The new analytical and numerical analysis of 2D stretching plates in the presence of a magnetic field and dependent viscosity

Abstract

This study explores heat transfer in a system involving Jeffery fluid of MHD flow and a porous stretching sheet. The mathematical representation of this system is initially described using a partial differential equation (PDE), which is then converted into an ordinary differential equation (ODE) through numerical techniques such as Lie similarity and transformation methods, along with the shooting approach. The results indicate that altering the variables of Jeffery fluid, heat source, porosity on a stretching sheet, and the physical characteristics of the magnetic field within the system leads to an upward trend. Implementing this enhanced heat transfer system can yield benefits across various domains, including industrial machinery, mass data storage units, electronic device cooling, etc., thereby enhancing heating and cooling processes. Furthermore, the study also utilized Akbari-Ganji’s Method, a new semi-analytical method designed to solve nonlinear differential equations of heat and mass transfer. The results obtained from this method were compared with those from the finite element method for accuracy, efficiency, and simplicity. This research provides valuable insights into heat transfer dynamics in complex systems and offers potential applications in various industrial settings. It also contributes to developing more efficient and effective heat transfer techniques.

Keywords

Akbari-Ganji’s method (AGM)porous media Jeffrey fluid Magnetic field stretching sheet

Introduction

Heat emitter and absorber describe how objects interact with heat radiation, a form of electromagnetic radiation emitted by a heated surface. Heat radiation can travel through a vacuum or a medium, such as air, and transfer energy from one place to another. Studying magnetohydrodynamics (MHD) in fluid dynamics is vital for simulating heat transfer systems. The need for enhanced heating and cooling systems extends to various areas, including industrial mass machinery, reservoirs, and microfabrication. Due to its heat resistance, the convective heat transfer system is limited in operating pressure and temperature and prone to leakages. Heat can result in corrosion, decreased reliability, and obstructions caused by particles in suspension. Advanced techniques in fluid dynamics allow for the analysis and assessment of heat transfer systems through different means, such as classification, computation, modeling, and parameter adjustment, to improve the convection performance. An advantage of analyzing heat and mass convection systems in fluid dynamics is the successful transformation of nonlinear equations, made possible using the Lie similarity technique to obtain FEM for solving nonlinear coupled differential equations. Also, this paper used the AGM solution to examine the governing equations and compare results to previous related work. Jeffrey fluid is shown to be remarkable in the heat transfer system due to its stress relaxation effects. The system’s mathematical analysis, incorporating MHD flow, takes the form of a Partial Differential Equation (PDE) and is examined using machine learning. Machine learning cannot solve PDEs, so the equation is transformed into an Ordinary Differential Equation (ODE) using FEM and the Lie similarity transformation technique.

The initial application of boundary layer theory to the motion of dynamic fluid on constant, flat-solid plates, using fundamental differential boundary layer equations, was first documented by Sakiadis.¹ The letter contains accounts of research into MHD and dynamic Jeffrey fluid models on expanding surfaces, coupled with the perks of using Lie-similarity and methods of transformation and shooting approach. The study by Agrawal et al.² explored MHD flow with expanding surface and permeable medium, utilizing lie similarity and transformation approaches to ascertain that the Prandtl number can increase heat transfer. Unsteady MHD heat and mass transfer is the study of how fluid, heat, and mass move under the influence of magnetic fields that change over time. Nazeer et al.³ utilized shooting and pseudo-spectral techniques to analyze Eyring-Powell fluid in an infinite cylindrical pipe, attaining precise results in the fourth and fifth orders of velocity and temperature measurements. According to His-similar,⁴ an analysis of Eyring-Powell fluid was performed using a perturbation method, showing that Brownian motion and thermophoretic force vary at particular concentrations and that viscous dissipation is observable. Jalili et al.⁵ examined the accuracy of three methods that simulate the thermal diffusivity profile in an artery with oblique stenosis and a hybrid nanofluid. The hybrid nanofluid consists of $A I_{2} O_{3}$ and Cu and is influenced by its volume fraction and heat sources. According to another publication,⁶ an analysis of a chaotic third-grade fluid with hafnium nanoparticles as a smooth wall flow was carried out, expressing a convergent channel and concluding that the volumetric flow parameter boosted the velocity profile. Enhanced heat transfer and reduced flow velocity were the outcomes of two non-isothermal flow models of Eyring-Powell fluid, as presented by Ali et al.⁷ For various characteristics, shooting technique was used to achieve more precise results. For solving through FEM, the Lie similarity technique was equipped by Kumar et al.⁸ and Ogunseye et al.⁹ to transform the differential equation of the heat transfer formula, including components for an electrically conductive fluid. Also, the fluid movements were examined in their reports. Jalili et al.¹⁰ explained the transient squeezing flow of 2D MHD involving Casson fluid in the presence of solar irradiance numerically and theoretically. A Casson fluid model is based on the assumption that the fluid consists of rigid spherical particles that are held together by a yield stress. The yield stress is the minimum shear stress required to initiate the fluid flow. Below the yield stress, the fluid behaves like a solid and does not deform. Above the yield stress, the fluid behaves like a viscous liquid and deforms proportionally to the shear rate. Rao and Sreenadh¹¹ and Mukhopadhyay and Layek¹² published two reports discussing the similarity transformation of Jeffrey fluid’s governing equations involving an expanding surface and permeable medium and also showed the falling temperature in a fluctuating profile. Jalili et al.¹³ have extensively studied the radiation heat transfer in non-Newtonian fluids under various conditions. The nanofluid stream through the microchannel heat sink and its heat transfer phenomena were studied by Jalili et al.¹⁴ Investigations into the heat transfer process with chemical reactions and heat sources in non-Newtonian fluid flows were conducted by Jalili et al.¹⁵

