RETRACTED ARTICLE: Impacts of soret and dufour possessions on micropolar fluid past a stretching sheet in a porous medium

44 min read Original article ↗

Introduction

Magnetohydrodynamics (MHD) focuses on the behavior of electrically conductive fluids influenced by magnetic fields, with wide-ranging applications in modern science and engineering. It plays a critical role in technologies such as electromagnetic pumps, nuclear reactor cooling, geothermal systems, and crystal fabrication. These practical implementations often involve the interplay of magnetic forces, temperature gradients, and mass transport, particularly in porous materials. With increasing industrial demands for precise thermal regulation and enhanced mass transport efficiency, the study of such complex phenomena under multiphysical conditions continues to attract significant attention.

In engineering applications like crystal fabrication, geothermal energy harvesting, and the design of generators and accelerators, MHD-based models have become indispensable. Several researchers have contributed to the development of MHD theory to understand its behaviour across various configurations. One such configuration involves the boundary layer flow generated by surface stretching, which is crucial for processes like polymer extrusion, wire drawing, liquid metal shaping, and paper and glass fiber production. These applications involve complex fluid behaviour that often cannot be captured accurately using traditional Newtonian assumptions.

Initial work in this area was conducted by Sakiadis1, who investigated the effects of continuous surface stretching on boundary layer development. Later, Tsou et al.2 analyzed heat and fluid transport in the boundary layer with variable thickness. Cortell3 expanded upon these foundations by incorporating thermal radiation and viscous dissipation into the thermal boundary layer analysis for stretching surfaces. Other studies examined magnetohydrodynamic and viscous flows over permeable, stretching surfaces. Research has more recently shifted to focus on boundary layers over shrinking surfaces due to their relevance in polymer processing. In4,5, the influence of a magnetic shear field on viscous fluid flow over deformable porous sheets was investigated. Bhattacharyya6 explored reactive solute dispersion in exponentially growing boundary layers, while thermal radiation effects on heat and mass transport over stretching sheets were considered by7. The influence of suction on MHD boundary layer flow over contracting sheets was examined by Muhaimin and Khamis8. Mandal and Mukhopadhya9 looked into surface heat flux impacts on exponentially stretching sheets, and Elbashbeshy10 analyzed heat and mass transfer along vertical surfaces under magnetic influence.

Current studies emphasize the impact of nonlinearly deformed sheets on thermal boundary layers. For instance, Cortell3 considered nonlinear stretching with radiative and viscous dissipation effects, while Muthukumaraswamy et al.11 analyzed heat distribution on vertically moving plates. Radiative energy transport along exponentially stretched surfaces was reported by Bidin and Nazar12, and Rashad13 discussed the role of radiation and viscosity variations in unstable MHD boundary flows in porous domains.

Micropolar fluids, a concept developed by Eringen14, account for the microstructural effects and the presence of microrotation and couple stresses. This makes them more accurate than Newtonian models for describing fluids such as colloidal suspensions, liquid crystals, and polymeric solutions. These fluids exhibit asymmetric stress tensors and are more representative in shear-dominated or particulate-laden systems. Abdel.

Rahman15 reported increased microscale velocities near the edge of rotating disks. Contributions from Sankara and Watson16, and Pal and Chatterjee17, have expanded the understanding of heat effects on micropolar flow in porous media. Other researchers, including Abd El-Aziz18 and Pal and Chatterjee19, have explored viscous dissipation, convection, and radiation effects on micropolar fluids. Studies20,21,22,23 considered additional effects such as slip, internal heating, and chemical reactions. Bejawada and Nandeppanavar24 evaluated radiation influence near upright plates, and Shankar Goud and Nandeppanavar25 analyzed chemical reactions within MHD micropolar systems. Heat generation effects on micropolar MHD flow over stretched surfaces were emphasized by Bejawada et al.26.

Additional research has examined flow structures involving porosity, suction, injection, and surface deformation27,28,29,30,31,32,33,34,35,36,37,38. For example, Reddy and Shankar Goud39,40 analyzed nanofluid behavior over stretched surfaces under MHD and radiative influences. Reddy et al.41 focused on heat generation and absorption in copper-based nanofluids over contracting sheets. Numerous studies42,43,44,45,46,47 have evaluated the combined effects of boundary motion, porosity, and heat/mass transfer in such complex systems.

In high-temperature industrial environments, the transport of heat and mass is heavily influenced by gradients in thermal and solute concentrations. The Dufour effect refers to heat flux caused by concentration gradients, while the Soret effect deals with mass diffusion driven by temperature gradients. Postelnicu48 and Cheng49 analyzed these phenomena in porous materials. Animasaun50 considered their roles in reactive Casson fluid flow, while M. Hayat et al.51 studied their combined effects in MHD Casson fluids over stretched surfaces. Additional simulation work from Motsa and Shateyi52, Kumaran and Sandeep53, and Abreu et al.54 validated the impact of these effects under varied boundary conditions.

