Free convection of a viscous electrically conducting liquid past a vertical stretching surface is investigated in the presence of a transverse magnetic field. Natural convection is driven by both thermal and solutal buoyancy. The original partial differential equations governing the problem are turned into a set of ordinary differential equations through a similar variables transformation. This alternate set of equations is solved through a Differential Transform Method (DTM) and the Pade approximation. The response of the considered physical system to the nondimensional parameters accounting for the relative importance of different effects is assessed considering different situations.
In recent the application of flow phenomena over a stretching surface is vital due to its uses in the production of materials in several industrial processes. However, these are composed with the heat and mass transfer phenomena and for the design of various equipment’s, the role of heat transfer is important. In particular, the production of plastic sheets, space vehicle aircrafts, gas turbines etc. the knowledge of radiating heat transfer is useful. To reveal the principle of buoyancy according to the engineering context its application is vast. The principle of buoyancy can be useful for the floating of the objects such as ships and boats, submarines, hydrometer, balloons and airships and many others. Further, in view of mathematics, the nonlinear equations are supreme factors for the studies relating to it. Many nonlinear equations don’t have an accurate analytic technique to solve for which numerical methods have been employed. There are some analytic methods like Adomian Decomposition Method (ADM), Differential Transform Method (DTM), Variation of Parameters Method (VPM), Perturbation method and Homotopy Perturbation Method (HPM) that can be employed to solve nonlinear equations easily.
Considering the free convection of air past a vertical plate in presence of earth’s gravitational field is one of the above cases was analyzed by Schmidt et al. [
For the current study, the collection is obtained for some special literature review [
There is an investigation by Usman et al. [
With reference to the aforesaid discussions, the aim of the present investigation is to reveal the impact of buoyancy forces on the twodimensional flow of conducting viscous fluid past an expanding surface. The present study overrides the earlier investigation by imposing an approximate analytical technique called Differential Transform Method and along with the enhanced solution is obtained by employing Pade approximant. The comparison also obtained and presented graphically.
A steady twodimensional flow of viscous fluid over a vertical stretching surface is undertaken in the present investigation. The flow occurs in the direction of xaxis and yaxis is transverse to it. The fluid becomes electrically conducting due to the applied magnetic field of strength B_{0} is acted along the flow direction i.e., normal to yaxis. The surface temperature and concentration are deployed as
Assuming the aforesaid conditions and following [
where
The corresponding boundary conditions are expressed as follows:
To transformed into the nondimensional form the followings [
Using the aforesaid equations,
where
Besides, the converted boundary conditions are prescribed as follows:
The physical quantities for the said problem are as follows:
Applying the nondimensional transformations (7) we have
Based upon the Taylor series expansion the methodology i.e., differential transform method, a semianalytical method, is illustrated here. To get a series solution in terms of polynomials the differential equations are converted into a set of recurrence relations. Zhou [
The transformation of k^{th} derivative of the defined function
The inverse transform of
Substituting
This is Taylor’s series expansion of
The method provide solutions in terms of convergent series with easily computable components. The aim of this article is to introduce the DTM as an efficient tools to solve the highly nonlinear differential equations. We choose a similarity transform variable to modify Navier–Stokes equation to a coupled highly nonlinear ordinary differential equation which then solved by using DTM along with Pade technique to get close form solution. To illustrate the simplicity and accuracy of its variants by comparing the results with numerical solution by using bvp5c technique, a MATLAB solver. This work introduces the DTM for two reasons:
This method gives a good sense of continuation to the Taylor series in the application of differentiation.
This method is gaining momentum among researchers due to its simplicity and has some pedagogical benefits.
This method constructs an analytical solution of linear as well as nonlinear differential equations in the form of a polynomial. Though it uses the general form of the traditional highorder Taylor series method, it overcomes the inefficient sides of Taylor series method which takes a long time for higher orders. Symbolic computations of the necessary derivatives of the data functions are computed by using iterative procedure for obtaining analytical solutions of differential equations. From
Original function  Transformed function 

The DTM transformations which we used in the present work are listed in
Original function  Transformed function 