The study of the Jeffrey fluid on MHD flow with stretching in porous sheets of a heat transfer system has been conducted by Jena et al.¹⁶ Another study conducted by Bouslimi et al.¹⁷ analyzed the effect of electromagnetic force and thermal radiation on the Williamson nanofluid on a stretching surface through a porous medium while considering heat generation/absorption and Joule heating. Thermal radiation and a heat source are two factors that influence the flow and heat transfer of a non-compressible Jeffrey fluid that deviates from Newton’s law of viscosity. Babu et al.¹⁸ examine how these factors interact with the fluid when it flows over a surface that can stretch or shrink. Agarwal et al.¹⁹ examined how a magnetic field affected the flow of a micropolar Jeffrey fluid, which had both fluid and solid-like properties when it flowed through a porous medium over a plate that could stretch. Yasmin et al.²⁰ investigated the mass and heat transfer in the flow of an electrically conducting non-Newtonian micropolar fluid. This flow has MHD effects and is caused by a curved stretching plate. The effects of power-law heat flux and heat source on the flow of a MHD fluid over a porous stretching sheet were investigated in detail by Ibrahim et al.²¹ The flow of a nanofluid over an inclined stretching sheet is unsteady and has MHD effects. The fluid properties in thermal conductivity and diffusion coefficient vary, and the flow is affected by thermal radiation and chemical reactions. Mjankwi et al.²² studied this flow. Jabeen et al.²³ compared the boundary layer fluid flow around a linearly stretching sheet with MHD effects. The surface had permeability, and the flow had radiative heat flux, heat generation/absorption, thermophoresis velocity, and chemical reaction effects. The MHD flow of a micropolar fluid over a stretched surface was investigated by Goud and Nandeppanavar.²⁴ They considered the effects of Ohmic heating and chemical reactions on the fluid.

Kumar et al.²⁵ investigated how a non-Newtonian Casson fluid flowed through a porous vertical plate in an unsteady MHD natural convective situation, considering the heat and mass transfer properties and chemical reaction. Ahmed et al.²⁶ examined the convective transport features of a fluid of the third grade with thermo-diffusion effects. The effects of multi-slip conditions on the steady-state flow of a fluid in a magnetic field with Soret and Dufour effects were investigated by Reddy et al.²⁷ over a stretching surface with variable temperature. The effect of a heat source on a fluid of Casson type in a magnetic field through a porous plate with oscillations was studied by Goud et al.²⁸ Srinivasulu and Goud²⁹ used numerical methods to study how a magnetic field aligned with a stretching surface affected the flow of a nanofluid of Williamson type with convective boundary conditions. Bafakeeh et al.³⁰ examined the unsteady flow of a porous medium with a vertical plate, where the fluid was viscous, incompressible, and electrically conductive, and where chemical reactions and thermal radiation affected the heat and mass transfer. A fluid with difficulty to flow and does not change its size when pressure is applied is called a viscous incompressible fluid. Chen³¹ investigated how ohmic heating and viscous dissipation influenced the fluid’s momentum, heat, and mass transfer in MHD natural convection flow over a porous, slanted surface with variable wall temperature and concentration. The effects of heat and mass transfer on an MHD flow in the presence of a chemical reaction over a porous, inclined plate were investigated by Sandhya et al.³² Using free convection flow analysis, Noor et al.³³ investigated how a heat source/sink effect and a radiating isothermal slanted plate influenced the thermophoresis and magnetism of the fluid. Systems involving unsteady free convection oscillation involve fluid motion due to the temperature and density difference between the fluid and its environment and the regular movement of a boundary or an external force. The heat and MHD flow characteristics of a generalized Maxwell fluid over a porous, canted plate under a tilted magnetic field were explored by Li et al.³⁴ Raptis et al.³⁵ examined the asymmetric flow of a fluid that can conduct electricity in a magnetic field over a stationary plate that extends infinitely, considering the effects of radiation. Ali examined how the second law of thermodynamics affected the flow of a nanofluid that can conduct electricity over a rotating disk with pores, considering the influence of a vertical magnetic field applied externally. The effects of an external vertical magnetic field on the flow of an electrically conducting nanofluid over a porous rotating disk were studied by Rashidi et al.³⁶ using the second law of thermodynamics. Moreover, we can also consider more related works such as A ternary hybrid nanofluid (Al–Cu – $F e_{2} O_{3}$ /Blood) flow through an inclined channel is analyzed for synthetic cilia entropy generation by Mishra et al.³⁷ A nonlinear vertically elongating surface of non-uniform thickness is used to study the sensitivity analysis of Jeffrey fluid flow with copper nanoparticles and polyvinyl alcohol water as the base fluid. The effects of source terms such as Joule heating, viscous dissipation, exponential heat source, and Arrhenius activation energy are considered in the analysis by Kumar et al.³⁸ The effects of entropy generation, heat transmission, and mass transfer on Jeffrey fluid flow are investigated under solar radiation by Sharma et al.³⁹ An incompressible Jeffrey fluid flow over a vertical nonlinear stretching surface of variable thickness is studied by Sharma et al.⁴⁰ under the influence of an electro-magneto-hydrodynamic (EMHD) field.

