MHD HEAT AND MASS FLOW OF NANO-FLUID OVER A NON-LINEAR PERMEABLE STRETCHING SHEET

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this study, we examine the Magneto-hydrodynamics (MHD) heat and mass flow of nano-fluid over a non-linearly permeable stretching sheet. The resulting partial differential equations are converted to a system of ordinary differential equations using the similarity transformation and solved numerically using shooting technique with fourth order Runge-Kutta method. The effect of some fluid parameters on the momentum, thermal and nanoparticle volume fraction boundary layers were expatiated for prescribed surface temperature and constant surface temperature through graphs. An excellent agreement was found when the results obtained in this study were compared with results in previous literature.


INTRODUCTION
In the past few years, MHD heat and mass flow has gained tremendous ground in engineering and scientific processes. Boundary layer flows due to a stretching sheet played important roles in technological processes such as paper production, hot rolling, glass-fibre production, extraction of polymer sheets etc. Sakiadis [1] did the pioneer work on continuous flow of a boundary layer.
Crane [2] analyzed boundary layer flow past a stretching plate. A stretching sheet with suction or blowing using the method of similar solution was studied by Gupta and Gupta [3].
Using numerical methods, Vajravelu [4] worked on heat transfer over a non-linear stretching sheet. He showed that the heat flow is always from the stretching sheet to the fluid. Wang [5] researched on suction and surface slip in a viscous flow on a stretching sheet. Cortell [6,7] investigated stretching surface of a fluid with internal heat generation through a porous medium.
Prasad et al [8] analyzed non-linear stretching sheet of mixed convection with variable fluid properties.
Lately, researchers began to show interest in nano-fluid due to its importance in many processes in heat transfer such as petroleum refining processes, pharmaceutical processes, engine cooling/vehicle thermal management, chiller, fuel cells, domestic refrigerator, in grinding etc.
Nano-fluid is a fluid containing nano-particles (less than 100nm). Common base fluids include water, ethylene glycol and oil (convectional mineral oils, crude oil and oils refined from crude oil). Khan and Pop [9] studied stretching sheet of a nano-fluid flow. Rana and Bhargava [10] investigated a non-linear stretching sheet of a nano-fluid flow and heat transfer. Ibrahim and Shankar [11] studied a nano-fluid transfer past a permeable stretching sheet with velocity, thermal and solutal slip. They concluded that local Nusselt number decreases with an increase in Brownian motion and thermophoresis parameter. Fauzi et al [12] worked on a non-linear shrinking sheet with slip effects at a stagnation point flow. Das [13] investigated a nano-fluid flow with partial slip over a non-linear permeable stretching sheet. He showed that an increase in the slip parameter and non-linear stretching parameter leads to decrease in the velocity of the nano-fluid. Eid and Mahny [14] researched on non-Newtonian nano-fluid flow of an unsteady MHD transfer over a permeable stretching wall with heat generation/absorption. Jahan et al [15] studied regression and stability analyzed on nano-fluid heat transfer past a convectively heated permeable stretching/shrinking sheet. Senge et al [16] investigated MHD flow over a stretching sheet in a thermally stratified porous medium.
The objective of this research work is to extend the work of Das [13] by examining the megnetohydrodynamics heat and mass flow of a nano-fluid over a non-linearly permeable stretching sheet. The ordinary differential equations are solved numerically using shooting method along with fourth order Runge-Kutta method. The effects of the parameters on the fluid velocity, temperature and concentration distributions is discussed and shown graphically.

MATHEMATICAL MODELING AND FORMULATION
Consider an incompressible, two-dimensional steady boundary layer flow of MHD heat and mass transfer of a nano-fluid over a non-linear permeable stretching sheet coinciding with the plane y = 0. The flow takes place at y ≥ 0 ( y is the coordinate measured normal to the surface of the stretching sheet in the vertical direction). The flow is generated as a result of a sheet which comes out of a slit at the origin (x = y = 0). The stretching velocity u w = ax n is assumed to vary non-linearly with the distance from the slit, where a > 0, n is a non-linear stretching parameter and x is the coordinate measured along the surface. The surface of the sheet is subjected to a Prescribed Surface Temperature (P.S.T.) and a Constant Surface Temperature (C.S.T.) which are given as: where b > 0, r is the surface temperature parameter in the Prescribed Surface Temperature (P.S.T.) boundary condition, T ∞ is the ambient temperature, T w is the temperature at the surface. The surface is also maintained at constant concentration, C w and its value is assumed to be greater than the ambient concentration, C ∞ which is given as: A uniform magnetic field of strength B o is assumed to be applied at y > 0 normal to the stretching sheet. The governing equations under this consideration are as follows: The associated boundary conditions are: In equations (4) -(9), u and v are the velocity components along the x -and y -axes respectively, ν is the kinematic viscosity, σ is the electrical conductivity, ρ is the fluid density, K is the constant permeability of the porous medium, α is the thermal diffusivity, τ = (ρc) p (ρc) f is the ratio between the effective heat capacity of the nano-particle material and heat capacity of the fluid, c is the volumetric volume expansion coefficient, ρ f is the density of the base fluid, D T is the thermophoretic diffusion coefficient, D B is the Brownian diffusion coefficient, v w is the suction/ injection and u s is the slip velocity which is assumed to be proportional to the local wall stress as follows: where l is the slip length as a proportional constant of the slip velocity.
Equations (5) -(7) are transformed into the non-dimensional ordinary differential equations by the following transformation where ψ is the stream function defined as: x The transformed ordinary differential equations are as follows: Subject to the boundary conditions as follows: is the slip parameter for liquids The physical quantities of main interest are the skin friction coefficient, C f , the local Nusselt number, Nu x , and the Sherwood number, Sh x , which are defined as follows: (18) is the surface shear stress is the wall mass flux µ is the dynamic viscosity and k is the thermal conductivity. By using similarity variables in (18), we have

METHOD OF SOLUTION
We solve the non-linear differential equations (11) − (13) for Prescribed Surfaced Temperature (P.S.T.) and equations (11), (14) and (15)    tends to blow the nano-particle volume fraction boundary layer away from the surface since hot surface repels the submicron-sized particles from it, thereby forming a relatively particle-free layer near the surface.

CONCLUSION
The effect of Magneto-hydrodynamics (MHD) heat and mass flow of nano-fluid over a nonlinearly permeable stretching sheet is investigated. From the analyses, the following conclusions may be drawn: (1) An increase in non-linear stretching parameter enhances velocity and temperature profiles for both P.S.T. and C.S.T. cases and also the nano-particle concentration profiles for P.S.T. case while it decreases for C.S.T. case.
(2) The temperature profiles increase and nano-particle concentration profiles decrease with increasing values of magnetic and permeability parameters whereas the velocity profiles decreases for P.S.T. case and increase for C.S.T. case.
(3) An increase in Prandtl number decreases and increases the thermal boundary layers for P.S.T. and C.S.T. cases respectively.
(4) The nano-particle concentration profiles increase with increasing Brownian motion parameters for both P.S.T. and C.S.T. cases and also the temperature profiles for P.S.T. case while it decreases for C.S.T. case.
(5) An increase in thermophoresis parameter enhances the temperature profiles and decreases the nano-particle concentration profiles.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.