## Introduction

A thorough review of the literature finds that three-dimensional developing mode models of thermal energy and mass transfer across a heated surface that is expanding vertically while also having hybrid-Prandtl nanofluid present have not yet been addressed. Due to Soret and Dufour effects' inclusion, the mathematical model is developed as being more sophisticated. With a heat source and Joule heating phenomena, a changing magnetic field is introduced. In addition, the hybrid nanofluid has collected variable features in terms of mass diffusion and thermal conductivity. The numerical calculations are done for Silver (Ag), Molybdenum Disulfide (MoS2) nanoparticles with Ethylene glycol (C2H6O2) as the base fluid using a boundary layer approach to the mathematical formulation. A finite element simulation is used to develop complex models. Since there are several potential solutions, this new inquiry is divided into five sections. Section "Analysis of flow" presents the problem formulation. Section "Galerkin finite element algorithm: a computational approach" provides an overview of the numerical approach. Section "Results and discussion" of the report discusses the results. This study is concluded in section "Core points and conclusions".

## Analysis of flow

Hybrid nanostructures with properties of heat conduction and solvent molecules in Prandtl liquid are inserted toward a heated area while being influenced by a dynamic magnetic field. A porous surface is used to examine the velocity and heat energy generated by the nanoparticles as well as the effects of Dufour and Soret when temperature variable mass transport and thermal conductivity are present. $$Ag$$ an is referred to as a nanoparticle, and the composite of $$Ag$$ and $$Cu$$ is known as a hybrid nanostructure. Table 1 provides examples of the thermal characteristics of $$Ag$$ and $$Cu$$. Figure 1 displays the general concept of the present system. It observed that the magnetic field is inserted along the y-direction, with the x-axis supposed to be in the vertical and the y-axis assumed to be in the horizontal.

Figure 2 displays the schematic chart representation of the mathematical model proposed in this study.

PDEs that characterize the issue include the following40,41,42