The utilization of a magnetohydrodynamics system with a Jeffrey fluid for heat transfer is discussed in this paper. Nonlinear equations with an expanding surface are restructured using lie similarity and transformations, and superior solutions are obtained using the shooting approach. The influence of the heat source and expanding surface on the dynamic characteristic of the Jeffrey fluid model is explored and shown in graphs.

Mathematical analysis

In the region where y is greater than or equal to zero, a symmetrical magnetic field can be created when a two-dimensional sheet with permeable material is heated or cooled. The sheet is stretched by applying two forces in opposite directions along the x-axis, as shown in Figure 1.

Figure 1.

Physical concepts depicted in a graphical format.

A magnetic field $B_{0}$ is generated symmetrically beside the y-axis, and the viscosity is dependent on the wall temperature, resulting in the governing equations of the MHD Jeffrey flow with stretching-permeable surfaces of the heat exchange system. We are examining a plane that expands in two dimensions, with heat being generated or absorbed within a permeable structure. Additionally, a magnetic field is symmetrically applied to the area where $y \geq 0$ . The surface is extended by applying two forces in contrasting directions along the x-axis. The temperature at the surface is not the same as the ambient fluid temperature, and a magnetic field is symmetrically positioned along the y-axis. As stated in Agrawal et al.,² the governing equations of the MHD Jeffrey flow with stretching-permeable surfaces of heat exchange are:

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0

(2.1)

\begin{matrix} u \frac{\partial u}{\partial x} + ν \frac{\partial u}{\partial y} = \frac{1}{ρ} \frac{\partial μ}{\partial T} \frac{\partial T}{\partial y} \frac{\partial u}{\partial y} + (\frac{ν^{*}}{1 + λ_{0}}) \frac{\partial^{2} u}{\partial y^{2}} \\ - \frac{μ}{ρ k} (\frac{1}{1 + λ_{0}}) - \frac{σ_{C} B_{0}^{2} u}{ρ} \end{matrix}

(2.2)

u \frac{\partial T}{\partial x} + ν \frac{\partial T}{\partial y} = \frac{\partial^{2} T}{\partial y^{2}} + \frac{Q_{*}}{ρ C_{p}} (T - T_{\infty})

(2.3)

u \frac{\partial C}{\partial x} + ν \frac{\partial C}{\partial y} = D \frac{\partial^{2} C}{\partial y^{2}}

(2.4)

When $ν^{*} = \frac{μ^{*}}{ρ}$

Boundary conditions become as follows:

T = {T_{w} |}_{y = 0}, C = {C_{w} |}_{y = 0}, u = {cx |}_{y = 0}

(2.5)

T \to {T_{\infty} |}_{y = \infty}, C \to {C_{\infty} |}_{y = \infty}, u \to {0 |}_{y = \infty} .

In this context, $u$ and $υ$ denote the flow velocity elements in the $x$ and $y$ directions of the cartesian coordinate system. $λ_{0}$ represents the dimensionless Jeffrey characteristic, $ρ$ indicates liquid density, $κ$ stands for vorticity, $T$ signifies liquid temperature, and $T_{\infty}$ , refers to either ambient or free stream temperature. $C_{p}$ , denotes specific heat, $μ$ represents dynamic viscosity, $Q_{*}$ indicates dimensional thermal generation or absorption, $σ_{c}$ , stands for the strong magnetic field generated by high voltage, and $Δ T$ is calculated as $T_{w} - T_{\infty}$ where $T_{w}$ , is the wall fluid temperature. $C$ represents fluid temperature, $C_{w}$ indicates wall concentration, $C_{\infty}$ , denotes free stream concentration, and $D$ signifies mass diffusivity. The stream function can be described in the following manner:

u = \frac{\partial ψ}{\partial y} ν = - \frac{\partial ψ}{\partial x}

(2.6)

Batchelor⁴¹ defined the temperature as a function of linear temperature.

\begin{matrix} θ (η) = \frac{T - T_{\infty}}{T_{w} - T_{\infty}}, μ (T) = μ^{*} (c + d (T_{w} - T)), \\ B = d (\begin{matrix} T_{w} - & T_{\infty} \end{matrix}), ϕ (η) = \frac{c - c_{\infty}}{c_{w} - c_{\infty}} . \end{matrix}

(2.7)

The fixed viscosity factor is $μ^{*}$ , and $c$ , $d$ are fixed parameters. $B$ is the viscosity characteristic. The equations (2.2)–(2.5) are changed into the following form using the stream function:

\frac{\partial ψ}{\partial y} \frac{\partial^{2} ψ}{\partial x \partial y} - \frac{\partial ψ}{\partial x} \frac{\partial^{2} ψ}{\partial y^{2}} = - B ν^{*} \frac{\partial θ}{\partial y} \frac{\partial^{2} ψ}{\partial y^{2}} + (\frac{ν^{*}}{1 + λ_{0}}) (1 + B (1 - θ))

