Viscoelastic fluid flow on channel in porous medium with Soret and Dufour along with the effect of activation energy and suction and blowing

Abstract

This study conducts a thorough numerical analysis of a complex problem involving magnetohydrodynamic flow, radiation, activation energy, Soret, the Dufour effect, and heat transfer in a permeable channel for a Maxwell fluid. Governing equations for momentum, concentration, and heat are transformed into ordinary differential equations through a similarity transformation, enabling focused analysis. Utilizing a fourth-order Runge-Kutta scheme, we compute dimensionless velocity, temperature, and concentration fields for the steady motion of a Maxwell fluid in a channel. To ensure accuracy, our results are validated against existing research, showing excellent agreement. The study investigates the impact of various physical parameters on fluid motion, revealing insights such as decreased velocity and concentration with increased chemical reaction, and rising fluid temperature. Additionally, concentration distribution decreases with a chemical reaction and a higher Schmidt number. This research enhances understanding of complex fluid dynamics, providing valuable insights for engineering applications, particularly in industries requiring precise fluid behavior control.

KEYWORDS:

• Channel
• Maxwell fluid flow
• activation energy
• chemical reaction
• Soret/Dufour

1. Introduction

Fluid flow in porous media channels represents an attractive and complex subset of fluid dynamics with significant applications in various fields, including geology, hydrogeology, petroleum engineering, and environmental science. In porous media, fluids such as water, oil, or gas flow through interconnected void spaces within a solid matrix, often composed of rocks, sediments, or soil. These flow channels exist at multiple scales, ranging from microscopic pores in rock formations to macroscopic conduits in geological structures, making the study of fluid flow in porous media both intricate and essential for a wide range of practical purposes. Permeable media channels are encountered in numerous natural and industrial settings, including underground aquifers, oil reservoirs, soil layers, and filtration systems. Understanding how fluids navigate through these porous environments is crucial for addressing challenges such as groundwater management, enhanced oil recovery, and pollutant transport. A combination of physical and geometrical factors influences the behaviour of fluid flow in porous media channels. These factors give rise to various flow phenomena, including Darcy's law, capillary forces, and multiphase flow, which collectively govern fluid movement in porous media. In this exploration of fluid flow in permeable media channels, we will delve into the fundamental principles that underlie this phenomenon. We will discuss the concept of permeability, which quantifies the ease with which fluids flow through a porous medium, and explore how it relates to Darcy's law, a fundamental equation that describes fluid flow in porous media.

Additionally, we will examine the role of capillary pressure, which arises from the interactions between fluid and solid surfaces, and its impact on fluid movement in porous materials. Understanding fluid flow in permeable media channels is not only of academic interest but also has significant practical implications. For example, it is crucial in predicting the movement of contaminants in groundwater, optimizing oil extraction techniques, and designing efficient filtration systems for water treatment. Whether you are a geologist, engineer, or environmental scientist, a grasp of fluid flow in porous media channels is essential for addressing real-world challenges and making informed decisions in various applications.

In the study shown by Abbas et al. [1], the focus was on examining the magnetohydrodynamic boundary-layer flow of a Maxwell fluid within a permeable channel. Hayat et al. [2] conducted an analysis that delved into the impact of wall properties on the MHD peristaltic flow of a Maxwell fluid. This analysis also considered energy transfer and the presence of a porous medium. Rassoulinejad et al. [3] engaged in an analytical examination of the flow characteristics of a Maxwell fluid as it passed through a permeable saturated channel. Their investigation encompassed several wall boundary conditions. Khan et al. [4] observed the impression of a magnetic field and heat source on the behaviour of a Maxwell fluid within a permeable channel. Fetecau et al. [5] discussed the mathematical aspects related to the flow of a Maxwell fluid through a permeable plate channel driven by a wall that was either constantly accelerating or oscillating. Lv et al. [6] delved into the effects of chemical reactions and thermal radiation on the flow of a nanofluid within a rotating channel. The study also considered the occurrence of hall currents. Devi and Srinivas [7] considered the intricate dynamics of two immiscible layers of viscoelastic liquid flowing within a perpendicular, permeable channel. Kashyap et al. [8] observed the mixed convective flow of a Maxwell fluid within a channel with variable permeability. This analysis also took into account the Soret effect. Hayat et al. [9] investigated the channel flow of a Maxwell fluid, with a specific focus on the impact of chemical reactions. Raizah et al. [10] explored the coupled possessions of Dufour and Soret on the flow of a hybrid nanofluid through a gyrating channel. They also considered chemically reactive Arrhenius activation energy. Abbas et al. [11] examined the influence of activation energy and the Soret–Dufour possessions. Rafiq et al. [12] investigated the impression of activation energy on peristaltic flow via a permeable wall channel. Ahmed et al. [13] performed an analysis based on Soret-Dufour to understand energy transport. These studies collectively contribute to our understanding of various fluid flow phenomena, encompassing a wide range of factors such as magnetohydrodynamics, heat transfer, chemical reactions, porous media, and more.

The study of fluid flow in porous media takes on a new dimension when we consider Maxwell fluids and their interaction with the Soret and Dufour effects. Maxwell fluids, a subclass of non-Newtonian fluids, exhibit viscoelastic behaviour characterized by a delayed stress response to deformation. When these fluids flow in channels within porous media, the interplay of their unique rheological properties and the Soret and Dufour effects introduces complexities that have important implications in various scientific and engineering applications. Instead, they exhibit a time-dependent stress response due to the incidence of both viscous and elastic components. This behaviour is described by the Maxwell constitutive equation, which incorporates a relaxation time, and it adds a layer of intricacy to fluid flow analysis. When Maxwell fluids flow within porous media channels, several phenomena come into play. First, the porous structure of the medium affects the fluid flow patterns, which may lead to variations in velocity, pressure, and concentration fields.