This research explores the combined impacts of radiation and viscous dissipation on the MHD Casson hybrid nanofluid flow over a sheet embedded within a porous medium. The study emphasizes how thermal and electromagnetic forces contribute to improving heat transfer performance in various industrial processes55,56,57,58,59.

Despite extensive progress, many existing models omit two critical thermal mechanisms: Ohmic heating (also known as Joule heating) and viscous dissipation. Ohmic heating refers to energy produced due to electrical resistance in a conducting fluid, while viscous dissipation accounts for the transformation of kinetic energy into thermal energy due to internal friction. These mechanisms become particularly important in applications such as electronic cooling, nuclear energy, and high-speed thermal processing, where ignoring them can lead to inaccurate predictions.

This study aims to bridge this gap by developing a comprehensive model that integrates Soret and Dufour effects, Ohmic heating, viscous dissipation, and micropolar fluid dynamics for MHD flow over a stretching sheet embedded in a porous medium. Using similarity transformations, the governing partial differential equations are reduced to a set of coupled nonlinear ordinary differential equations, which are then numerically solved using MATLAB’s BVP4C solver.

In recent years, several investigations have addressed the interplay between thermal, magnetic, and chemical processes in complex fluid systems. Ashraf et al.60 explored catalyst-assisted exothermic chemical reactions on curved surfaces, offering important insights into localized heat generation in catalytic environments. Building on this, Rafiq et al.61 extended the analysis by incorporating magnetic field effects, demonstrating the combined influence of electromagnetic forces and chemical reaction mechanisms. Ali et al.62 examined the peristaltic motion of a micropolar Casson fluid, with a particular focus on thermal transport in the presence of viscous dissipation. Similarly, Li and Kumar63,64 analyzed the combined effects of viscous dissipation and magnetohydrodynamics on periodic energy transfer along a cone embedded within a porous medium. The understanding of MHD and micropolar fluid dynamics under different physical effects has been greatly enhanced by recent contributions65,66,67,68,69,70,71,72,73 in addition to the previously stated works. In slip-flow regimes, elastico-viscous MHD fluid flow with Soret and Dufour effects was studied by Samuel and Ajayi70. Entropy formation in radiative MHD nanofluid flow with viscous dissipation was investigated by Adegbie et al.71. A thorough examination of micropolar fluid motion with varying thermal conductivity was provided by Siahchehreghadikolaei et al.72. Lin et al.73 investigated temperature-dependent viscosity and thermal conductivity in MHD convective flows, whereas74,75,76,77,78 investigated stagnation-point MHD flow with heat and mass transfer under the influence of chemical reactions. Authors79 examined magnetohydrodynamic flow and mass transfer within Casson fluid systems featuring chemical reactivity.

Ohmic heating (Joule heating) and viscous dissipation have emerged as key mechanisms influencing thermal energy distribution in magnetohydrodynamic flows. Ohmic heating arises due to electrical resistance in conducting fluids, contributing to internal energy generation and altering thermal boundary layers. Viscous dissipation, on the other hand, converts mechanical energy into heat through viscous friction, becoming particularly important in high-shear or high-viscosity flows. Recent works have shown that when these effects are incorporated together, they significantly modify temperature distributions, heat transfer rates, and in some cases, the stability of flow solutions. Several numerical studies have demonstrated that the combined influence of Ohmic heating and viscous dissipation can enhance or suppress thermal transport depending on competing effects such as radiation, porous drag, and cross-diffusion phenomena.

Despite these advancements, most previous studies have examined either Ohmic heating or viscous dissipation in isolation, or in conjunction with only a limited set of other mechanisms. Few have attempted a comprehensive integration of Ohmic heating, viscous dissipation, Soret and Dufour effects, chemical reaction, and micropolar fluid dynamics in magnetohydrodynamic flows through porous media. The present study addresses this gap by developing a unified model that incorporates all these physical effects, thereby providing a more complete understanding of their combined influence on momentum, thermal, and mass transport characteristics.

However, few studies have integrated all these mechanisms into a unified framework. This research seeks to fill that void and offer valuable insights for real-world applications such as enhanced oil recovery, geothermal energy, membrane filtration, and advanced thermal management systems.

Real time application

The investigation of Soret and Dufour effects in micropolar fluid flow through a porous medium is relevant to several industrial and environmental processes. In geothermal systems, the interaction between thermal and concentration gradients within porous structures plays a key role in maximizing heat extraction. In the case of enhanced oil recovery (EOR), understanding the influence of temperature and mass diffusion on micropolar fluid behavior can contribute to more efficient fluid injection techniques. Additionally, such studies provide valuable insights for designing advanced filtration units for environmental cleanup, where adsorption and desorption processes are strongly affected by thermal and solutal variations. These examples highlight the practical importance of analyzing coupled heat and mass transfer phenomena in complex fluid systems.

