 is the time constant, A 1 is the first Rivlin-Ericson tensor and  g is defined as follows; where π is the second invariant strain tensor. Here we consider the case in which μ ∞ ≠ 0 and  g G < 1. Then we obtain is the ratio of viscosities.
The components of stress tensor are In the absence of body force the governing equations can be described in Cartesian system as listed below (See Aamir Hamid et al [28]).
The appropriate boundary conditions are (See Aamir Hamid et al [27]) is the ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid, c is the volumetric volume expansion coefficient and r p is the density of the particles, L is the velocity slip length with dimension of length, h f is heat transfer slip, T f is the temperature of the fluid v w is suction velocity at the wall. We can write the thermal radiation using Rosseland approximation for heat flux q r as follows (See Hayat et al [40] and Ramesh et al [44]) where s* is the Stefan-Boltzman constant, k* is the absorption constant. Assuming the temperature difference within the flow such that T 4 may be expanded in a Taylor series about ¥ T and neglecting higher orders. We get = -  [5]) and Rahman et al [11] ) Rahman et al [15] ), m is a positive constant related to the wedge angle which is )   0 m 1 and β is Hartree pressure gradient parameter which resembles to b = p W for a total angle W of the wedge. The constant moving parameter g is defined as g = , u u w e so that g > 0 corresponds to stretching wedge, g < 0 corresponds to a shrinking wedge, and g = 0 corresponds to flat plate.
In order to solve equations (11)-(13) subjected boundary condition (14) and (15) we introduce similarity transformations as follows (See Sattar [5] and Rahman et al [15]); We choose the stream function ( ) y x, y such that Equations (11)-(13) are transformed into the non-dimensional ordinary differential equation by applying the similarity transformation equation (18) as follows: With corresponding boundary conditions where ¢ f is dimensionless velocity, q is dimensionless temperature, f is dimensionless concentration and h is the similarity variable. The prime denotes differentiation with respect to h. The overall governing parameters are defined as; ( is Lewis number, is the wall mass transfer coefficient.
The skin friction C , f local Nusselt number Nu x and the Sherwood number Sh x are the important physical quantities of interest in this problem which are defined as is local Reynolds number.

Keller box method
The transmuted ordinary differential equations (15), (16) and (17) subject to boundary conditions (18) and (19) are solved numerically using an implicit finite difference method (Keller box) in combination with the Newton linearization techniques. The key features of this method are: 1. Only to some extent more arithmetic to solve than the Crank-Nicolson method.
2. Second-order correctness with arbitrary (non-uniform) x and y spacing's. 4. Solve the linear system by the block-tridiagonal-elimination method.
We consider the net rectangle in the h x plane shown in figure 2 and the net points defined as below where k i is the Dx-spacing and h j is the h D -spacing. Here i and j are the sequence of numbers that indicate the coordinate location, not tensor indices or exponents.
Since only first derivatives appear in the governing equations, centered differences and two-point averages can be constructed involving only the four corner nodal values of the 'box'. For example, if g represents any of the dependent variables then    Figure 3 revealed that the impact of the slip parameter on the velocity profiles within the boundary layer. From the figure it can be observed that the velocity profiles increase with a growing in values of slip parameter. The horizontal line which approaches to one (converges to 1) is an asymptote to the graph as h tends to infinity ( ) ¥ . The boundary layer thickness of velocity profile is an increasing function for slip parameter (ℓ). The boundary layer thickness decreases as the fluid velocity increases.
The influence of the stretching parameter g > 0 on the velocity profile is displayed in figure 4. From the figure it can be observed that the velocity profiles within the boundary layer intensified with the rising values of the stretching parameter. Moreover, the velocity boundary layer thickness increases with an increasing stretching parameter. Figure 5 displayed that the effect of shrinking parameter on the velocity profile. The figure reveals that the velocity boundary layer width declines when the shrinking parameter g rises from --0.01 to 0.1.   figure 18 it is found that as the values of Pr rises the temperature at the wedge surface decreases and the thermal boundary layer width shrinks. This shows that a fluid with higher Pr has relatively low thermal conductivity, which results in heat conduction and there by the thermal boundary layer and Ec.  thickness, and temperature drops. The concentration profile within the boundary layer increases with an increasing the value of Prandtl number ( ) Pr was displayed in figure 19. Near the wall of the wedge, the concentration growths with a growth in Pr and at some point away from the wall it starts falling before coming to a point far away from where it becomes to a stable position. The effect Eckert number ( ) Ec (viscous dissipation parameter) on temperature is plotted in figure 20. It is found that the temperature profile is increased with an increase of Eckert number ( ) Ec . This is due to the fact that heat energy is stored in the liquid due to the frictional heating. The impact of rising Eckert number is to increase the temperature at any point. Figure 21 is plotted to indicate the effect of radiation parameter ( ) R on temperature profiles. It is observed that the temperature profile   decreases for increasing values of R. This is due to the fact that an increase in the radiation parameter R leads to decrease in the boundary layer thickness and enhances the heat transfer rate. A ratio of the hot fluid side convection resistance to the cold fluid side convection resistance on a surface is the heat convection parameter (Biot number). From this fact we observed that when the values of heat   convection parameter upsurge the hot fluid side convection resistance reduces and as a result the surface temperature rises. Accordingly, figure 22 reveals that with an increase of Biot number the temperature profiles within the boundary layer raises. The impact of heat convection parameter on concentration profiles is plotted in figure 23. As the values of heat convection parameter raises the concentration profiles upsurges within boundary layer.