(2.8)

\frac{\partial^{3} ψ}{\partial y^{3}} - \frac{v^{*}}{k} (\frac{1}{1 + λ_{0}}) (1 + B (1 - θ)) \frac{\partial ψ}{\partial y} - \frac{σ_{C} B_{0}^{2}}{ρ} \frac{\partial ψ}{\partial y}

\frac{\partial ψ}{\partial y} \frac{\partial θ}{\partial x} - \frac{\partial ψ}{\partial x} \frac{\partial θ}{\partial y} = κ \frac{\partial^{2} θ}{\partial y^{2}} + \frac{Q_{*}}{ρ C_{ρ}} θ

(2.9)

\frac{\partial ψ}{\partial y} \frac{\partial ϕ}{\partial x} - \frac{\partial ψ}{\partial x} \frac{\partial ϕ}{\partial y} = D \frac{\partial^{2} ϕ}{\partial y^{2}}

(2.10)

The conditions at the boundaries are as follows:

{\frac{\partial ψ}{\partial y} |}_{y = 0} = cx, {\frac{\partial ψ}{\partial x} |}_{y = 0} = 0, {θ |}_{y = 0} = 1, {ϕ |}_{y = 0} = 1

(2.11)

{\frac{\partial ψ}{\partial y} |}_{y = \infty} \to 0, {θ |}_{y = \infty} \to 0, {ϕ |}_{y = \infty} \to 0 .

Examination of similarity through the Lie group method

The Lie group similarity transformation can be defined and introduced in the following manner⁴²:

Γ : x^{*} = x e^{ε γ_{1}}, y^{*} = y e^{ε γ_{2}}, ψ^{*} = ψ e^{ε γ_{3}}, u^{*} = u e^{ε γ_{4}}

(2.12)

v^{*} = v e^{ε γ_{5}}, θ^{*} = θ e^{ε γ_{6}}, ϕ^{*} = ϕ e^{ε γ_{7}}

Using the Lie group transformation (2.12), the coordinates ( $u$ , $v$ , $ψ$ , $x$ , $y$ , $θ$ , $ϕ$ ) are changed into ( $u^{*}$ , $v^{*}$ , $ψ^{*}$ , $x^{*}$ , $y^{*}$ , $θ^{*}$ , $ϕ^{*}$ ). After exchanging this transformation into equations (2.8)–(2.10), the resulting equations are as follows:

\begin{matrix} e^{ε (γ_{1} + 2 γ_{2} - 2 γ_{3})} (\frac{\partial ψ^{*}}{\partial y^{*}} \frac{\partial^{2} ψ^{*}}{\partial x^{*} \partial y^{*}} - \frac{\partial ψ^{*}}{\partial x^{*}} \frac{\partial^{2} ψ^{*}}{\partial y^{* 2}}) \\ = - B ν^{*} e^{ε (3 γ_{2} - γ_{3} - γ_{6})} \frac{\partial θ^{*}}{\partial y^{*}} \frac{\partial^{2} ψ^{*}}{\partial y^{* 2}} + (\frac{ν^{*}}{1 + λ_{0}}) (1 + B) \\ e^{ε (3 γ_{2} - γ_{3})} \frac{\partial^{3} ψ^{*}}{\partial y^{* 3}} - (\frac{ν^{*}}{1 + λ_{0}}) B θ^{*} e^{ε (3 γ_{2} - γ_{3} - γ_{6})} \frac{\partial^{3} ψ^{*}}{\partial y^{* 3}} \end{matrix}

(2.13)

\begin{matrix} - \frac{ν^{*}}{k} (1 + B) (\frac{1}{1 + λ_{0}}) e^{ε (γ_{2} - γ_{3})} \frac{\partial ψ^{*}}{\partial y^{*}} + \frac{ν^{*}}{k} (\frac{1}{1 + λ_{0}}) \\ B e^{ε (γ_{2} - γ_{3} - γ_{6})} θ^{*} \frac{\partial ψ^{*}}{\partial y^{*}} - \frac{σ_{c} B_{0}^{2}}{ρ} e^{ε (γ_{2} - γ_{3})} \frac{\partial ψ^{*}}{\partial y^{*}} \\ e^{ε (γ_{1} + γ_{2} - γ_{3} - γ_{6})} (\frac{\partial ψ^{*}}{\partial y^{*}} \frac{\partial θ^{*}}{\partial x^{*}} - \frac{\partial ψ^{*}}{\partial x^{*}} \frac{\partial θ^{*}}{\partial y^{*}}) = κ e^{ε (2 γ_{2} - γ_{6})} \frac{\partial^{2} θ^{*}}{\partial y^{* 2}} \\ + \frac{Q_{*}}{ρ C_{P}} e^{- ε γ_{6}} θ^{*} \end{matrix}

(2.14)

e^{ε (γ_{1} + γ_{2} - γ_{3} - γ_{7})} (\frac{\partial ψ^{*}}{\partial y^{*}} \frac{\partial ϕ^{*}}{\partial x^{*}} - \frac{\partial ψ^{*}}{\partial x^{*}} \frac{\partial ϕ^{*}}{\partial y^{*}}) = D e^{ε (2 γ_{2} - γ_{7})} \frac{\partial^{2} ϕ^{*}}{\partial y^{* 2}}