Additionally, the Soret and Dufour effects become significant factors. The Soret outcome, also known as thermal diffusion, arises due to a temperature gradient in the fluid. It causes species (e.g. solute or gas) to diffuse along the temperature gradient, impacting concentration distribution within the porous medium. The Dufour outcome, on the other hand, deals with mass transfer driven by a concentration gradient in a thermally conducting medium, leading to changes in temperature distribution. Together, the Soret and Dufour effects introduce a coupling between energy and mass transfer, altering the energy and concentration profiles within the porous medium. The study of Maxwell fluid flow in permeable media channels with the Soret and Dufour possessions has profound implications in various fields. For instance, it is crucial in modelling energy and mass transfer courses in geothermal reservoirs, petroleum reservoirs, and environmental remediation scenarios. Understanding how these complex interactions affect fluid flow and transport phenomena is essential for optimizing resource extraction, predicting contaminant transport, and developing sustainable energy solutions. In this exploration of Maxwell fluid flow in channels within porous media with the Soret and Dufour effects, we will delve into the mathematical modelling, governing equations, and numerical techniques employed to analyze and simulate these intricate phenomena. We will also explore practical applications and the significance of this research in addressing real-world challenges associated with fluid flow, energy transfer, and mass transport in porous media environments.

Harikrishna et al. [14] showed a study focusing on the examination of the Soret and Dufour possessions in the situation of magnetohydrodynamic natural convective flow of Maxwell nanofluid past a sheet. Muhammad et al. [15] explored mathematical modelling within the domain of magnetohydrodynamics for a Maxwell fluid past a horizontal plate. Abbas et al. [16] delved into the transportation behaviour of flow with energy slip, considering the stimulus of Soret–Dufour effects and second-order slips. Khan et al. [17] investigated the phenomenon of chemotaxis bioconvection in a swirling flow of Maxwell fluid. Kumar and Reddy [18] examined the possessions of radiation with the combined effect of chemical impact on the magnetohydrodynamic boundary layer flow of a Maxwell fluid past a stretching sheet. Khan et al. [19] calculated the effect of double diffusion on the radiative flow of a thin film Maxwell fluid over a channel. Sudarmozhi et al. [20] considered the characteristics of MHD and energy convective flow of a Maxwell fluid crosswise an inclined perpendicular plate which is embedded in a porous medium. Iranian et al. [21] analyzed the implication of energy generation and the effects of suction/blowing on Maxwell fluid flow past a flat plate, taking into account the stimulus of radiation. Mamatha et al. [22] showed a study that involved the application of multi-linear regression techniques, including suction and injection effects. Tayebi [23] investigated the similarity solution of fractional MHD fluid flow using a porous stretching sheet. Khan et al. [24] explored the impact of entropy generation on Powell Eyring fluid within a permeable channel. Khan and Rafaqat [25] delved into the effects of radiation and MHD on compressible Jeffrey fluid with peristalsis. Khan and Tariq [26] analyzed the peristaltic flow of second-grade dusty fluid through a porous medium in an asymmetric channel. Turkyilmazoglu and Duraihem [27] examined the fully developed flow in a long triangular channel under the influence of an applied magnetic field. Siddiqui and Turkyilmazoglu [28] elucidated the slit flow and thermal analysis of micropolar fluids in a symmetric channel with dynamic and permeable boundaries. Turkyilmazoglu [29] investigated analytical solutions for the magnetohydrodynamic moving liquid plug within a microchannel.

1.1. Literature gap and novelty of the gap

After an inclusive review of the remaining literature, it becomes clear that numerous researchers have extensively explored boundary value problems involving various fluids and geometries. However, a noticeable research gap emerges when it comes to investigating Maxwell fluid flow within a channel while simultaneously considering the combined effects of Magnetohydrodynamics (MHD), radiation, energy generation, activation energy, Soret and Dufour within a porous medium. To bridge this research gap, our study initiates by formulating the governing equations, which initially appear as complex PDEs. Subsequently, we employ a transformation process that utilizes similarity techniques to convert these intricate PDEs into ODEs. The primary aim of this transformation is to achieve linearity in the equations. This transformation holds paramount importance as it facilitates the utilization of the Runge-Kutta method for solving the resulting ODEs. The research findings are presented in the form of graphical plots and tabular representations, providing clear insights into the contours of velocity, energy, and concentration. These contours are thoroughly examined in the context of MHD, radiation, and heat generation, considering both suction and injection scenarios. The learning of Maxwell fluid flow in channels within porous media, regarding the Soret and Dufour possessions, radiation, heat generation, and MHD interactions, is a highly interdisciplinary field with the potential to impact various industries and address complex engineering and environmental challenges. Its novelty lies in the integration of these multifaceted phenomena, and its objectives include developing accurate models, conducting numerical simulations, and applying findings to practical solutions.

2. Problem statement

Researchers and engineers can gain valuable insights into intricate fluid flow phenomena by examining the complex interplay of various factors, including a Maxwell fluid, radiation, magnetohydrodynamics (MHD), activation energy, Soret/Dufour effects, and chemical reactions. To analyze the combined impact of these elements, computational techniques such as numerical simulations and mathematical modelling are often applied to solve modified equations. This understanding has the potential to drive advancements in diverse fields, including the design and optimization of energy exchangers, chemical processes, and energy systems, particularly in the context of channel flows operating under complex conditions. The study focuses on the investigation of the 2D non-time dependent laminar flow of a Maxwell fluid in a channel under the inspiration of a transverse magnetic field. The x-axis is aligned along the channel; here, the y-axis is perpendicular to it, resulting in symmetric flow behaviour along the x-axis. The velocity mechanisms in the x and y directions are represented as u and v, respectively. The study incorporates boundary conditions that consider suction or injection effects at the channel boundaries.