$$\frac{\partial u}{{\partial x}} + \frac{\partial v}{{\partial y}} + \frac{\partial w}{{\partial z}} = 0,$$
(1)
\begin{aligned} & u\frac{\partial u}{{\partial x}} + v\frac{\partial u}{{\partial y}} + w\frac{\partial u}{{\partial z}} = \left( {\beta_{hnf} } \right)_{T} g^{*} \left( {T - T_{\infty } } \right) + \left( {\beta_{hnf} } \right)_{C} g^{*} \left( {C - C_{\infty } } \right) \\ & \quad - \frac{{\sigma_{hnf} }}{{\rho_{hnf} }}B_{o}^{2} A^{2} \left( {x + y} \right)^{{ - \frac{2}{3}}} u - \mu_{hnf} \frac{u}{{K_{1} }} + \nu_{hnf} \left[ {\frac{A}{C}\frac{{\partial^{2} u}}{{\partial z^{2} }} + \frac{A}{{2C^{3} }}\frac{{\partial^{2} u}}{{\partial z^{2} }}\left( {\frac{\partial u}{{\partial z}}} \right)^{2} } \right], \\ \end{aligned}
(2)
\begin{aligned} & u\frac{\partial v}{{\partial x}} + v\frac{\partial v}{{\partial y}} + w\frac{\partial v}{{\partial z}} = \nu_{hnf} \frac{{\partial^{2} v}}{{\partial z^{2} }} + \left( {\beta_{hnf} } \right)_{T} g^{*} \left( {T - T_{\infty } } \right) + \left( {\beta_{hnf} } \right)_{C} g^{*} \left( {C - C_{\infty } } \right) \\ & \quad - \frac{{\sigma_{hnf} }}{{\rho_{hnf} }}B_{o}^{2} A^{2} \left( {x + y} \right)^{{ - \frac{2}{3}}} v - \mu_{hnf} \frac{v}{{K_{1} }} + \nu_{hnf} \left[ {\frac{A}{C}\frac{{\partial^{2} v}}{{\partial z^{2} }} + \frac{A}{{2C^{3} }}\frac{{\partial^{2} v}}{{\partial z^{2} }}\left( {\frac{\partial v}{{\partial z}}} \right)^{2} } \right], \\ \end{aligned}
(3)
\begin{aligned} & u\frac{\partial T}{{\partial x}} + v\frac{\partial T}{{\partial y}} + w\frac{\partial T}{{\partial z}} = \frac{1}{{\left( {\rho c_{p} } \right)_{Thnf} }}\frac{\partial }{\partial z}\left( {K_{Thnf} \left( T \right)\frac{\partial T}{{\partial z}}} \right) \\ & \quad + \frac{{Q_{0} }}{{\left( {\rho c_{p} } \right)_{hnf} }}\left( {T - T_{\infty } } \right) + \frac{{DK_{T} }}{{C_{s} C_{p} }}\frac{{\partial^{2} C}}{{\partial z^{2} }} + \frac{{\sigma_{hnf} B_{o}^{2} A^{2} \left( {x + y} \right)^{{ - \frac{2}{3}}} }}{{\left( {\rho c_{p} } \right)_{hnf} }}\left( {u^{2} + v^{2} } \right), \\ & u\frac{\partial C}{{\partial x}} + v\frac{\partial C}{{\partial y}} + w\frac{\partial C}{{\partial z}} = \frac{\partial }{\partial z}\left( {D_{hnf} \frac{\partial T}{{\partial z}}} \right) + \frac{{D_{T} }}{{T_{\infty } }}\frac{{\partial^{2} T}}{{\partial z^{2} }}, \\ \end{aligned}
(4)

