Assuming the two dimensional MHD laminar boundary layer heat and mass transfer flow past an impermeable stretching wedge with the influence of thermal radiation, heat generation and chemical reactionand moving with the velocity u
w
(x) in a nanofluid, and the free stream velocity is u
e
(x), where x is the coordinate measured along the surface of the wedge. The sketch of the physical configuration and coordinate system are shown in Figure 1 by following Khan and Pop [[27]].
Here u
w
(x) >0 corresponds to a stretching wedge surface velocity and u
w
(x) < 0 corresponds to a contracting wedge surface velocity, respectively. Instantaneously at time t > 0, temperature of the plate and species concentration are raised to T
w
( > T
∞) and C
w
( > C
∞) respectively, which are thereafter maintained constant, where T
w
, C
w
are temperature and species concentration at the wall and T
∞, C
∞ are temperature and species concentration far away from the plate respectively. A strong magnetic field B = (0, B
0, 0) is applied in the y- direction. Under the above assumptions and usual boundary layer approximation, the MHD Mixed convective nanofluid flow governed by the following equations (see [[21]] and [[22]]);
(1)
(2)
(3)
(4)
Here in equation (2) the 3rd term on the right hand side is the convection due to thermal expansion and gravitational acceleration, the 4th term on the right hand side is the convection due to mass expansion and gravitational acceleration and the 5th term generated by the magnetic field strength because a strong magnetic field B = (0, B
0, 0) is applied in the y- direction.
Again in equation (3) the 2nd term on the right hand side is the effect of heat generation [[40]] on temperature flow, 3rd term on the right hand side expressed the radiative [[32]] heat transfer flow, and the last term indicates the Brownian motion due to nanofluid heat and mass transfer flow.
Also in equation (3) the 2nd term on the right hand side is the thermophoresis diffusion term due to nanofluid flow and the 3rd term is the rate of chemical reaction on the net mass flows.And the boundary condition for the model is;
(5)
In order to conquers a similarity solution to eqs. (1) to (4) with the boundary conditions (5) the following similarity transformations, dimensionless variables are adopted in the rest of the analysis;
(6)
For the similarity solution of equation (1) to (4) with considering the value (from the properties of wedge, see reference [[27]]) u
e
(x) = ax
m, u
w
(x) = cx
m, where a, c and m (0 ≤ m ≤ 1) are positive constant. Therefore, the constant moving parameter λ is defined as λ = c/a, whereas λ < 0 relates to a stretching wedge, λ > 0 relates to a contracting wedge, and λ = 0 corresponds to a fixed wedge, respectively.
From the above transformations the non-dimensional, nonlinear, coupled ordinary differential equations are obtained as;
(7)
(8)
(9)
The transformed boundary conditions are as follows;
(10)
where the notation primes denote differentiation with respect to η and the parameters are defined as:
Magnetic parameter,
Pressure gradient parameter,
Grashof number,
Modified Grashof number,
Thermal convective parameter,
Mass convective parameter,
Local Reynolds number,
Prandtl number,
Heat source parameter,
Lewis number,
Brownian motion parameter,
Thermophoresis parameter,
Radiation parameter,and
Chemical reaction parameter,
Significant prominent that numerous of non-dimensionalized thermo-fluid constraints are known as “local parameters”. This methodology is effective and has been in practice for a number of years. It is an effectual methodology which engrosses the x-dependence into correctly scaled dimensionless numbers. The solutions remain valid and correct and the reviewer is referred to the following references corroborating this approach- Khan and Gorla [[44]] and Mahmoodet al. [[45]]. In this context, M is a local magnetic body force number (Mahmoodet al. [[45]]) and λ
T
is therefore a function of local thermal Grashof number and λ
m
is a function of local species Grashof number (Khan and Gorla [[44]]).