Additionally, Let's analyze the heat transfer properties of a magnetohydrodynamic (MHD) 2D flow involving an incompressible Maxwell fluid within a channel featuring permeable walls situated at y = ±H/2, as illustrated in the image below. The porous material occupies the region between the channel walls, and both walls maintain a uniform temperature of T_H. At the channel's midpoint (y = 0), the temperature gradient ?T/?y is zero. We are interested in examining the boundary layer equations governing this particular flow scenario. The governing equations for this complex scenario, which include radiation, Soret/Dufour effects, activation energy, and chemical reactions, were established in a prior study directed by Hayat et al. [30].

Flow model (Hayat et al. [30])

The continuity equation remains a fundamental principle in describing the conservation of mass in fluid flow, even when dealing with complex scenarios like Maxwell fluid flow in porous media channels. However, the specific constitutive equations and boundary conditions applied to the problem will depend on the characteristics of the Maxwell fluid, the properties of the porous medium, and the overall system under investigation. (1) $u_x+v_y=0.$(1) Here, u_x and u_y represent the partial derivatives of the velocity components u and v with respect to the x and y directions, respectively.

The momentum equation serves as the foundation for understanding how the viscoelastic properties of Maxwell fluids, along with other factors like porous medium structure and external forces, influence the fluid's velocity distribution and behaviour. Depending on the specific scenario and boundary conditions, researchers and engineers can apply this equation to analyze and predict the flow characteristics of Maxwell fluids within porous channels, taking into account the unique rheological properties that distinguish them from Newtonian fluids. From Hayat et al. [30], with the addition of Maxwell fluid flow, the momentum equation can be written as,

PDE (2) $\begin{array}{rl}& u{u}_x+v{u}_y+\lambda [u^2{u}_{\mathrm{xx}}+v^2{u}_{\mathrm{yy}}+2\mathrm{uv}{u}_{\mathrm{xy}}]\\ & =\vartheta u_{\mathrm{yy}}-\frac{\sigma {B_o}^2}{\rho }[\mathrm{\lambda v}{u}_y+u]-\frac{\vartheta }{k_0}u.\end{array}$(2) Incorporating the relaxation time (λ), kinematic viscosity (ϑ), fluid density (ρ), porosity (k0), electrical conductivity (σ), and magnetic field (B₀), and applying similarity transformations along with nondimensional quantities, Hayat et al. [30] derived dimensionless equations. (3) $\begin{array}{rl}{y}^{\ast }& =\frac{y}{H},v=\mathrm{Vf}(y^{\ast }),u=-V{x}^{\ast }{f}^{\mathrm{\prime }}(y^{\ast }),x^{\ast }=\frac{x}{H},\end{array}$(3) (4) $\begin{array}{rl}M& =\frac{\sigma B_0^{2}H}{\mathrm{\rho V}},k_p=\frac{H^2}{k_0},\mathrm{De}=\frac{\lambda V^2}{\vartheta },\mathrm{Re}=\frac{\mathrm{VH}}{\vartheta }.\end{array}$(4) ODE (5) $\begin{array}{rl}& f^{\mathrm{\prime \prime \prime }}(1-\mathrm{De}{f}^2)+2\mathrm{De}f{f}^{\mathrm{\prime }}{f}^{\mathrm{\prime \prime }}-M(\mathrm{Re}{f}^{\mathrm{\prime }}+\mathrm{De}f{f}^{\mathrm{\prime \prime }})\\ & -k_p{f}^{\mathrm{\prime }}+\mathrm{Re}(f^{\mathrm{\prime }2}-f{f}^{\mathrm{\prime \prime }})=0.\end{array}$(5) Here, De denotes the Deborah number, M represents the Hartman number, k_p indicates the porosity parameter, Re is the Reynolds number, and The Reynolds number (Re) is a dimensionless quantity used to predict the flow patterns in fluid dynamics. It is defined as the ratio of inertial forces to viscous forces within a fluid flow. Here in this research, Re > 0 is referred to as suction, and Re < 0 means blowing effect.

Dimensional boundary condition. (6) $\begin{array}{rl}\text{At }y& =0,u_y=0,v=0,\end{array}$(6) (7) $\begin{array}{rl}\text{At }y& =\frac{H}{2},v=\frac{V}{2},u=0.\end{array}$(7) Here, uy is Referred to as a partial differentiation of u with respect to y. u, v is the velocity of the fluid in which V > 0 corresponds to suction and V < 0 indicates blowing.

The dimensionless boundary condition at (8) $\begin{array}{rl}\text{at }y& =0,f^{\mathrm{\prime \prime }}=0,f=0,\text{ (7)}\\ f& =0.5,f^{\mathrm{\prime }}=0\text{ at y = 0}\text{.5}\text{.}\end{array}$(8) The energy equation with these additional effects and heat generation provides a comprehensive framework for analyzing and predicting temperature distributions within a fluid under various conditions. Engineers and researchers use this equation to understand the impact of these factors on energy transfer and temperature profiles, which is crucial in fields such as combustion, environmental engineering, and materials processing.

PDE (9) $\begin{array}{rl}u{T}_x+v{T}_y& =\frac{k}{\rho c_p}{T}_{\mathrm{yy}}-\frac{1}{\rho c_p}\frac{\mathrm{\partial }{q}_r}{\mathrm{\partial }y}+\mu {u_y}^2+\frac{D_B{K}_T}{C_s{C}_p}{C}_{\mathrm{yy}}\\ & +\frac{Q_0}{\rho c_p}(T-T_H).\end{array}$(9) Q ₀ denotes heat generation; K_T denotes Dufour number, T is temperature; ρc_p denotes specific heat, and q_r is the radiative heat flux. In accordance with Rosseland [31], the heat flux qr can be expressed as q_r = −(4 σ * /3 k*) ∂T ⁴/∂ y here σ* represents the Stephan-Boltzmann coefficient and k* stands for the mean absorption coefficient. As per the approach outlined by Raptis and Perdikis [32], it is assumed that the energy differences within the flow are suitably minor. Consequently, T ⁴ can be approximated as a linear function of temperature by expanding it around the ambient temperature T_∞ and subsequently disregarding the square and higher-order terms, yielding the following expression $T^4\cong 4T{T}_{\mathrm{\infty }}^3-3T_{\mathrm{\infty }}^3$.