System of Eqs. (1)–(4) BCs are43,44

\begin{aligned} u & = U_{w} \left( { = a\left( {x + y} \right)^{\frac{1}{3}} } \right),v = V_{w} \left( { = b\left( {x + y} \right)^{\frac{1}{3}} } \right), w = 0 \\ T & = T_{w} \left( { = cT_{o} \left( {x + y} \right)^{\frac{2}{3}} + T_{\infty } } \right),\quad C = C_{w} \left( { = dC_{o} \left( {x + y} \right)^{\frac{2}{3}} + C_{\infty } } \right)\quad as\quad y = 0 \\ u & = 0, v = 0,T \to T_{\infty } ,C \to C_{\infty } \quad as\quad y \to \infty \\ \end{aligned}
(5)

Correlations among hybrid nanostructures and nanomaterial in ethylene glycol are 43

\begin{aligned} \rho_{hnf} & = \left[ {\left( {1 - \phi_{2} } \right)\left\{ {\left( {1 - \phi_{1} } \right)\rho_{f} + \phi_{1} \rho_{s1} } \right\}} \right] + \phi_{2} \rho_{s2} ,\rho_{nf} = \left( {1 - \phi } \right)\rho_{f} + \phi \rho_{s} \\ \left( {\rho C_{p} } \right)_{nf} & = \left( {1 - \phi } \right)\left( {\rho C_{p} } \right)_{f} + \phi \left( {\rho C_{p} } \right)_{s} , \\ \left( {\rho C_{p} } \right)_{hnf} & = \left[ {\left( {1 - \phi_{2} } \right)\left\{ {\left( {1 - \phi_{1} } \right)\left( {\rho C_{p} } \right)_{f} + \phi_{1} \left( {\rho C_{p} } \right)_{s1} } \right\}} \right] + \phi_{1} \left( {\rho C_{p} } \right)_{s2} \\ \end{aligned}
(6)
\begin{aligned} \frac{{k_{nf} }}{{k_{f} }} & = \left\{ {\frac{{k_{s} + \left( {n + 1} \right)k_{f} - \left( {n - 1} \right)\phi \left( {k_{f} - k_{s} } \right)}}{{k_{s} + \left( {n - 1} \right)k_{f} + \phi \left( {k_{f} - k_{s} } \right)}}} \right\} , \quad \mu_{nf} = \frac{{\mu_{f} }}{{\left( {1 - \phi } \right)^{2.5} }} \\ \mu_{hnf} & = \frac{{\mu_{f} }}{{\left( {1 - \phi_{2} } \right)^{2.5} \left( {1 - \phi_{1} } \right)^{2.5} }}, \; - \frac{{\sigma_{hnf} }}{{\sigma_{f} }} = \left( {1 + \frac{{3\left( {\sigma - 1} \right)\phi }}{{\left( {\sigma + 2} \right) - \left( {\sigma - 1} \right)\phi }}} \right) \\ \end{aligned}
(7)
\begin{aligned} \frac{{k_{hnf} }}{{k_{bf} }} & = \left\{ {\frac{{k_{s2} + \left( {n - 1} \right)k_{bf} - \left( {n - 1} \right)\phi_{2} \left( {k_{bf} - k_{s2} } \right)}}{{k_{s2} + \left( {n - 1} \right)k_{bf} - \phi_{2} \left( {k_{bf} - k_{s2} } \right)}}} \right\} \\ \frac{{\sigma_{hnf} }}{{\sigma_{f} }} & = \left( {\frac{{\sigma_{s2} + 2\sigma_{f} - 2\phi_{2} \left( {\sigma_{bf} - \sigma_{s2} } \right)}}{{\sigma_{s2} + 2\sigma_{f} + \phi_{2} \left( {\sigma_{bf} - \sigma_{s2} } \right)}}} \right) \\ \end{aligned}
(8)

Thermal conductivity and mass diffusion based on temperature are defined as43

$$K_{hnf} \left( T \right) = K_{hnf} \left( {1 + \varepsilon_{1} \frac{{T - T_{\infty } }}{{T_{w} - T_{\infty } }}} \right), \quad D_{hnf} \left( T \right) = K_{hnf} \left( {1 + \varepsilon_{2} \frac{{T - T_{\infty } }}{{T_{w} - T_{\infty } }}} \right),$$
(9)

Next, the similarity transformation is40

\begin{aligned} u & = a\left( {x + y} \right)^{\frac{1}{3}} ,\;v = a\left( {x + y} \right)^{\frac{1}{3}} , \;\eta = \sqrt {\frac{a}{{\nu_{f} }}} \left( {x + y} \right)^{{ - \frac{1}{3}}} z, \\ w & = - \sqrt {a\nu_{f} } \left( {x + y} \right)^{{ - \frac{1}{3}}} \left( {\frac{2}{3}\left( {f + g} \right) - \frac{1}{3}\eta \left( {f^{\prime } + g^{\prime } } \right)} \right),\quad \theta = \frac{{T - T_{\infty } }}{{T_{w} - T_{\infty } }},\quad \phi = \frac{{C - C_{\infty } }}{{C_{w} - C_{\infty } }} \\ \end{aligned}
(10)

