Dynamics of Triple Diffusive Free Convective MHD Fluid Flow: Lie Group Transformation

: This analysis is interested in the dynamic flow of incompressible triple diffusive fluid flowing through a linear stretched surface. The current study simulates when Boussinesq approximation and MHD are significant. As for originality, a comparative study of all the results for opposing and assisting flow cases is provided. Lie-group transformation is utilized to determine symmetry depletions of partial differential equations. The transformed system of ordinary differential equations is solved using the Runge-Kutta shooting technique. The impacts of magnetic parameter, buoyancy ratio parameter of temperature and concentration, and Lewis number on velocity, temperature, and concentration are depicted through graphs. We observed that the magnetic field parameter decelerates in velocity distribution for both fluid flow cases. Additionally, the same phenomenon was noticed with the buoyancy ratio parameters on both salt concentration distributions. Finally, the influence of heat and mass transfer rates decreases for both fluid flow cases with an increase in Lewis number.


Introduction
The flow of mass and heat transfer has many applications in science, industry, technical processes, and many theoretical or experimental disciplines.As examples, take aerospace, power generation, automotive, materials, and chemical processing industries [1][2][3].Magneto-hydro dynamics (MHD) is one of the branches of physics used to analyze the fluids dynamics with the help of magnetic effects.Its applications have been extensive in numerous disciplines ranging from the study of solar winds [4] to MHD-driven biomedical sensors [5] and actuators [6,7].Massoudi et al. [8] examined the impacts of a nanofluid's thermal radiation and magnetic field inside a nonagon inclined cavity embedded in a porous medium.Massoudi et al. [9] used a MHD W-shaped inclined cavity saturated with Ag/Al2O3 hybrid nanofluid for uniform heat generation/absorption.Lie theory is a field of group theory that deals with continuous symmetry.It has a component which is close to identity transformation.Sophie Lie devised symmetry analysis, also known as Lie group analysis, to identify point transformations that allow a differential equation to be transferred to itself.Almost every known accurate integration technique for ordinary and differential equations is gathered in this transformation [9].The Lie group transformation is a method of determining all aspects of a differential equation that does not require any extempore hypotheses or foreknowledge of the equation.
For most nonlinear differential equations, the approximate solutions are numerical due to the complexity of the problems [10].He et al. [11] discussed on new analytical methods for cleaning up the solution of nonlinear equations.Massoudi et al. [12] studied numerical techniques on magneto natural convection of SWCNT nanofluid inside a T-inverted cavity.There are many general methods for solving linear and nonlinear partial differential equations.The Lie group theory, explained in [13], is the usual approach.Symmetry groups are invariant transformations that do not change the structure of the equations.Using this transformation, one can find the exact solution of differential equations developed by Sophie Lie about one century ago.Nowadays, the method is used widely.It creates a new solution from an existing one, rather than looking for so-called similar solutions [14].This transformation has an n-independent variable partial differential system that can be converted to a system of 1 n − independent variables, and if 2 n = , then the situation is considered best.It is one of the most effective tools for developing similarity transformations.Scaling transformation is the most common tool used to apply to the boundary layer equation in fluid dynamics.By reducing the number of independent variables, this transformation converts a system of nonlinear coupled partial differential equations regulating fluid motion into a system of coupled ordinary differential equations.Using nanofluid, Uddin et al. [15] utilized this transformation to analyze the boundary layer in MHD fluid flow on a stretched surface.MHD heat transfer in thermal slip using a semi-infinite domain and Carreau fluid was discussed by Rehman et al. [16].Using this transformation, many researchers gave exact solutions to their problems [17,18].
The fluid motion created by the buoyancy forces is called free convection.Heat transfer in free convection depends on fluid circulation over and around the object, which is induced by temperature gradients, which create density gradients.In heat transfer, many applications were developed, such as food heating processing and cooling systems in which free convection is assertive.The sterilization process of food in cans and meat freezing are controlled by free convection.In the presence of a chemical reaction, researchers [19] analyzed an unsteady free convection MHD flow around a vertical cone in porous media with variable heat and mass flux.A lot of research is carried out in the area of free convection MHD flow [20][21][22][23][24][25].
In general, free convection is merely due to heat transfer and not in the mode of forced flow.In gas or liquid, the density differences are induced by temperature differences.Due to its numerous applications, the convective process has grown important over the last century.Bernard started the convective process experiments after that carried out by Rayleigh [26].Triple diffusive free convection [27] develops when a fluid is applied to three density gradients with different diffusion rates.In this study, the triple diffusion is formulated by the diffusion of heat and species concentrations, a linear stability analysis for triple diffusive convection in Oldroyd-B fluid is carried out, and the expression of Rayleigh number for stationary and oscillatory convection is achieved [28].Patil [29] examined a quadratic mixed convective nanofluid flow over a wedge by considering viscous dissipation.He observed that the rate of heat transfer increases with an increase in Biot number.In the normal mode theory, introducing a Maxwell fluid to a saturated porous layer causes triple diffusive convection [30].In a time-dependent model, Khan et al. [31] investigated a triple diffusive natural convective flow along with a vertical plate.The heat transfer rate increased as the volume fraction distribution of nanoparticles increased.Triple diffusive convective importance was analyzed [32][33][34][35][36][37][38].
Few works have been performed using Lie group transformation analysis, as per the authors' knowledge of the literature in most practical uses; however, the components of mass and heat are inextricably linked.The heat and mass diffusion components, on the other hand, are invariably coupled in most real-world applications.This fact prompted us to investigate the combined impacts of mass diffusion and heat on magnetohydrodynamic free-convective flow in the boundary layer.As far as we can tell, the findings of this work appear to be perfectly consistent with previous research, and due to their simplicity, easily transferable to relevant applications.