Thus, by making use of the above. (10) $\begin{array}{rl}\frac{\mathrm{\partial }{q}_r}{\mathrm{\partial }y}& =-\frac{4{\sigma }^{\ast }}{3k^{\ast }}\frac{{\mathrm{\partial }}^2{T}^4}{\mathrm{\partial }{y}^2}\approx -\frac{16{\sigma }^{\ast }{T}_{\mathrm{\infty }}^3}{3k^{\ast }}\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}\end{array}$(10) (11) $\begin{array}{rl}{x}^{\ast }& =\frac{x}{H},v=\mathrm{Vf}(y^{\ast }),y^{\ast }=\frac{y}{H},\\ \theta & =\frac{T-T_H}{T_w-T_H},u=-V{x}^{\ast }{f}^{\mathrm{\prime }}(y^{\ast }),\end{array}$(11) Here, Pr is the Prandtl number, Q represents heat generation, and Rd is radiation. Ec represents Eckert number; D_f is the Dufour effect, and θ is the temperature profile. (12) $\begin{array}{rl}Pr& =\frac{\mu c_p}{k},Q=\frac{H^2{Q}_0}{\rho c_p},\mathrm{Rd}=\frac{16{\sigma }^{\ast }{T}_{\mathrm{\infty }}^3}{3k{k}^{\ast }},\\ \mathrm{Ec}& =\frac{V^2{x}^2}{C_p(T_w-T_H)},D_f=\frac{D_B{k}_T(C_w-C_H)}{\vartheta C_s{C}_p(T_w-T_H)}.\end{array}$(12) ODE (13) $\begin{array}{rl}& (1+\mathrm{Rd}){\theta }^{\mathrm{\prime \prime }}-Pr\mathrm{Ref}{\theta }^{\mathrm{\prime }}+Pr\mathrm{Q\theta }+Pr\mathrm{Ec}{f}^{\mathrm{\prime \prime }2}\\ & +Pr{D}_f{\mathrm{\Phi }}^{\mathrm{\prime \prime }}=0.\end{array}$(13) Dimensional boundary condition (14) $\begin{array}{rl}\text{at }y& =0,T_y=0,\end{array}$(14) (15) $\begin{array}{rl}\text{at }y& =\frac{H}{2},T=T_H.\end{array}$(15) Both the walls have the same temperature, T_H, T is the temperature of the flow, and T_{y is} referred to as the partial differentiation of T with respect to y.

The dimensionless boundary condition. (16) $\begin{array}{rl}{\theta }^{\mathrm{\prime }}& =0,\text{ at }y=0,\end{array}$(16) (17) $\begin{array}{rl}\theta & =1,\text{ at y = 0}\text{.5}\text{.}\end{array}$(17) In practical applications, solving the mass equation with activation energy and the Soret effect is a complex task, as it involves coupling with other equations, such as the energy equation and reaction rate expressions, to fully describe combustion or chemical reaction processes. Computational methods, including numerical simulations, are often employed to solve these coupled equations and analyze the behaviour of reacting mixtures under varying temperature and concentration gradients. Understanding the mass equation with activation energy and the Soret effect is crucial for optimizing combustion processes, predicting pollutant emissions, and designing efficient chemical reactors in various industrial applications.

PDE (18) $\begin{array}{rl}& u{C}_x+v{C}_y\\ & =D_B{C}_{\mathrm{yy}}+\frac{D_B{K}_T}{T_m}{T}_{\mathrm{yy}}-k_r^2\frac{T}{T_H}\exp (-E_a/\mathrm{kT})(C-C_H).\end{array}$(18) Here, D_B denotes mass diffusion; K_T denotes Soret number, k_r is a chemical reaction, E_a indicates activation energy, and C is concentration. (19) $\begin{array}{rl}v& =\mathrm{Vf}(y^{\ast }),x^{\ast }=\frac{x}{H},u=-V{x}^{\ast }{f}^{\mathrm{\prime }}(y^{\ast }),y^{\ast }\\ & =\frac{y}{H},\varphi =\frac{C-C_H}{C_w-C_H},\end{array}$(19) (20) $\begin{array}{rl}\mathrm{Sc}& =\frac{v}{D_B},\sigma =\frac{k_r^2{H}^2}{\vartheta },\delta =\frac{T_w-T_H}{T_H},\\ E& =\frac{E_a}{k{T}_H},S_r=\frac{D_B{k}_T(T_w-T_H)}{T_m\vartheta (C_w-C_H)}.\end{array}$(20) Here, Sc is Schmith number, σ is the chemical reaction, δ denotes the temperature difference parameter, S_r is Soret number, E means activation energy, and ϕ indicates the concentration profile.

ODE (21) $\begin{array}{rl}& {\varphi }^{\mathrm{\prime \prime }}-\mathrm{ScRe}f{\varphi }^{\mathrm{\prime }}-\mathrm{Sc}\sigma (1+\mathrm{\delta \theta }{)}^n\exp (-E/1+\mathrm{\delta \theta })\\ & +\mathrm{Sc}{S}_r{\theta }^{\mathrm{\prime \prime }}=0.\end{array}$(21) Dimensional boundary condition (22) $\begin{array}{rl}{C}_y& =0,\text{ at }y=0,\end{array}$(22) (23) $\begin{array}{rl}C& =C_H,\text{ at }y=\frac{H}{2}.\end{array}$(23) Both the walls have the same concentration of C_H, C is the concentration of the flow, and C_y is referred to as the partial differentiation of C with respect to y.