(2.15)

When the Lie transformation group $Γ$ is applied, the set of equations does not change, and the parameters include the following elements:

\begin{matrix} γ_{1} + 2 γ_{2} - 2 γ_{3} = 3 γ_{2} - γ_{3} - γ_{6} = γ_{2} - γ_{3} \\ = - γ_{6} + γ_{2} - γ_{3} \end{matrix}

(2.16)

γ_{1} - γ_{3} + γ_{2} - γ_{6} = - γ_{6} + 2 γ_{2} = - γ_{6}

(2.17)

γ_{2} - γ_{3} + γ_{1} - γ_{7} = 2 γ_{2} - γ_{7}

(2.18)

Solving the relations (2.16)–(2.18) yields the following:

γ_{1} = γ_{3}, γ_{2} = 0 = γ_{6}, γ_{7} = 0

(2.19)

The boundary conditions yield the following results:

γ_{1} = γ_{4}, γ_{5} = 0

(2.20)

By inserting the results above into the Lie group transformation, the value of $γ$ is decreased to the number of parameters in one Lie transformation group.

\begin{matrix} ψ^{*} = ψ e^{ε γ_{1}}, u^{*} = u e^{ε γ_{1}}, x^{*} = x e^{ε γ_{1}}, y^{*} = y, \\ v^{*} = v, θ^{*} = θ, ϕ^{*} = ϕ . \end{matrix}

(2.21)

When the relation mentioned above is expanded using Taylor’s series and terms of second and higher order are neglected, the result is as follows:

x^{*} - x = ε γ_{1} x, ψ^{*} - ψ = ε γ_{1} ψ, y^{*} - y = 0,

(2.22)

u^{*} - u = u ε γ_{1}, v^{*} - v = 0, θ^{*} - θ = 0, ϕ^{*} - ϕ = 0 .

By reducing the previous relations, we can obtain differential equations,

\frac{d x}{γ_{1} x} = \frac{dy}{0} = \frac{d ψ}{γ_{1} ψ} = \frac{d u}{γ_{1} u} = \frac{d v}{0} = \frac{d θ}{0} = \frac{d ϕ}{0}

(2.23)

Integrating the equations as mentioned above yields the following terms: $y = η$ , $θ = θ (η)$ , $ϕ = ϕ (η)$ , $ψ = xG (η)$ . Substituting these relations into (2.13)–(2.15) simplifies them to the following as Thenmozhi et al.⁴² did:

{G'}^{2} - G G ″ = - B ν^{*} θ' G ″ + ν^{*} (1 + (1 - θ) B) (\frac{1}{1 + λ_{0}}) G ″'

(2.24)

\begin{matrix} - \frac{ν^{*}}{k} (\frac{1}{1 + λ_{0}}) (1 + (1 - θ) B) G' - \frac{σ_{c} B_{0}^{2}}{ρ} G' \\ κ θ ″ + θ' G + \frac{Q_{*}}{ρ C_{p}} θ = 0 \end{matrix}

(2.25)

D ϕ ″ + G ϕ' = 0

(2.26)

The adjusted boundary conditions are converted into the following:

G' = c, G = 0, θ = 1, ϕ = 1 when η = 0

(2.27)

G' \to 0, θ \to 0, ϕ \to 0 when η \to \infty .

The connections between $η, G, θ and ϕ$ can be established in the following manner:

η = ν^{* γ} c^{β} η^{*}, G = ν^{* γ'} c^{β'} G^{*}, θ = ν^{* γ ″} c^{β ″} θ^{*}, ϕ = ν^{* γ ″'} c^{β ″'} ϕ^{*} .

(2.28)

From the previous connections, we obtained the following results:

γ = \frac{1}{2} = γ', β' = \frac{1}{2} = - β, γ ″ = 0, β ″ = 0,

(2.29)

γ ″' = 0, β ″' = 0, G^{*} = g, θ^{*} = θ, ϕ^{*} = ϕ .

When the aforementioned relation is inserted into (2.24)–(2.27), the equations are simplified to the following:

(\frac{1 + B (1 - θ)}{1 + λ_{0}}) g ″' - B θ' g ″ + g g ″ - g^{' 2} - M g'

(2.30)

\begin{matrix} - K (\frac{1}{1 + λ_{0}}) (1 + B (1 - θ)) g' = 0 \\ θ ″ + \Pr (g θ' + λ θ) = 0 \end{matrix}

(2.31)

ϕ ″ + scg ϕ' = 0

(2.32)

About the boundary constraints:

g' = 1, g = 0, θ = 1, ϕ = 1 when η^{*} = 0

(2.33)

g' \to 0, θ \to 0, ϕ \to 0 when η^{*} \to \infty .

To solve (2.30)–(2.32), we have to apply the boundary conditions (2.33) when $η^{*} = 0$ and $η^{*} \to \infty$ . In this research, alongside the FEM, we also use the AGM approach as a novelty solution to obtain results.