Mathematical Model and Formulation
The rate of mass change and heat transfer performance of an electrically conducting viscous fluid in a steady, triple-diffusive, two-dimensional hydro-magnetic flow follow.
A uniformly strong magnetic field B is introduced perpendicular to the flow direction.
Except for the influences of density variation on concentration and temperature, all liquid properties are assumed to be uniform.The surface temperature should be maintained at w T , higher than the constant.With the help of Lie group transformation analysis, we studied a triple-diffusive, free-convective, 2D, steady laminar flow through a stretching surface and its incompressible fluid flow model.
Under the pre-defined assumptions mentioned above, the governing equations are expressed as shown below [18].

0, uv xy
Along the boundary conditions To non-dimensionalize the above-mentioned Equations from (1) to (5) together with Equation ( 6), the following similarity transformations are considered.
We substitute above non-dimensional quantities into Equations ( 1)-( 5), along with the boundary conditions (6), and then complete simplification results in the following non-dimensional differential equations: With the boundary conditions where 12 12 12 ,, are the thermal, salt_1, and salt_2 solutal buoyancy ratios.

Scaling Transformations
The stream function  is introduced before scaling modifications are applied, as demonstrated below [17]: Replace the above Equation ( 14) in to ( 8)-( 12) together with the boundary conditions (13).Obviously, Equation ( 8) is satisfied by them.Additionally, remaining equations are transformed as below: With the boundary conditions We suppose that  , as a smaller scale transformation parameter.Then, the transformation F (a specified set of Lie-group transformation analysis) is reflected as shown below: , , , , , r r r r r r are real numbers.The point of the transformation defined through (21) is to convert the coordinates as ( ) Equations ( 22)-( 25) will remain invariant after the translation, if the exponent of this converted system of equations which satisfies the resulting linear equations is as follows: By solving above linear Equations ( 26)-( 29) simultaneously, we find: Substituting the above values (30) The Taylor series expansions are: x x x r y y r A simple algebraic expression for the above transformations (31) with the help of Taylor series expansion leads to a mono parametric group of transformations in the form of the characteristic equation given below: From Equation (32), we can easily obtain new similarity transformations as: where  is the similarity variable and 12 , , , f    are the dependent variables.Now, we substitute the above quantities specified in Equation (33) into PDEs ( 16)-( 18) in boundary conditions mentioned in (19).We obtain the set of ordinary differential equations as: Together with the boundary conditions: with the predefined parameters

Numerical Results and Discussion
The effects of triple diffusive free convective magnetohydrodynamic fluid flow in a linearly stretching sheet were studied numerically.Lie group transformation analysis converts a collection of nonlinear partial differential equations and boundary situations into a set of ODEs.BVP4C resolved the system of the reduced nonlinear ODEs with corresponding boundary conditions in MATLAB.Table 1 shows a great correlation between the present results and the results of Ferdows et al. [17].decreases in assisting and opposing flows.It can be seen that increasing the value of the magnetic field parameter decreases the momentum boundary layer thickness.This is because the strong magnetic field inside the boundary layer increases the Lorentz force, which strongly resists flow in the opposite direction.It is also worth noting that the presence of a magnetic field reduces velocity near the wall while increasing velocity far away.The effects of magnetic parameters on temperature and concentration distributions are shown in Figures 2-4.As the value rises, the fluid becomes warmer, raising the temperature.The thickness of the thermal boundary layer is always increased by the presence of a magnetic field, and the concentration field is likewise increased by the magnetic parameter.M on salt_2 concentration.
The influences of the Prandtl number are seen in Figures 5-8.The dimensional velocity profile grows as the Prandtl number increases in the opposing flow case and reduces in the helping flow case, as seen in Figure 5.In contrast, in the natural convective flow of a regular fluid over a vertical surface, the thickness of the hydrodynamic boundary layer reduces as the Prandtl number grows, and the vertical flow velocity drops.It is worth noting that, unlike in the classical analysis of natural convection on a vertical surface, where the Prandtl number appears in the energy equation, the Prandtl number appears in the momentum equation in the present analysis and thus has a different effect on the velocity profiles than in the classical velocity profiles.The same phenomenon can be seen in the salt_1 and salt_2 concentration profiles (see Figures 7 and  8).The temperature profile falls as the Prandtl number grows, as shown in Figure 6.An increase in the Prandtl number reduces the thermal boundary layer thickness, since the Prandtl number signifies the ratio of momentum diffusivity to thermal diffusivity.In heat transfer problems, Pr controls the thicknesses of the momentum and thermal boundary layers.both the flow circumstances.In helping flows, the dimensionless concentration increases with increasing Lewis numbers inside the concentration boundary layer, whereas buoyancy opposes and resists flows.Lewis number determines the thickness of the concentration boundary layer physically.That is, the greater the Lewis number, the thinner the concentration boundary layer will be.For Lewis numbers, the dimensionless wall concentrations decline quickly to zero in both buoyancy assisting and opposing flows, and the inside wall concentration is equivalent to the ambient concentration.In fluid flow instances, the thickness of the thermal boundary layer decreases while the momentum boundary layer increases.1.The friction factor rate is enhanced in both fluid flow cases for increases in Lewis numbers  1 and  2 .2. The friction factor coefficient decreases with increases in buoyancy ratio parameters  1 and  2 , but the opposite behavior can be observed for the Nusselt and Sherwood numbers.3.In the assisting flow case, the dimensionless concentration rises as the Lewis number rises; for buoyancy, the opposing flow case also remains the same.4. The magnetic field parameter decreased the velocity distribution by the effect of Lorentz force. 5.In both assisting and opposing flow instances, the Prandtl number increases the skin friction coefficient in assisting flow and decreases it in opposing flow in heat and mass transfer rates.