The dimensionless boundary condition. (24) $\begin{array}{rl}{\varphi }^{\mathrm{\prime }}& =0,\text{ at }y=0,\end{array}$(24) (25) $\begin{array}{rl}\varphi & =1,\text{ at y = 0}\text{.5}\text{.}\end{array}$(25)

3. Method of solution

When using bvp4c or similar solvers for BVPs, the linearization process can involve approximating nonlinear terms with linear ones around a given operating point. This local linearization allows for the use of well-established linear analysis tools and facilitates the application of numerical methods for solving the resulting linearized system. Keep in mind that the effectiveness of linearization depends on the nature of the problem and the desired level of accuracy. In some cases, higher-order methods or techniques designed explicitly for nonlinear issues may be more appropriate. $\begin{array}{rl}{r}_1& =f,r_2=f^{\mathrm{\prime }},{r^{\mathrm{\prime }}}_4=f^{\mathrm{\prime \prime \prime }},r_3=f^{\mathrm{\prime \prime }},\\ r_5& =\theta ,r_6={\theta }^{\mathrm{\prime }},{r^{\mathrm{\prime }}}_6={\theta }^{\mathrm{\prime \prime }},\\ r_7& =\varphi ,r_8={\varphi }^{\mathrm{\prime }},{r^{\mathrm{\prime }}}_8={\varphi }^{\mathrm{\prime \prime }},\\ {r_4}^{\mathrm{\prime }}& =\frac{\begin{array}{c}-2\mathrm{De}f{f}^{\mathrm{\prime }}{f}^{\mathrm{\prime \prime }}+M(\mathrm{Re}{f}^{\mathrm{\prime }}+\mathrm{De}f{f}^{\mathrm{\prime \prime }})\\ -\mathrm{Re}({f^{\mathrm{\prime }}}^2-f{f}^{\mathrm{\prime \prime }})+k_p{f}^{\mathrm{\prime }}\end{array}}{(1-\mathrm{De}{f}^2)},\\ {r_6}^{\mathrm{\prime }}& =\left(\frac{Pr\mathrm{Re}f{\theta }^{\mathrm{\prime }}-PrQ\theta +Pr\mathrm{Ec}{f^{\mathrm{\prime \prime }}}^2-Pr{D}_f{\varphi }^{\mathrm{\prime \prime }}}{1+\frac{4\mathrm{Rd}}{3}}\right),\\ {r_8}^{\mathrm{\prime }}& =\mathrm{Sc}\mathrm{Re}f{\varphi }^{\mathrm{\prime }}+\mathrm{Sc}\sigma (1+\mathrm{\delta \theta })\exp (-\frac{E}{(1+\mathrm{\delta \theta })})\\ & -\mathrm{Sc}{S}_r{\theta }^{\mathrm{\prime \prime }}.\end{array}$

4. Validation of current work with existing

Table 1: Validation of current work with published work for different parameter values for local Nusselt number for the case of blowing $\mathrm{Re}<0$.

Table 1. Validation of current results with published work.

Re	M	Kp	De	R	Pr	Ec	Published work [29]	current study
−5	1	1	1	0.5	0.5	0.5	1.373870	1.373879
−10							1.252644	1.252643
−20							1.029914	1.029914
−5							1.377779	1.377778
	4						1.427874	1.427876
	6						1.578043	1.578042
	8						1.933996	1.933995
	1						1.373832	1.373837
		4					1.376067	1.376068
		8					1.383122	1.383129
		16					1.407723	1.407721
		1					1.373870	1.373875
			1.5				1.374013	1.374014
			2				1.374365	1.374363
			2.5				1.375137	1.375132
			1				1.295588	1.295587
				0.2			1.335939	1.335938
				0.4			1.363268	1.363269
				0.6			1.382998	1.382998
				0.5			0.580176	0.580175
					0.4		1.118977	1.118974
					0.6		1.619599	1.619592
					0.8		2.084982	2.084983
					0.5		1.373870	1.373871
						1	2.747741	2.747747
						1.5	4.121611	4.121613
						2	5.495482	5.495485

5. Result and discussion

This investigation endeavours to conduct numerical analyses on a complex problem involving magnetohydrodynamic flow, radiation, activation energy, sort, Dufour effect and energy transfer within a permeable channel for a Maxwell fluid. The governing equations of momentum, concentration, and power are changed into ODEs through a similarity transformation, facilitating concentrated analysis. Using a fourth-order Runge-Kutta scheme, we numerically determine dimensionless velocity, temperature, and concentration fields corresponding to the steady motion of Maxwell fluid on a channel. To ensure the accuracy of our results, we conducted a validation exercise against existing work, achieving an excellent agreement between the obtained results and prior research. The effect of physical parameters on fluid movement is plotted and debated. The standard literature shows that the parameter range is K_p= 0-3, D_f= 0-2, S_r= 0.1-0.4, σ =  1-4, E = 1-4, Sc = 1-4, M = 1-3, De = 0-2, R = 0-2, Ec = 0.2-0.5, Q = 0-1 and Pr = 1-4.