In Eqs. (1)–(5), similarity transformation is used, we have

\begin{aligned} & \frac{{\nu_{hnf} }}{{v_{f} }}\left( {\alpha_{1} f^{\prime \prime \prime } + \alpha_{2} f^{\prime \prime 2} f^{\prime \prime \prime } } \right) - \frac{1}{3}\left( {f^{\prime } + g^{\prime } } \right)f^{\prime } + \frac{2}{3}\left( {f + g} \right)f^{\prime \prime } + \left( {Gr} \right)_{t} \theta \\ & \quad + \left( {Gr} \right)_{c} \phi - \left( {\frac{{\sigma_{hnf} }}{{\sigma_{f} }}} \right)\left( {\frac{{\rho_{f} }}{{\rho_{hnf} }}} \right)Mf^{\prime } - \left( {\frac{{\mu_{hnf} }}{{\mu_{f} }}} \right)K^{*} f^{\prime } = 0 \\ &f^{\prime } \left( 0 \right) = 1, \;f\left( 0 \right) = 0, \; f^{\prime } \left( \infty \right) \to 0, \\ \end{aligned}
(11)
\begin{aligned} & \frac{{\nu_{hnf} }}{{v_{f} }}\left( {\alpha_{1} g^{\prime \prime \prime } + \alpha_{2} g^{\prime \prime 2} g^{\prime \prime } } \right) - \frac{1}{3}\left( {f^{\prime } + g^{\prime } } \right)g^{\prime } + \frac{2}{3}\left( {f + g} \right)g^{\prime \prime } + \left( {Gr} \right)_{t} \theta \\ & \quad + \left( {Gr} \right)_{c} \phi - \left( {\frac{{\sigma_{hnf} }}{{\sigma_{f} }}} \right)\left( {\frac{{\rho_{f} }}{{\rho_{hnf} }}} \right)Mg^{\prime } - \left( {\frac{{\mu_{hnf} }}{{\mu_{f} }}} \right)K^{*} g^{\prime } = 0 \\ & g^{\prime } \left( 0 \right) = \beta , \;g\left( 0 \right) = 0, \; g^{\prime } \left( \infty \right) \to 0, \\ \end{aligned}
(12)
\begin{aligned} & \frac{{K_{hnf} }}{{K_{f} }}\left[ {\left( {1 + \varepsilon_{1} \theta } \right)\theta^{\prime \prime } + \varepsilon_{1} \left( {\theta^{\prime } } \right)^{2} } \right] + \left( {\frac{{\left( {\rho c_{p} } \right)_{hnf} }}{{\left( {\rho c_{p} } \right)_{f} }}} \right)\frac{2}{3}Pr\left( {f + g} \right)\theta^{\prime } - \left( {\frac{{\left( {\rho c_{p} } \right)_{hnf} }}{{\left( {\rho c_{p} } \right)_{f} }}} \right)\frac{2}{3}Pr\left( {f^{\prime } + g^{\prime } } \right)\theta \\ & - Pr\beta^{*} \theta + \left( {\frac{{\left( {\rho c_{p} } \right)_{hnf} }}{{\left( {\rho c_{p} } \right)_{f} }}} \right)DuPr\phi^{\prime \prime } + \left( {\frac{{\sigma_{hnf} }}{{\sigma_{f} }}} \right)MPrEc\left( {f^{\prime } + g^{\prime } } \right)^{2} = 0 \\ & \quad \theta \left( 0 \right) = 1, \;\theta \left( \infty \right) \to 0, \\ \end{aligned}
(13)
\begin{aligned} & \frac{{D_{hnf} }}{{D_{f} }}\left[ {\left( {1 + \varepsilon_{1} \varphi } \right)\varphi^{\prime \prime } + \varepsilon_{2} \varphi^{\prime } \theta^{\prime } } \right] + \frac{2}{3}Sc\left( {f + g} \right)\phi^{\prime } - \frac{2}{3}Sc\left( {f^{\prime } + g^{\prime } } \right)\phi + SrSc\theta^{\prime \prime } = 0 \\ & \phi \left( 0 \right) = 1, \;\phi \left( \infty \right) \to 0, \\ \end{aligned}
(14)

