Comparative study of chemically reacting Blasius and Sakiadis unsteady MHD radiated flow with variable conductivity

This study aims to investigate the effects of unsteady MHD chemically reacting radiated flow with variable conductivity about a flat plate in a uniform stream of fluid (Blasius flow), and about a sheet in a quiescent ambient fluid(Sakiadis flow) both under a convective surface boundary condition. Numerical solutions are offered graphically and tabular form with the aid of shooting approximation. Results for Blasius and Sakiadis flow cases are exhibited through plots for the parameters of concern. It is observed that the heat and mass transfer rate is high in Blasius flow when compared with Sakiadis flow.

Sekhar et al. [15] discussed MHD Maxwell fluid Blasius and Sakiadis flows of exponentially decaying heat source/sink. Radiation effects of Blasius and Sakiadis flows of nanofluid have been demonstrated by Uddin et al. [16]. Hady et al. [17] presented Blasius and Sakiadis slip flow in a thermal radiation convective surface boundary condition in the presence of nanofluids with porous medium. Later on, Anuar et al. [18] discussed nanofluid properties of Buongiorno model and thermo-physical properties with Blasius and Sakiadis flow problem of nano-liquids. Mustafa et al. [19] investigated magnetic field and convective boundary conditions in the presence of Sakiadis slip flow with Maxwell fluid. An analysis of Blasius and Sakiadis slip flow of MHD Jeffrey fluid in the presence of non-uniform heat source/sink has been deliberated by Prasad et al. [20]. Cattaneo-Chrisov heat flux model with heat transfer analysis slip effects of Blasius and Sakiadis flow of MHD radiative Maxwell fluid have been analysed by Vinod et al. [21]. Sekhar et al. [22] investigated mixed convection Couette flow of a nanofluid through a vertical channel. Nadeem et al. [23] studied a numerical investigation for steady flow of a nanofluid past a stretching sheet due to Brownian motion.Several researches utilized convective conditions as a passive technique [24][25][26][27][28][29][30].
Motivated by the above investigations, the present analysisis focused on the study of unsteady MHD heat and mass transfer flow the effects of unsteady MHD chemically reacting radiated flow with variable conductivity about a flat plate in a uniform stream of fluid (Blasius flow), and about a sheet in a quiescent ambient fluid (Sakiadis flow) both under a convective surface boundary condition. Numerical solutions are offered graphically and tabular form with the aid of shooting approximation. Results for Blasius and Sakiadis flow cases are exhibited through plots for the parameters of concern. It is observed that the heat and mass transfer rate is high in Blasius flow when compared with Sakiadis flow. The numerical results of skin friction, Nusselt number, and Sherwood number are presented in tabular form whereas the graphical results are presented and discussed for various physical parameters influencing the fluid flows and heat and mass transfer characteristics.Exact solutions obtained in this paper are useful for explainingthe flow physics in detail.

Mathematical formulation
Consider the unsteady laminar two-dimensional boundary layer flow of a viscous incompressible fluid past a semi-infinite porous stretching sheet coinciding with the plane 0 y  . The Cartesian coordinate system has its origin located at the leading edge of the sheet with the positive xaxis extending along the sheet in the upwards direction, while the y-axis is measured normal to the surface of the sheet and is positive in the direction of the sheet to the fluid. We assume that for time Subject to the boundary conditions i) Blasius problem , 0, , Where u and v are the velocity components in the x and y directions, respectively, T is the fluid temperature inside the boundary layer and  is the kinematic viscosity. Ishak et al [32] assume that the stretching velocity Where a and c constants (with 0 a  and 0 c  where 1 ct  ), and both have dimension 1 t  , we have a as the initial stretching rate 1 a ct  and it is increasing with time. In the context of polymer extrusion, the material properties, in particular, the elasticity of the extruded sheet may vary with time even though the sheet is being stretched by a constant force. With unsteady stretching, however, 1 a  becomes the representative time scale of the resulting unsteady boundary layer problem. We assume the surface temperature   , w T x t and concentration   , w C x t of the stretching sheet to vary with distance x and inverse square law for its decrease with time in the following form: Here b is the constant and has dimension temperature or length, with 0 b  and 0 b  corresponding to the assisting opposing flow, respectively, and 0 b  is for the forced convection limit (absence of buoyancy force). These particular forms of

been chosen in order
to obtain a new similarity transformation, which transforms the governing equations (1) to (4) into as a set of coupled ordinary differentiable equations, thereby facilitating the exploration of the effects of the controlling parameters.
We introduce now the following dimensionless functions f , and  with the similarity variable  (see Vajravelu Where primes denote differentiation with respect to . is the suction or injection velocity and r q is the radiative heat flux. The second and third terms in Eq.(2)are due to the buoyancy force. The "+" and "-" signs refer to the buoyancy assisting and buoyancy opposing flow situations, respectively.
In the present study, the thermal conductivity is assumed to vary linearly with temperature as