Figure 1 illustrates the impact on the velocity profile. As the porosity value surges, the velocity profile decreases for suction and blowing effects. The physical reason behind the observed behaviour in Figure 1, where a swelling porosity leads to a decreasing velocity contour for both suction and blowing effects, can be understood through the principles of fluid flow through porous media. Porosity represents the fraction of a porous medium's volume that consists of void space or pores. It characterizes the degree of porosity in the material and affects the flow of fluid on the porous medium. In fluid dynamics, “suction” and “blowing” refer to the introduction or removal of fluid at a boundary. Suction involves removing fluid from the boundary while blowing involves introducing fluid into the boundary layer. The velocity contour describes how the fluid flow varies across the boundary layer or within the fluid flow. It shows how the fluid's velocity changes from the boundary of a channel or surface into the bulk of the fluid. An increasing porosity value indicates that the porous medium has a higher fraction of void space or more prominent pores. This means that the porous medium is more permeable, allowing fluid to flow more easily through it. With an increasing porosity, there is less resistance to fluid flow over the porous medium. This reduced resistance leads to a higher flow rate of the fluid and, therefore, a decline in the velocity contour. The fluid moves more quickly over the porous medium, resulting in lower velocity gradients near the boundary. Whether you have suction (removing fluid) or blowing (introducing fluid), the porosity value influences the behaviour of the velocity profile because it affects the ease of fluid movement through the porous medium, which, in turn, moves the rate of fluid flow. As the porosity of the porous medium increases, the medium converts more permeable to fluid flow, reducing resistance and allowing the fluid to move more quickly. This leads to a lower velocity profile, regardless of whether suction or blowing is applied, as the porosity value rises.

Figure 1. For increasing the value of the porosity parameter, the effect of suction/blowing on the velocity profile.

Figure 2 illustrates how the porosity parameter affects the temperature contour. As the permeability parameter rises, the energy contour rises for suction and blowing effects. The physical reason behind the observed behaviour in Figure 2, where a swelling porosity parameter leads to an increasing temperature contour for both suction and blowing effects, can be understood through the principles of energy transfer in a fluid flow over porous media. The porosity parameter quantifies the fraction of a porous medium's volume that consists of void space or pores. It characterizes the degree of porosity in the material and influences the flow of fluid over the porous medium. In fluid dynamics, “suction” and “blowing” refer to the introduction or removal of fluid at a boundary. For example, suction involves removing fluid from the boundary, while blowing involves introducing fluid into the boundary layer. The temperature profile describes how the temperature within a fluid varies across the boundary layer or within the fluid flow. It illustrates how the temperature changes from the boundary of a channel or surface into the bulk of the fluid. As the porosity parameter increases, it indicates that the porous medium has a higher fraction of void space or more prominent pores. This results in a greater ease of fluid flow over the porous medium. With an increasing porosity parameter, there is reduced resistance to fluid flow over the porous medium.

Figure 2. For the swelling value of the porosity parameter, the effect of suction/blowing on the temperature profile.

As a consequence, heat is transported more efficiently within the fluid, leading to a more uniform temperature distribution. This results in a higher temperature near the boundary (wall), leading to an increased temperature profile. Regardless of whether you have suction (removing fluid) or blowing (introducing fluid), the porosity parameter influences the behaviour of the temperature profile because it affects the ease of fluid movement through the porous medium, which, in turn, influences the rate of energy transfer. As the k_p increases, the porous medium becomes more permeable to fluid flow, which enhances heat transport within the fluid.

Figure 3 illustrates how the porosity parameter influences the concentration contour. As the porosity parameter upsurges, the concentration profile decreases for suction and blowing effects. The physical reason behind the observed behaviour in Figure 3, where a growing porosity parameter leads to a declining concentration profile for both suction and blowing effects, can be understood through the principles of fluid flow over porous media. The porosity parameter represents the fraction of a porous medium's volume that is void space or filled with pores. It quantifies the extent to which a material is porous, and it affects how fluid flows through the medium. In fluid dynamics, “suction” and “blowing” refer to the introduction or removal of fluid at a boundary. For example, suction involves removing fluid from the boundary, while blowing involves introducing fluid into the boundary layer. The concentration profile describes how the concentration of a specific substance (e.g. a solute in a fluid) varies across the boundary layer or within the flow. It shows how the concentration changes from the boundary of a channel or surface into the bulk of the fluid. As the porosity parameter increases, it signifies that the porous medium has a higher fraction of void space or more prominent pores. This results in a greater ease of fluid flow through the porous medium. With an increasing porosity parameter, there is less hindrance to fluid flow over the porous medium.

Figure 3. For increasing the value of the porous effect, the effect of suction/blowing on the concentration profile.

Consequently, the substance being transported (e.g. a solute) within the fluid experiences less resistance as it moves through the porous material. This reduced resistance leads to a more efficient dispersion of the substance away from the boundary, resulting in a lower concentration near the wall. Whether you have suction (removing fluid) or blowing (introducing fluid), the porosity parameter influences the behaviour of the concentration profile because it affects the ease of fluid movement through the porous medium, which, in turn, influences the rate of mass transport. As the k_p increases, the porous medium becomes more permeable to fluid flow, allowing the transported substance to disperse more readily away from the boundary. This phenomenon is consistent with the observation that the concentration profile decreases for both suction and blowing effects as the porosity parameter rises.

Figure 4 depicts how the Dufour number impacts the temperature contour. As the Dufour number surges, the temperature outline rises for suction and blowing effects. The physical reason behind the observed behaviour in Figure 4, where a growing Dufour number leads to a swelling temperature contour for suction and blowing impact, can be understood through the principles of heat and mass transfer in fluid dynamics. The Dufour number is a nondimensional parameter that characterizes the relative rates of energy transfer and species transfer in a fluid mixture. It quantifies the effect of species concentration gradients on heat transfer within a fluid. In fluid dynamics, “suction” and “blowing” refer to the introduction or removal of fluid at a boundary. For example, suction involves removing fluid from the boundary, while blowing involves introducing fluid into the boundary layer. The temperature profile describes how the temperature within a fluid varies across the boundary layer or within the fluid flow. It shows how the temperature changes from the boundary of a channel or surface into the bulk of the fluid. A higher Dufour number indicates a more substantial impact of concentration gradients on heat transfer within the fluid. In other words, it signifies that the concentration gradients within the fluid have a more pronounced effect on the temperature distribution. When the Dufour number is elevated, concentration gradients within the fluid play a more significant role in driving heat transfer. Higher Dufour numbers mean that variations in solute concentration lead to more significant variations in temperature. As a result, the temperature near the wall of the channel rises more rapidly, leading to a higher temperature profile. Whether you have suction (removing fluid) or blowing (introducing fluid), the Dufour number governs the behaviour of the temperature contour because it affects the fluid's sensitivity to concentration gradients, which in turn influences the energy transfer rate. As the Dufour number increases, concentration gradients become more influential in driving heat transfer within the fluid, causing the energy near the boundary to increase more rapidly. This phenomenon is consistent with the observation that the temperature contour increases for suction and blowing effects as the Dufour number rises.

