EFFECT OF CHEMICAL REACTION ON MAGNETOHYDRODYNAMIC WILLIAMSON NANOFLUID WITH VARIABLE THICKNESS AND VARIABLE THERMAL CONDUCTIVITY

The effect of chemical reaction on an MHD boundary layer flow with variable thickness and variable thermal conductivity on a stretching sheet has been examined. Here is an investigation done on the chemical reaction parameter along with the slip parameter and velocity index parameter to a non-flat sheet. Similarity transformations are applied to convert the PDEs into a system of non-linear ODEs. The nonlinear coupled ODEs are boundary value problems. These are reduced to IVP by the shooting technique. Finally, the IVPs are solved by using the Adam-Bashforth Moulton method. The velocity, concentration, temperature profiles have been discussed for different parameters.


INTRODUCTION
Nanotechnology has been quickly growing in the recent years. The liquid containing nanometer-sized particles are called nanofluids. Nanoparticles range is between 1 and 100nm.  [1] introduced the fundamental vision of consolidating the nanoparticles inside the base fluid to upgrade the thermal conductivity. Buongiorno [2] built up a model with analytic solution for convective heat transfer in a Brownian diffusion of nanofluid. He watched the impact of diffusion and thermophoresis in nanofluid. Daungthongsuk and Wongwises [3], Wang [4], Mujumdar [5] and Kakac and Pramuanjaroenkij [6] have presented excellent results on the flow of nanofluids.
A.V.Kuznetsov [7] studied the convective nanofluid in a vertical plate, later they extended their work for a permeable medium. The model is utilized for the nanofluid joins the impacts of Brownian motion and thermophoresis. N. Bachok, A. Ishak [8] described a nanofluid on a moving semi-infinite plate in a uniform stream to the steady boundary layer. In this plate, it is accepted to be moved in the equivalent or backwards path to the free stream. W. A. Khan, I. Pop [9] both investigated numerically the boundary layer laminar fluid stream from the extending of a level surface in nanofluid. In this nanofluid, it contains the effects of Brownian motion and thermophoresis. P.
Rana and R. Bhargava [10] have studied laminar boundary layer steady state flow of a nanofluid over a nonlinearly extending sheet. In this additionally nanofluid, it consolidates the impacts of thermophoresis and Brownian motion. G. K. Ramesh et. al. [11] have explored the magneto hydrodynamic stagnation point flow on a non-level plate.It is stretched in this plate.And the effects of magnetic field and thermal radiation have also been investigated. In this paper, it has been observed that temperature and nanoparticles fraction distribution has increased due to the raising value of wall thickness parameter. Mahapatra & Gupta [12] have analyzed heat transfer in the stagnation point flow induced by a stretching sheet. The transient MHD laminar free convection stream of nanofluid has been examined by S. Mahmud et. al. [13]. The flow is past on a vertical surface which is permeable and extendable under increasing velocity. In this process, it has taken nanoparticles which are 23 Al O and Cu . Consequently, it has come to the modulus of skin friction coefficient.
M.M. Rashidi et. al. [14] have studied the boundary layer flow of nanofluids over a continuously moving surface. They researched the velocity and temperature profiles. R.V. Williamson [15] described the flow of pseudo plastic materials. S. Nadeem, S.T.Hussain [16] inspected the 2D flow of the Williamson fluid over an extending sheet under the impacts of nano-sized particle additionally depicted the nano Williamson fluid. Shagaiya et. al. [17] introduced nano fluid flow on non-linear extended surface with variable thickness in the occurrence of electric field.

EFFECT OF CHEMICAL REACTION ON MHD WILLIAMSON NANOFLUID
In the present article, the numerical outcomes of Shazwani Md. Razi et. al. [18] have been replicated with a convincing agreement and extended by considering the chemical reaction.

MATHEMATICAL FORMULATION
Consider a 2D, laminar, incompressible MHD Williamson nanofluid flow through a plate in a porous medium. The starting point is here situated at cleavage from which the sheet is drawn through the fluid medium. The sheet has been stretched exponentially with velocity m is an index of velocity power, 0 U and b are dimensional constants. The sheet is moving along x − axis direction and it is a non-flat. The thickness of a sheet varies as ( ) Here, A is a very small quantity so that the sheet is sufficiently thin.
Magnetic field is applied to the perpendicular direction of fluid flow.
The governing equations for the above mathematical model is as follows.
Where  is fluid density, Г is material parameter, σ is electrical conductivity, k is porous medium permeability,  is Stefan-Boltzmann constant and * k mean absorption coefficient. Boundary conditions can be written as In Roseland approximation, Expand 4 T as a Taylor's series expansion and neglect the higher differentials then *32 *2 16 3 The following process was implemented to convert the PDEs to ODEs The final dimensionless form of the governing model is  The boundary condition (2.5) getting into the following dimensionless form

SOLUTION METHODOLOGY
The equations (2.7), (2.8), (2.9) are coupled nonlinear boundary value ordinary differential equations. Their numerical solution has been obtained by converting into the first order initial value ODEs using shooting method. In this, it shoots our trajectories until it is discovered a trajectory that has the desired boundary value. In this process, the equations (2.7), (2.8) and (2.9) are rewritten and stated below.
If it is used the following notations ,, r s t are missing conditions and can be calculated by Newtoon-Raphson method until the following conditions are met.

NUMERICAL SOLUTIONS
In the present paper, the profiles of velocity, temperature and concentration have been analyzed.
These profiles are expressed in terms of graphs for different choice of flow parameters. Effect of kp , R, Le,  , Pr,  on velocity, temperature and nanoparticle volume fraction profiles are discussed.  n which permits more fluid to fall past the sheet. Fig. 3 shows the increasing of temperature profile for an increasing of n1 which prompts to the reducing of density of thermal boundary layer.

CONCLUSION
On the basis of the analysis of solution, the following results are obtained.
• The velocity profile has been decreased for an increase of 1 2 3 , , , n n n  with 1 m  and slightly increased for increasing of m, increased for more than one of m.
• The variation of temperature has been decreased for an increase of its slip parameter, wall thickness parameter for less than one of m, but increased for the raising values of m .
• The profile of nanoparticle volume fraction has been decreased by increasing value of  .
• The concentration profile has been reduced for the raising the value of  .