The dimensionless numbers and defined here

\begin{aligned} \left( {Gr} \right)_{t} & = \frac{{\left( {\beta_{hnf} } \right)_{T} g^{*} cT_{0} }}{{a^{2} }} ,\left( {Gr} \right)_{c} = \frac{{\left( {\beta_{hnf} } \right)_{C} g^{*} dC_{0} }}{{a^{2} }}, M = \frac{{\sigma_{f} }}{{\rho_{f} }}\frac{{B_{0}^{2} A^{2} }}{a}, K^{*} = \frac{{\mu_{f} }}{{ak_{1} }}, \\ Ec & = \frac{1}{{(c_{p} )_{f} }}\frac{{a^{2} }}{{cT_{0} }} , \; \beta^{*} = \frac{{Q_{0} }}{{a\left( {\rho c_{p} } \right)_{f} }} , \;Du = \frac{{DK_{T} dC_{0} }}{{C_{s} C_{p} V_{f} cT_{0} }}, \;Sc = \frac{{V_{f} }}{{d_{f} }} ,\;Sr = \frac{{D_{T} T_{0} }}{{(T_{\infty } C_{0} )V_{f} }} . \\ \end{aligned}
(15)

Table 1 describes the set of parameters that have been used in this investigation for practical purposes43,44.

Surface-based forces are described as

$$C_{fx} = \frac{{\left. {\frac{\partial u}{{\partial z}}} \right|_{z = 0} }}{{\rho_{f} \left( {U_{w} } \right)^{2} }} = \frac{{\left( {1 - \phi_{1} } \right)^{ - 2.5} }}{{\left( {1 - \phi_{2} } \right)^{2.5} \left( {Re} \right)^{1.5} }}\left[ {\alpha_{1} f^{\prime \prime } \left( 0 \right) + \alpha_{2} \left( {f^{\prime \prime \prime } \left( 0 \right)} \right)^{3} } \right],$$
(16)
$$C_{gy} = \frac{{\left. {\frac{\partial v}{{\partial z}}} \right|_{z = 0} }}{{\rho_{f} \left( {U_{w} } \right)^{2} }} = \frac{{\left( {1 - \phi_{1} } \right)^{ - 2.5} }}{{\left( {1 - \phi_{2} } \right)^{ - 2.5} \left( {Re} \right)^{1.5} }}\left[ {\alpha_{1} g^{\prime \prime } \left( 0 \right) + \alpha_{2} \left( {g^{\prime \prime \prime } \left( 0 \right)} \right)^{3} } \right].$$
(17)

Nusselt number is

$$Nu = - \frac{{\left. {\left( {x + y} \right)K_{hnf} \frac{\partial T}{{\partial y}}} \right|_{y = 0} }}{{k_{f} \left( {T - T_{\infty } } \right)}} = - \frac{{K_{hnf} }}{{k_{f} \left( {Re} \right)^{1.5} }}\theta^{\prime } \left( 0 \right),$$
(18)

the mass flux is

$$Sh = \frac{{\left. {\left( {x + y} \right)D_{hnf} \frac{\partial C}{{\partial y}}} \right|_{y = 0} }}{{D_{f} \left( {C - C_{\infty } } \right)}} = - \frac{{D_{hnf} }}{{D_{f} \left( {Re} \right)^{1.5} }}\phi^{\prime } \left( 0 \right),$$
(19)

where $$Re = \frac{{xU_{w} }}{{\nu_{f} }}$$, the Reynolds number.

## Galerkin finite element algorithm: a computational approach

The provided problem is solved using the Galerkin finite element algorithm (G-FEA). The FEMs explain the method are listed here45,46,47,48,49. Some limitations on finite element method are listed below.

• Analysis of finite elements is perceived as more complex in view of understanding rather than others numerical methods;

• Finite element method can be expensive in term of computational cost as compared to other methods;

• Large data is needed for mesh free analysis.

• Construction of the residual equations is done.

• The residual is integrated across a conventional discrete time domain component.

• Stiffness matrices are generated after calculating the weighted residual integrals using by G-FEM technique.

• By following the restrictions of element assembly, the nonlinear equations are modeled. Under the constraints for calculation, the linearized system is solved $$10^{ - 3}$$.