Figure 4. For the swelling value of the Dufour effect, the effect of suction/blowing on the temperature profile.

Figure 5 illustrates how the Dufour number impacts the concentration contour. As the Dufour number increases, the concentration contour decreases for suction and blowing effects. The physical reason behind the observed behaviour in Figure 5, where an increasing Dufour number leads to a diminishing concentration profile for suction and blowing impacts, can be understood through the principles of energy and species transfer in fluid dynamics. The Dufour number is a nondimensional parameter that characterizes the relative rates of energy transfer and mass transfer in a liquid mixture. It quantifies the outcome of thermal gradients on the mass transfer (species concentration) in a fluid. In fluid dynamics, “suction” and “blowing” refer to the introduction or removal of fluid at a boundary. For example, suction involves removing fluid from the boundary, while blowing involves introducing fluid into the boundary layer. The concentration profile describes how the concentration of a specific substance (e.g. a solute in a fluid mixture) varies across the boundary layer or within the fluid flow. It shows how the concentration of the substance changes from the boundary of a channel or surface into the bulk of the fluid. In other words, it signifies that the temperature gradients within the fluid have a more pronounced effect on the solute's diffusion. When the Dufour number is elevated, temperature gradients within the fluid play a more noteworthy role in driving the diffusion of the solute. Higher temperature gradients lead to more efficient mass transport, causing the solute to disperse away from the boundary more rapidly. As a result, the concentration of the solute near the wall decreases. Whether you have suction (removing fluid) or blowing (introducing fluid), the Dufour number dictates the behaviour of the concentration contour because it impacts the solute's sensitivity to temperature gradients, which, in turn, influences its diffusion rate. as the D_f increases, temperature gradients become more influential in driving the solute's diffusion, causing it to disperse away from the boundary more rapidly. This phenomenon is consistent with the observation that the concentration profile decreases for both suction and blowing effects as the Dufour number rises.

Figure 5. For the swelling value of the Dufour effect, the effect of suction/blowing on the concentration profile.

Figure 6 illustrates how the Soret number influences the concentration profile. As the S_r surges, the concentration profile decreases for suction and blowing effects. The physical reason behind the observed behaviour in Figure 6, where a growing Soret number leads to a decreasing concentration contour for both suction and blowing effects, can be understood through the principles of Soret diffusion and species transport. The S_r is a nondimensional parameter that characterizes the relative rates of energy transfer and species transfer in a fluid mixture. It quantifies the effect of thermal gradients on the diffusion of a solute or species within a fluid. In fluid dynamics, “suction” and “blowing” refer to the introduction or removal of fluid at a boundary. For example, suction involves removing fluid from the boundary, while blowing involves introducing fluid into the boundary layer. The concentration profile describes how the concentration of a specific substance (e.g. a solute in a fluid mixture) varies across the boundary layer or within the fluid flow. It shows how the concentration of the substance changes from the boundary of a channel or surface into the bulk of the fluid. In other words, a surge in the S_r means that the solute's diffusion is more sensitive to temperature variations. When the Soret number is elevated, temperature gradients within the fluid have a more pronounced effect on the diffusion of the solute. Specifically, higher temperature gradients can drive more efficient mass transport. This results in a more rapid dispersion of the solute away from the boundary and a corresponding decrease in its concentration near the wall. Whether you have suction (removing fluid) or blowing (introducing fluid), the Soret number governs the behaviour of the concentration profile because it affects the solute's sensitivity to temperature variations, impacting its diffusion rate. As the Soret number increases, temperature gradients become more influential in driving the solute's diffusion, causing it to disperse away from the boundary more rapidly. This phenomenon is consistent with the observation that the concentration profile decreases for both suction and blowing effects as the Soret number rises.

Figure 6. For increasing the value of the Soret effect, the effect of suction/blowing on the concentration profile.

Figure 7 illustrates how chemical reactions impact the concentration profile. As the extent of the chemical reaction rises, the concentration profile decreases for suction and blowing effects. The physical reason behind the observed behaviour in Figure 7, where a growing extent of chemical reaction leads to a decreasing concentration profile for both suction and blowing effects, can be understood through the principles of chemical kinetics and mass transport. In chemical kinetics, the extent of a chemical reaction is a measure of how far a reaction has progressed toward its equilibrium state. It is often expressed in terms of reaction conversion, which represents the fraction of reactants that have been converted into products. In fluid dynamics, “suction” and “blowing” refer to the introduction or removal of fluid at a boundary. For example, suction involves removing fluid from the boundary, while blowing involves introducing fluid into the boundary layer. The concentration profile describes how the concentration of a specific substance (e.g. reactants or products in a chemical reaction) varies across the boundary layer or within the fluid flow. It shows how the concentration of a substance changes from the boundary of a channel or surface into the bulk of the fluid. When the extent of a chemical reaction is increased, it implies that more reactants are being converted into products. In practical terms, it means that as the fluid flows along the boundary, there is a greater consumption of the substance involved in the chemical reaction (e.g. reactants), resulting in a diminution in its concentration. With an increasing extent of chemical reaction, the concentration of the substance near the boundary (wall) decreases more rapidly because a more significant fraction of it is being consumed to form products. This leads to a lower concentration near the boundary and a steeper decrease in the concentration profile as you move away from the wall. Regardless of whether there is suction (removing fluid) or blowing (introducing fluid), the extent of the σ governs the behaviour of the concentration profile because the reaction rate determines how quickly the relevant substance is consumed or produced. When the extent of the chemical reaction increases, more reactants are converted into products, leading to a faster decrease in the concentration of the relevant substance near the boundary. This phenomenon is consistent with the observation that the concentration profile decreases for both suction and blowing effects as the extent of the chemical reaction rises.