Limitations and Future Scope
6.1.Limitations 1.The flow was considered as not time-dependent and incompressible.
2. The turbulence due to the hematite nanoparticles interaction was neglected.
3. Away from the object's surface, viscous effects can be considered negligible, and potential flow can be assumed.4. The viscosity, conductivity, and density properties were constant.

Future Scope
1. Triple diffusion analysis can include variable viscosity and nonlinear convection flow properties.2. Variable conductivity and unsteady nonlinear flow characteristics of various flow characteristics with chemical species in the triple diffusion process.3. The investigations mentioned above could also be extended by using various techniques, such as mess-free methods, the finite-difference scheme, the spectral element method, the finite element method, the Keller-box method, homotopy analysis, and spectral methods.
flow and buoyancy opposing flow, respectively.

Figures 9 -
Figures 9-16 illustrate the effects of buoyancy ratio parameters on velocity, temperature, and salt_1 and salt_2 concentration distributions.Increasing the values of buoyancy ratio parameters increases the velocity for smaller values of  , as shown in

Figures 9 and 13 .
Figures 9 and 13.From the graphical representations of Figures 10-12 and 14-16, it is noticed that an increase in buoyancy ratio decreases the temperature and salt_1 and salt_2 concentration profiles in both assisting and opposing cases.
for both the flow circumstances.In helping flows, the dimensionless concentration increases with increasing Lewis numbers inside the concentration boundary layer, whereas buoyancy opposes and resists flows.Lewis number determines the thickness of the concentration boundary layer physically.That is, the greater the Lewis number, the thinner the concentration boundary layer will be.For Lewis

Figure 25 .
Figure 25.Surface plots of the

Figure 26 .
Figure 26.Surface plots of the

Figure 27 .
Figure 27.Surface plots of the

Figure 28 .
Figure 28.Surface plots for the

Figure 29 .
Figure 29.Surface plots for the

Figure 30 .
Figure 30.Surface plots for the

Figure 31 .
Figure 31.Surface plots for the

Figure 32 .
Figure 32.Surface plots for the

Figure 33 .
Figure 33.Surface plots for the .
Figure 34  shows that when  2 increases, the skin friction coefficient decreases at a rate of −0.31689 ( 0)   or −0.32663 ( 0)   .For assisting and opposing flow, the Nusselt and Sherwood numbers escalate rates of 0.019164 and 0.022813, and 0.022813 and 0.026919, respectively.

Figure 34 .
Figure 34.Surface plots for the

Figure 35 .
Figure 35.Surface plots for the

Figure 37 .
Figure 37. Surface plots for the

Figure 38 .
Figure 38.Surface plots for the

Figure 39 .
Figure 39.Surface plots for the

Figure 40 .
Figure 40.Surface plots for the

Figure 41 .
Figure 41.Surface plots for the

Figure 42 .
Figure 42.Surface plots for the

Figure 43 .
Figure 43.Surface plots for the

Figure 44 .
Figure 44.Surface plots for the

A
steady, incompressible, and laminar boundary layer was used to investigate the effects of triple diffusion on boundary layer flow, heat, and mass transfer through a linearly stretching sheet.Numerical findings for surface ( ) various values of the governing parameters.The following are the investigation's principal findings: (21) the scaling transformations given in(21), we get:

Table 1 .
Comparison values of Nusselt number