• Results are obtained that are grid independent after the convergence is validated. It utilizes the error analysis criterion.

$$\left| {\frac{{\eta^{i + 1} - \eta^{i} }}{{\eta^{i} }}} \right| < 10^{ - 5} .$$
(20)

Examples of the parametric research are provided to demonstrate the effects of heat generation, porous media, mass diffusion, thermal diffusivity, the rate of heat flow and mass diffusion on the study of thermal energy and mass transfer in 3D Newtonian fluid flow. Table 2 shows 300 element mesh-free issue analysis results.

## Results and discussion

To investigate the physics of the issue described in the previous part, parametric research has been presented. The fractionated finite element method is used to generate a numerical solution. Using FEM, the mathematical model for mass and thermal energy transfer in non-Newtonian flows beyond a surface with thermal and wall density gradients is numerically solved.

As yield stress is the property that prevents fluid from deforming until a specific applied stress is reached. The fluid must oppose the applied tension in order to reach the equilibrium condition, the yield stress must increase. As a result, a drop in the velocity profile (in both $$x$$ and $$y$$ -components) is seen (see Figs. 3 and 4). Figures 3 and 6 have indeed been produced to illustrate how fluid parameters affect velocity curves. It is noticed that fluid becomes thin versus the higher impacts of fluid parameter.

The numerous numerical experiments are run using various samples of customizable elements. The numerical experiments yield a few significant findings. It is significant to notice that solid curves are concerned with flow, heat exchange, and mass transfer in hybrid nanofluid, whereas dashed curves are connected with flows, heat exchange, and mass transfer in MoS2-Ag-hybrid nanofluid. Consequently, the flow in both the $$x$$- and $$y$$ directions slow down (see Figs. 5 and 6). Moreover, Figs. 7 and 8 shows the parameter $$k^{*}$$ related to the resistance of a porous media to fluid flow and how it affects how fluid particles move. These figures likewise show declining velocities. Additionally, these figures demonstrate that compared to mono nano-Casson fluid, hybrid nano-Casson fluid encounters greater resistance from the porous media. When compared to hybrid nano-Casson fluid, the mono, nano-Casson fluid has a wider viscosity region.

### Fluid flow versus the magnetic field's function

The magnetic field and the Lorentz force are directly related. The evolution of $$M$$ can be used to calculate the Lorentz force's influence on flow. The adverse impact of the Lorentz force increases with increasing values of $$M$$. As a result, the Lorentz force causes flow to slow down. (See Figs. 7 and 8). As a result, change in the magnetic field is used to reduce boundary layer thickness (the intensity of applied). The Lorentz force for the flow of MoS2-Ag-hybrid nanofluid is also reported to be greater than the Lorentz force for the flow of MoS2-nanofluid.

### Temperature field in relation to changes in key model parameters

For both MoS2 and Ag nanofluid, the effects of $$M$$, $$Ec$$ $$Pr$$,$$\beta^{*}$$, and $$Du, \left( {Gr} \right)_{t}$$, versus thermal energy are studied. Figures 9 through 13 demonstrate the observed influence of these parameters, accordingly. The Dufour number refers to the input variable $$Du$$. When transcript of heat energy resulting from gradient of concentration is taken into account it shows in the non—dimensional the energy equation's form. The heat transport is examined due to compositional variations brought on by nanoparticles and soluble compounds distributed throughout the fluid. Figure 9 illustrates how $$Du$$ affects the temperature of MoS2-nanofluid and MoS2-Ag-hybrid nanofluid. As a factor of $$Du$$, the temperature of both types of fluids tends to rise.