Figure 7. For increasing the value of the chemical reaction, the outcome of the concentration profile for suction/blowing impact.

Figure 8 depicts how the activation energy influences the concentration contour. As the activation energy surges, the concentration contour rises for suction and blowing effects. The physical reason behind the observed behaviour in Figure 8, where a growing activation energy leads to a swelling concentration profile for both suction and blowing effects, can be understood through principles related to chemical reactions and mass transport. Activation energy is a fundamental concept in chemistry and physics. It represents the minimum energy required for a chemical reaction to occur. Specifically, it's the energy needed to break or overcome the energy barrier for reactant molecules to transform into products. In fluid dynamics, “suction” and “blowing” refer to the introduction or removal of fluid at a boundary. For example, suction involves removing fluid from the boundary, while blowing involves introducing fluid into the boundary layer. The concentration profile describes how the concentration of a specific substance (e.g. a reactant or product in a chemical reaction) varies across the boundary layer or within the fluid flow. It illustrates how the concentration of a substance changes from the boundary of a channel or surface into the bulk of the fluid. A higher activation energy means that the chemical reactions involved in the process require more energy to proceed. In practical terms, it implies that reactions will occur at a slower rate because a more significant fraction of the molecules have insufficient energy to strengthen the activation energy barrier. When the activation energy is elevated, chemical reactions involved in mass transport are slower. As a result, the consumption or production of the substance being transported (e.g. reactants or products) near the boundary proceeds at a slower pace. This leads to a higher concentration of the substance near the boundary since it has less time to react and is less likely to be consumed. Regardless of whether there is suction (removing fluid) or blowing (introducing fluid), the concentration profile exhibits the same behaviour because the rate of chemical reactions, driven by the activation energy, is the governing factor in determining the concentration near the boundary. When activation energy is increased, chemical reactions occur more slowly, leading to a higher concentration of the relevant substance near the boundary. This phenomenon is consistent with the observation that the concentration profile increases for both suction and blowing effects as the activation energy rises.

Figure 8. Increasing value of activation energy constant on concentration profile for suction/blowing impact.

Figure 9 illustrates how the Schmidt number influences the concentration contour. As the Sc surges, the concentration profile declines for both suction and blowing effects. The physical reason behind the observed behaviour in Figure 9, where growing the Schmidt number results in a decreasing concentration profile for both suction and blowing effects, can be understood through the following principles. For example, suction involves removing fluid from the boundary, while blowing involves introducing fluid into the boundary layer. The concentration profile represents how the concentration of a specific substance (e.g. a solute in a fluid) varies across the boundary layer or within the flow. It can describe how the concentration of a substance changes from the boundary of a channel or surface into the bulk of the fluid. A higher Schmidt number indicates that mass diffusivity is relatively low compared to dynamic viscosity. This implies that it is more difficult for the substance to diffuse through the fluid, and its concentration gradient is steeper. When the Schmidt number is high, the substance being transported (e.g. a solute) has a reduced ability to diffuse away from the boundary due to the lower diffusivity. As a result, the concentration near the boundary remains relatively high, leading to a steeper decrease in concentration away from the boundary. This results in a more pronounced reduction in the concentration profile as you move away from the wall, whether you have a suction effect (removing fluid) or a blowing effect (introducing fluid). The higher Schmidt number hinders the diffusion of the substance away from the boundary, making the concentration gradient steeper near the wall. This behaviour is consistent with the observation that the concentration profile decreases for both suction and blowing effects as the Schmidt number increases.

Figure 9. The increasing value of the Schmith number on the concentration profile.

6. Conclusion

In conclusion, our numerical analysis of the magnetohydrodynamic flow, radiation, activation energy, Soret effect, Dufour effect, and energy transfer within a porous channel for a Maxwell fluid has yielded significant insights into the underlying physical mechanisms. Through the conversion of governing equations into ODEs via similarity renovation and subsequent application of a fourth-order Runge-Kutta scheme, we obtained dimensionless velocity, temperature, and concentration fields, providing a comprehensive understanding of the steady motion of Maxwell fluid in a channel. The validation exercise against existing work demonstrated a robust agreement between our obtained results and prior research, affirming the reliability of our numerical approach. The key conclusions drawn from this investigation include:

1. We are elevating the porous parameter, which results in a reduction in both velocity and concentration contours. Interestingly, the temperature contour experiences an increase when subjected to suction and blowing effects.

2. Increasing the Dufour effect leads to a rise in the temperature profile, accompanied by a decrease in the concentration profile. This behaviour remains consistent for both suction and blowing effects.

3. The concentration contour exhibits a decrease as the values of the Soret effect, temperature difference parameter, and Schmidt number increase. This trend holds regardless of the presence of suction or blowing effects.

4. An increase in the activation energy value corresponds to a rise in the concentration profile, irrespective of the presence of suction or blowing effects.

5. These findings not only enhance our understanding of the complex interplay of multiple factors in Maxwell fluid flow within porous channels but also provide valuable insights for practical applications in diverse fields, including materials processing, heat exchanger design, and chemical reactor optimization. The observed behaviours offer guidance for manipulating flow characteristics in real-world scenarios, contributing to advancements in engineering and technology.

Acknowledgements

This work was supported by Princess Nourah Bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R59), Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia.: [Grant Number number (PNURSP2024R59)].

Disclosure statement

No potential conflict of interest was reported by the author(s).