( 

Taking the differential transforms of
where
Solving
To proceed with our work we have taken
After applying DTM to get an accurate solution of the boundary value problem, we have applied Pade Approximant.
The Pade approximant of
The power series form of the function
The notation
which has a Maclaurin expansion, agrees with
Equating the coefficients of
If
From the above expression,
Thus,
Pade approximant is used to bring the infinite boundary layer behaviour of the solution obtained by DTM. The Pade approximant gives a better closed form solution when applied after DTM, and it may still work where the DTM as well as Taylor series does not converge. For these reasons we have used Pade approximants. Since Pade approximant is a rational function, it can be used to get the initial values of the function of higher order which may contain an artificial singular point, but this can be avoided.
Following, we have calculated diagonal Pade approximant of order [2/2] of
Solving
We get,
So, the desired solutions of
An electrically conducting viscous liquid past a vertical stretching surface via the influences of thermal ad buoyancy is characterized in the present analysis. As a novelty of the investigation, we aim to solve the transformed differential equations using the Differential transform method, and the refinement of the solution is obtained by Pade approximant. Finally, the numerical results are compared with the earlier investigation carried out by the help of the numerical method. The variation of the physical quantities affecting the flow phenomena is presented via graphs and the physical significance of each parameter is deployed. At the time of computation the variation of parameters in the corresponding figure is displayed for the fixed values of other pertinent parameters laid down as;
The present section displays the validation graphs of transverse and longitudinal velocities, energy, and solutal profiles using the current methodology and the earlier numerical method via
In
The comparison plots for the fluid temperature and concentration in all these methods are displayed via
The computational behavior of the physical parameters, i.e., magnetic parameter, Prandtl number, thermal, and mass buoyancy parameters are exhibited in
The efficiency of the physical parameters such as magnetic parameter, Prandtl number, thermal, and mass buoyancy on the fluid temperature is displayed via
The computation of the physical quantities of interest, i.e., the coefficients of shear stress, rate of heat, and mass transfer for several values of contributing parameters are described in
The behavior of thermal and mass buoyancy parameters on the rate of mass transfer is presented in
The approximate analytical approach for the study of the steady twodimensional flow of viscous fluid in the presence of the magnetic field is analyzed in the present investigation. The transformed governing equations are tackled by using the Differential transform method and its refinement using the Pade approximant method for the various values of contributing parameters. The validation with the earlier numerical method was also established. However, the conclusive remarks for various physical parameters described earlier are presented below.
The validation with earlier study presents road map for the further investigation on the physical properties of contributing parameters.
The Pade approximant shows a better accuracy than that of the numerical as well as the DTM applied for the said problem.
Heavier viscous diffusivity produces higher Prandtl number retards the velocity distribution at all points within the flow domain.
Buoyant driven forces enhances the velocity distributions whereas impact is opposite in the case of temperature distributions.
Mass buoyancy parameter is counterproductive than that of thermal buoyancy for the enhancement in the shear rate coefficient.
Heavier species favours is to boost up the rate of mass transfer with increasing magnetic parameter.
constant
ambient temperature
magnetic field strength
velocity of the fluid along
fluid concentration
characteristic velocity
nondimensional skin friction
velocity of the fluid along
concentration of the plate
coordinate axes along and
ambient concentration perpendicular to the plate
molecular diffusivity
dimensionless stream function
thermal diffusivity
dimensionless velocity profile
kinematic viscosity
acceleration due to gravity
coefficient of thermal expansions
vortex viscosity
coefficient of solutal expansions
magnetic parameter
similarity variable
Nusselt number
dimensionless temperature
Prandtl number
dimensionless concentration
mass transfers from the plat
stream function
heat transfers from the plat
coefficient of viscosity
Reynold number
magnetic permeability of fluid
Schmidt number
fluid density
Sherwood number
thermal buoyancy parameter
fluid temperature
solutal buoyancy parameter
constant temperature of the plate
the wall shear stress
We are very much thankful to learned reviewers for their useful suggestions for the improvement in the quality of our manuscript.