As a function of $$Du$$, the temperature of the both types of fluids tend to rise. $$Du$$ has less of an impact on the temperature of MoS2-nanofluid than it does on the temperature of MoS2-Ag-hybrid nanofluid. Figure 10 depicts the effects of fluid particles on the temperature of MoS2-Ag-hybrid nanofluid. When the flow is enhanced by a positive drag force, the situation is $$\left( {Gr} \right)_{ \in } > 0$$. If buoyancy force is negative, however, as it is in the situation in $$\left( {Gr} \right)_{t} < 0$$, the flow is referred to as opposed flow. The Heat and mass transfer effect occurs when heat is produced during conversion and is added to a medium, such as fluid. Consequently, Fig. 11 displays the temperature as a result of Joule heating. Additionally, it is found that the hybrid nanofluid exhibits a stronger Joule heating phenomena than the MoS2 does (mono-fluid). Additionally, the parameter $$\beta^{*}$$ arises as a result of the energy equation's energy equation's heat generation part not being dimensioned. The fluid absorbs the heat that is produced, which raises the fluid's temperature. Figure 12 provides evidence to support this observation. The temperature of fluids considerably increases as a result of fluid motion (nanofluid and MoS2-Ag-hybrid nanofluid). Simulations reveal that the fluid velocity in the MoS2 is larger than that in the MoS2-Ag-hybrid nanofluid. These findings are evident from Fig. 13.

### Role of mass diffusion

The parameters $$Sr$$, $$\left( {Gr} \right)_{c}$$, and $$Sc,$$ respectively, determine the impact of temperature gradient, Buoyancy force due to concentrations difference and diffusion coefficient on concentration field. Their influence on concentrations can be seen from Figs. 14, 15 and 16. Hence an increasing effect of $$Sr$$ and $$\left( {Gr} \right)_{c}$$ can be noticed in Figs. 14 and 15. On the other hand, concentration field decreases as a function $$Sc$$ (Fig. 16).

### Mass flux, heat transfer rate, and wall shear stresses

Investigations are conducted into the relationship between numerical data on wall stresses in the $$x and y$$ directions, wall heat transfer rate, and wall mass flow rate for both fluids, MoS2-fluid (mono nanofluid) and MoS2-Ag-fluid (hybrid nanofluid) (see Table 3). Table 3 provides an overview of the numerical results. The $$k^{*}$$ appears to be negatively correlated with the number of voids in the porous medium. As a result, the stress (or resistive force) per unit area rises. Wall shear stresses are therefore increasing functions of $$k^{*}$$ in both the x and y directions. Both the mass-flux and the temperature gradient are diminishing effects of $$k^{*}$$. Additionally, it has been found that increasing $$Du$$ causes an increase in wall shear stress. However, a surge in the wall mass transfer coefficient against $$Du$$ is observed. Lastly, $$Sr$$ determines the temperature difference on solute particles, and an increase in $$Sr$$ causes a reduction in wall shear stress. For $$Sc$$, the opposite tendency is shown.

## Core points and conclusions

The vertical 3D melting interface is used to characterize the thermal energy and mass transport characteristics that have a substantial impact on nanoparticles and hybrid nanoparticles. On a Newtonian fluid, the cumulative effects of heat transfer, a porous medium, heat gradient, rates of mass transport, and heat conduction are considered. Along with the phenomenon of heat generation, non-Furrier’s law is used in the energy equation. To determine numerical and graphical results related to velocity and temperature by different factors, G-FEA (Galerkin finite element algorithm) is used. The following is a list of the study's principal conclusions:

• Convergence study is tested observing by 300 elements;

• Approach of Hybrid nanoparticles is estimated as efficient to achieve maximum production of energy into fluidic particles as compared for nanofluid;

• The magnetic field parameter slows down particle velocity;

• As thermal energy reaches its maximum, in contrast to the given values of the Eckert number, bouncy forces, and magnetic parameter.

• Role of variable thermal conductivity number rises growth of heat energy;

• In comparison to higher values of the heat source number, the non- Fourier's results in decreased thermal dispersion and reduced heat transfer rate.

• 300 elements are needed for mesh free analysis.

Future applications of the Galerkin finite element algorithm (G-FEA) could include a range of physical and technological difficulties11,50,51,52,53,54,55,56,57,58,59. According to60,61,62,63,64,65,66,67,68,69,70, there have been several recent advancements that explore the importance of the research domain under consideration.