The Prandtl number, which is a dimensionless number that measures the relative importance of momentum and heat diffusion in fluid flow, denoted by $\Pr = \frac{ν^{*}}{κ}$ , the heat emission and absorption by $λ = \frac{Q_{*}}{ρ C_{p}}$ , the permeable parameter by $K = \frac{ν^{*}}{κ}$ the Schmidt number is a dimensionless quantity that compares a fluid’s momentum and mass transport rates. It indicates how well the fluid mixes under the influence of a velocity gradient. It is defined as the ratio of the kinematic viscosity, which is a measure of momentum diffusivity, to the mass diffusivity, which is a measure of mass transfer by $Sc = \frac{ν^{*}}{D}$ , and the magnetism field characteristic by $M = \frac{σ_{c} B_{0}^{2}}{ρ}$ .

The default values of the following are considered according to Thenmozhi et al.⁴² $λ = 0.1, λ_{0} = - 0.7, M = 0.1, B = 1, Sc = 0.22, \Pr = 0.733$ . The ( $λ$ ) is stated for heat emitter, ( $λ_{0})$ for jeffrey value, $(M)$ for the field of magnetism, ( $B$ ) viscosity characteristic, ( $Sc$ ) Schmidt number, and ( $\Pr$ ) is for Prandtl number.

Analysis of FEM

The FEM breaks down the problem domain into smaller finite elements, each with distinct mathematical equations that describe its shape and system behavior. This technique makes it easier to solve complex governing equations that would otherwise be difficult to solve by hand. FEM uses shape functions to interpolate nodal values and calculate element-level solutions. In this study, FEM is employed as a numerical method for solving flow and heat transfer equations on a permeable vertical flat plate using finite elements. The general steps of FEM include the following:

Discretization: The problem domain is broken down into smaller sections, known as finite elements, which can be triangles or rectangles.

Formulation: Each finite element is associated with a series of linear algebraic equations that illustrate the fluctuation of unknown variables over that specific element.

Assembly: A universal system of equations is established by integrating the separate linear algebraic equations from all finite elements, depicting the system’s complete domain.

Solution: The unknown variables at all finite elements, for instance, velocity and temperature, are derived by finding a solution to the global equation system.

Post-processing: The flow and temperature distribution over the permeable vertical flat plate can be comprehended by analyzing and depicting the solution using graphical methods such as contour plots.

This study employs FEM to validate the results obtained from a semi-analytical technique used to simplify equations into ordinary differential equations. By applying FEM, a more accurate and detailed solution for the flow and heat transfer within the permeable vertical flat plate can be attained, enabling the optimization and design of thermal systems using the hybrid nanofluid. Additionally, the precision and consistency of FEM outcomes can be contrasted with those derived from the AGM to confirm the findings.

Application of Akbari-Ganji’s method (AGM)

Considering the approach and the interconnected nonlinear differential equations within the system, we can restructure equations (2.30)–(2.32) in this order:

G (x) = (\frac{1 + B (1 - θ)}{1 + λ_{0}}) g ″' - B θ' g ″ + g g ″ - g^{' 2} - M g'

(3.1)

\begin{matrix} - K (\frac{1}{1 + λ_{0}}) (1 + B (1 - θ)) g' = 0 \\ θ (x) = θ ″ + \Pr (g θ' + λ θ) = 0 \end{matrix}

(3.2)

ϕ (x) = ϕ ″ + scg ϕ' = 0

(3.3)

A finite series with unvarying coefficients of polynomials was utilized as a solution for the differential equation examined in this research, founded on the essential principle of AGM in the following manner:

\begin{matrix} G (x) = \sum_{k = 0}^{7} d [k] \cdot x^{k} = d_{0} + d_{1} x + + d_{2} x^{2} + d_{3} x^{3} \\ + d_{4} x^{4} + d_{5} x^{5} + d_{6} x^{6} + d_{7} x^{7} \end{matrix}

(3.4)

θ (x) = \sum_{k = 0}^{4} c [k] \cdot x^{k} = c_{0} + c_{1} x + + c_{2} x^{2} + c_{3} x^{3} + c_{4} x^{4}

(3.5)

\begin{matrix} ϕ (x) = \sum_{k = 0}^{5} b [k] \cdot x^{k} = b_{0} + b_{1} x + + b_{2} x^{2} \\ + b_{3} x^{3} + b_{4} x^{4} + b_{5} x^{5} \end{matrix}

(3.6)

Applying boundary condition

The constant coefficients of equations (3.4)–(3.6) are calculated in AGM. by implementing boundary conditions derived from these techniques.

(a) By applying the boundary condition to equations (3.4)–(3.6), they can be expressed in the following manner:

u = u (B \cdot C)

Based on the above clarifications, the boundary conditions are applied to equations (3.4)–(3.6) in this manner, with BC being the abbreviation for boundary condition.

g (0) = 0 \to d_{0} = 0

(3.7)

\begin{matrix} g (10) = 1 \to 10000000 d_{7} + 1000000 d_{6} + 100000 d_{5} \\ + 1000 d_{4} + 1000 d_{3} + 100 d_{2} + 10 d_{1} + d_{0} = 1 \end{matrix}}

(3.8)

g' (0) = 1 \to d_{1} = 1

(3.9)

\begin{matrix} g' (10) = 0 \to 7000000 d_{7} + 600000 d_{6} + 50000 d_{5} \\ + 4000 d_{4} + 300 d_{3} + 20 d_{2} + d_{1} = 0 \end{matrix}}

(3.10)

θ (0) = 1 \to c_{0} = 1

(3.11)