In figures 24-26 the effect of wedge angle parameter was plotted. From figure 24 we observed that as the value wedge angle parameter raises the velocity increased within boundary layer. This displays that velocity    Figure 27 indicates that when Brownian motion ( ) Nb parameter increases, the movement of nanoparticles from the hot surface to cold surface is happened and ambient fluid occurred. Due to this the   temperature and thermal boundary layer thickness rises. From figure 28 we see that when Brownian motion ( ) Nb upsurges volume fraction of nanoparticles within the boundary layer upsurges. It is interesting to note that Brownian motion of the nanoparticles at the molecular and nanoscale levels is a key mechanism in governing their thermal behavior. In nanofluid system, due to the size of the nanoparticles, Brownian motion affects the   heat transfer properties. As the particle size scale approaches to the nano-meter scale, Brownian motion on the surrounding liquids play an important role in heat transfer. The influence of Thermophoresis on concentration and temperature was displayed in figures 29-30. The graphs indicated that when the value of Thermophoresis increases both temperature and concentration profiles and boundary layer thickness are increased. This is         because of the fact that temperature difference between surface and ambient growths for larger thermophoresis parameter and hence the fluid temperature accelerates for both cases. Figure 31 displays the effects of Chemical reaction parameter on the nanoparticles volume fraction profile. From this figure we have observed that increasing the value of the chemical reaction reduces the concentration   Le may be because of a larger thermal diffusivity of the fluid for a constant mass diffusivity. This causes an increase of the flow within the boundary layer. Therefore, the graph reveals that when Lewis number ( ) Le increases the concentration upsurges at the wedge wall and at some point away from the wall it starts falling before coming to a stable position.

Conclusions
This study presents a mathematical model of unsteady boundary layer flow of Williamson nanofluid past a stretching/Shrinking permeable wedge with thermal radiation, viscous dissipation and chemical reaction. The study extended the previous work of Wubshet and Ayele [19] by taking into consideration the effects of unsteadiness parameter, Williamson fluid, chemical reaction, slip effect\and the passively controlled boundary condition. Fluid suction/injection is imposed on the wedge surface. The governing unsteady non-linear ordinary deferential equations(ODEs) are reduced to a set of a non-linear differential equations(DEs) by introducing appropriate similarity transformations and then solved numerically by using Keller Box scheme with MATLAB software(R2013a) for different values of the parameters. The obtained numerical results are in good agreement with the previously published data in limiting condition. The numerical results of velocity, temperature and concentration for the dimensionless parameters are presented graphically. As well as the numerical values of local skin friction, local Nusselt number and local Sherwood number are presented in tabular form. Depending on these the result of the study is summarized as follows:  1. Increasing suction A and unsteady l 1 parameters raises both the velocity and concentration profiles near the surface of the wedge but it diminutions the thermal boundary layer thickness.
2. Both temperature and concentration profiles are declining function within the boundary layer but the velocity boundary layer thickness enhanced for raising the values of β parameter.
3. For growing the values of Bi and Nt the thermal boundary layer thickness and the concentration profiles growths near the wedge surface.
4. The temperature profile declines within the boundary layer and the concentration profile enhanced near the wedge surface for rising values of Pr. 5. When the values of thermal radiation and Eckert number rises, the temperature profiles are amplified.
6. As the values of chemical reaction upsurges the concentration profiles diminutions but the concentration profiles grows with a rise of Lewis number.
7. For increasing the values of Nb both the temperature and concentration profiles are increased.
8. The skin friction coefficient rises with growing values of l , 2 b and A. But it declines for enlargement values of l . 1 9. The heat transfer rate ( ( )) q -¢ 0 and the mass transfer rate ( ( )) f -¢ 0 enhances with rising values of b A , and l . 1 10. The heat transfer rate is improved for rising values of Ec but it declines for rising values of l . 2 11. The mass transfer rate reduces for an increase of l 2 and Ec.