θ (10) = 0 \to 10000 c_{4} + 1000 c_{3} + 100 c_{2} + 10 c_{1} + c_{0} = 0

(3.12)

ϕ (0) = 1 \to b_{0} = 1

(3.13)

\begin{matrix} ϕ (10) = 0 \to 100000 b_{5} + 10000 b_{4} + 1000 b_{3} \\ + 100 b_{2} + 10 b_{1} + b_{0} = 0 \end{matrix}

(3.14)

(b) Following the substitution of equations (3.4)–(3.6) into the primary differential equations, the boundary conditions are imposed on the main differential equations (equations (3.1)–(3.3)) and their derivatives in this manner:

G (g (x)) \to G (g (B \cdot C)) = 0, G' (g (B \cdot C)) = 0, \dots

(3.15)

Θ (θ (x)) \to Θ (θ (B \cdot C)) = 0, Θ' (θ (B \cdot C)) = 0, \dots

(3.16)

Φ (ϕ (x)) \to Φ (ϕ (B \cdot C)) = 0, Φ' (ϕ (B \cdot C)) = 0, \dots

(3.17)

Utilizing the above equations, we enforce the boundary conditions on the differential equation. In finding a solution, the AGM involves obtaining a series of polynomials by utilizing a certain quantity of constant coefficients in trial functions. Basic arithmetic allows us to acquire these polynomials. In this issue, we have three trial functions that contain 19 constant coefficients and eight equations according to equations (3.7)–(3.14). We need to create 11 additional equations from equations (3.15)–(3.17) to achieve a set of polynomials that contains 19 equations and 19 constants.

By the above clarifications, we generated supplementary equations, equations (3.15)–(3.17), in this order:

I. Four equations are the result of calculating obtained equations: $G (0) = 0, G (10) = 1, G' (0) = 1$ , $G' (10) = 0$

II. Four equations are the result of calculating obtained equations: $Θ (0) = 1, Θ (10) = 0, Θ' (0) = 0$ , $Θ' (10) = 1$

III. Three equations are the result of calculating obtained equations: $Φ (0) = 1, Φ (10) = 0, Φ' (0) = 0$

The equations in subsections (I) through (III) are too big for visual display. Using the procedures above, we got a group of polynomial expressions with 19 equations and 19 constants. After resolving these polynomials, equations (3.4)–(3.6) were attainable and can be effortlessly obtained in the following manner:

\begin{matrix} g (x) = 1.732292131 \times 10^{- 6} x^{7} - 0.00008181687017 x^{6} \\ + 0.001561245723 x^{5} - 0.01565453405 x^{4} \\ + 0.09336983807 x^{3} - 0.374551210 x^{2} + 1 x \end{matrix}

\begin{matrix} θ (x) = - 0.0001531539220 x^{4} + 0.004533926803 x^{3} \\ - 0.03665000000 x^{2} - 0.03373875830 x + 1 \end{matrix}

\begin{matrix} ϕ (x) = - 0.00005506211180 x^{5} + 0.001452251553 x^{4} \\ - 0.01032608696 x^{3} + 1 + 0.03097826087 x \end{matrix}

Results and discussions

The FEM calculations are presented using various parameters such as viscosity, porosity, magnetic field, Prandtl number, and heat source with temperature. Figures 2 and 3 illustrate that modifications to viscosity cause corresponding changes in the convection system’s fluid velocity and temperature with different values of ( $B$ ). As depicted in Figure 3, the velocity profile experiences a substantial increase beyond a specific boundary point when the velocity boundary stiffness rises along with the viscosity parameter, which means that the fluid near the wall is more resistant to deformation and has a higher momentum than the fluid away from the wall. This results in a sharper velocity gradient near the wall and a flatter velocity profile in the outer layer of the boundary layer. The heat transfer system’s ability to impose limits is denoted by its boundary values. An increase in the viscosity parameter range causes a decrease in boundary layer stiffness, as evidenced by this observation.

Figure 2.

The velocity profile versus variable viscosity characteristic $B$ .

Figure 3.

The temperature profile versus variable viscosity characteristic $B$ .

Figure 4 illustrates that the temperature profile ( $λ$ ) decreases with an increase in the heat sink variable. Concurrently, the heat lineation increment range also diminishes, indicating a corresponding decrease in the thermal layer’s stiffness. As a result, a lower heat sink variable means that the electronic device can dissipate heat more efficiently to the surrounding air. The temperature profile ( $λ$ ) shows how evenly the heat is distributed along the heat sink, and a smaller value means a better balance. The heat lineation increment range shows how steady the heat transfer rate is along the heat sink, and a smaller value means a more consistent rate. The thermal layer’s stiffness shows how much the heat sink can bend due to thermal expansion, and a smaller value means a more pliable material. Both fluid velocity and temperature undergo a reduction as the porosity parameter ( $K$ ) rises, as shown in Figures 5 and 6. The heat exchange between the solid and the fluid phase is improved by the enlargement and interconnection of the pores. The porosity parameter ( $K$ ) is a nondimensional number that expresses the proportion of the permeability to the thermal diffusivity of the porous medium. A larger value of $K$ implies that the fluid can move more freely through the pores. This observation is made in the convection system, where the porosity of the stretching plate $K$ is being increased.

Figure 4.

The temperature profile versus variable viscosity characteristic $λ$ .

Figure 5.

The velocity profile versus variable porosity characteristic $K$ .

Figure 6.

The temperature profile versus variable porosity characteristic $K$ .

Figures 7 and 8 illustrate that an increase in the $(λ_{0})$ Jeffrey parameter, which corresponds to an increase in the amount of Jeffrey fluid in the convection system, both velocity and temperature profile decrease. The impact of the Jeffrey parameter is far-reaching, resulting in a reduction in the stiffness of the thermal boundary layer, which indicates a higher Jeffrey parameter means that the fluid has a higher elasticity and a lower viscosity, which implies that the fluid can deform more easily under stress and recover more quickly after stress removal. The lineation decreases, and the boundary layer stiffness is impacted when the magnetic field $(M$ ) parameter increases, causing a reduction in velocity, as depicted in Figure 9. Figure 10 provides a visual representation of the changes that take place in the temperature profile when the magnetic field $(M$ ) parameter is increased. The velocity experience decrease when magnetic field (M) parameter is increased. However, the temperature increases due to magnetic field ( $M$ ) parameter growth. The magnetic field $(M$ ) parameter is a measure of the strength of the magnetic field, and as it increases, it can have a significant impact on various physical properties, including the temperature profile.

Figure 7.

The velocity profile versus variable Jeffrey characteristic $λ_{0}$ .

Figure 8.

The temperature versus variable Jeffrey characteristic $λ_{0}$ .

Figure 9.

The velocity profile versus variable magnetic field characteristic $M$ .

Figure 10.

The temperature profile versus variable magnetic field characteristic $M$ .

Figure 11 shows that the temperature lineation decreases as the Prandtl number increases. Temperature lineation refers to the alignment of temperature gradients in a particular direction, which means that the heat transfer by convection is more dominant than the heat transfer by conduction in the fluid.

Figure 11.

The temperature profile versus variable Prandtl $\Pr$ .

Figure 12 visually represents the relationship between the Sc number and the concentration curve. The Sc number, also known as the Schmidt number, is a dimensionless quantity used in fluid dynamics to represent the ratio of momentum diffusivity to mass diffusivity, which is a measure of how much of a substance is dissolved or dispersed in a fluid.

Figure 12.

The concentration versus variable Schmidt $Sc$ .

Validation

The following compares the temperature profiles with FEM in this study and the analysis of Thenmozhi et al.⁴² did. The graphs show a good agreement between the results obtained with previous research and the validation that has been done (Figure 13).

Figure 13.

Comparison of temperature profile between FEM and Thenmozhi et al.⁴²

In Figures 14 to 16, the following comparison is made between the accuracy of the AGM and the FEM:

Figure 14.

FEM and AGM solution results for $g' (ζ)$ .

Figure 15.

FEM and AGM solution results for $θ (ζ) .$

Figure 16.

FEM and AGM solution results for $ϕ (ζ)$ .

The optimizations of $g'$ , $θ$ , and $ϕ$ are shown in Figures 17 to 19.

Figure 17.

Optimization results for $g' (η)$ .

Figure 18.

Optimization results for $θ (η)$ .

Figure 19.

Optimization results for $ϕ (η)$ .

Conclusions

A partial differential equation represents the mathematical model of the heat transfer system with Jeffery fluid MHD flow on a permeable stretching plate. Since machine learning methods cannot solve PDEs, the FEM is employed to solve the equations. A better solution for the ODE and an analysis of the heat transfer system are achieved through the AGM. The mathematical model considers various parameters, such as fluid, heat source, velocity, concentration, and boundary layer. The Jeffery fluid dynamic model is used to analyze the heat transfer system, considering how these parameters impact the heat transfer rate. The following cases and observations are reported:

The temperature in the heat transfer system drops with the flow velocity when the input parameters of Jeffery fluid and heat source are increased. This implies that increasing Jeffery fluid improves the heat transfer rate in the system and leads to more convection.

Increasing both the magnetic field’s input parameter and the stretching plate’s porosity reduces the temperature in the heat-transferring system with dynamic fluid velocity.

The convection system’s temperature drops as the Prandtl number’s value increases. The Prandtl number is a ratio of momentum, thermal diffusivity, and a coefficient.

The magnetic field ( $M$ ) parameter rising causes the fluid velocity to decline and the temperature to ascend as a consequence of the magnetic field ( $M$ ) parameter increase.

The heat lineation increment range also decreases, which implies that the thermal layer’s stiffness also reduces.

An increase in permeability, magnetic field, and Prandtl number is characteristic of the boundary layer stiffness, as deduced from the data.

A higher heat flux and a higher temperature gradient are achieved by increasing the thermal boundary layer thickness and decreasing the fluid velocity near the plate. This can be done by increasing the magnetic field parameter, the heat source/sink parameter, or the Prandtl number. These parameters enhance the heat transfer rate from the plate to the fluid.

The thermal convection in the heat transfer system is mathematically improved through the use of Jeffery fluid, porosity, and a stretching plate. When AGM and the FEM are compared using various parameter values, the error figure data indicates a slight difference between the solutions from AGM and the exact solutions. The convergence figure implies that increasing the number of terms of AGM can lead to more accurate solutions.

Footnotes

Appendix

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Davood Domiri Ganji

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