Novelty

This work presents an in-depth review of recent progress in magnetohydrodynamic (MHD) boundary layer flows, focusing on the effects of stretching and shrinking surfaces, micropolar fluid characteristics, and the roles of thermal gradients and magnetic forces. It outlines important developments in the field, including how surface motion affects boundary layer behavior, the influence of radiation and viscous dissipation on heat transfer, and emerging studies on the importance of Soret and Dufour effects. By providing this broad assessment, the study seeks to pinpoint existing research gaps and suggest potential directions for exploring complex interactions in MHD flow and micropolar fluid studies.

Mathematical analysis

This investigation focuses on the steady, two-dimensional flow of an incompressible, electrically conducting micropolar fluid over a stretching surface situated within a porous medium. The stretching of the surface occurs along the x-axis, with velocity increasing proportionally with distance from a fixed origin, thereby inducing a stretching-type flow. The assumption of incompressibility and steady-state conditions is valid under low Mach number regimes, where variations in fluid density are negligible. This assumption simplifies the governing equations of motion without compromising the physical realism of the boundary layer formulation.

The flow is assumed to be laminar, and the fluid under consideration is capable of exhibiting microrotation and electrical conductivity. Given the small magnetic Reynolds number, magnetic induction effects are insignificant and thus omitted from the analysis. As a result, the externally applied magnetic field B0 is considered weak, and the Hall current effects are also neglected. These assumptions streamline the MHD model while retaining its essential physical attributes.

The porous medium is modeled as homogeneous and isotropic, implying uniform structural properties throughout. The analysis incorporates the Soret and Dufour effects, which are vital for accurate modeling of thermo-diffusive phenomena. The Soret effect accounts for the influence of temperature gradients on species diffusion, whereas the Dufour effect represents the contribution of concentration gradients to heat transfer. These effects become particularly important in micropolar fluids, where thermal and concentration fields interact more intricately due to microstructural motion.

Radiative heat transfer in the medium is modeled using the Rosseland diffusion approximation, which is appropriate for optically thick fluids where radiation behaves like a diffusive process rather than free propagation. This technique is widely adopted in thermal boundary layer analyses and is well supported by existing studies (e.g., Int. J. Hydrogen Energy, 44 (2019) 17,072–17,083; J. Taiwan Inst. Chem. Eng., 97 (2019) 12–23).

By considering the combined effects of magnetic field, thermal radiation, porous structure, and cross-diffusion, the model provides insights into complex transport processes found in engineering systems such as geothermal devices, chemical processing equipment, energy-efficient technologies, and environmental systems. The physical configuration of the problem—depicting a steady micropolar fluid flow over a stretching surface embedded in a porous domain—is illustrated in Fig. 1.

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

(1)

$$u\frac{\partial u}{{\partial x}} + v\frac{\partial u}{{\partial y}} = \left( {v + \frac{k}{\rho }} \right)\frac{{\partial^{2} u}}{{\partial y^{2} }} + \frac{k}{\rho }\frac{\partial N}{{\partial y}} - \frac{{\sigma B_{0}^{2} }}{\rho }u - \frac{v}{{k_{p}^{*} }}u + g\beta_{T} \left( {T - T_{\infty } } \right) + g\beta_{C} \left( {C - C_{\infty } } \right)$$

(2)

$$u\frac{\partial N}{{\partial x}} + v\frac{\partial N}{{\partial y}} = \frac{\gamma }{j\rho }\frac{{\partial^{2} N}}{{\partial y^{2} }} - \frac{k}{j\rho }\left( {2N + \frac{\partial N}{{\partial y}}} \right)$$

(3)

$$u\frac{\partial T}{{\partial x}} + v\frac{\partial T}{{\partial y}} = \frac{{k_{f} }}{{\rho C_{p} }}\frac{{\partial^{2} T}}{{\partial y^{2} }} + \frac{{\left( {\mu + k} \right)}}{{\rho C_{p} }}\left( {\frac{\partial u}{{\partial y}}} \right)^{2} + \frac{{\sigma B_{0}^{2} }}{{\rho C_{p} }}u^{2} - \frac{1}{{\rho C_{p} }}\frac{{\partial q_{r} }}{\partial y} + \frac{{D_{M} K_{T} }}{{C_{s} C_{p} }}\frac{{\partial^{2} C}}{{\partial y^{2} }}$$

(4)

$$u\frac{\partial c}{{\partial x}} + v\frac{\partial c}{{\partial y}} = D\frac{{\partial^{2} c}}{{\partial y^{2} }} - Kr{\prime} \left( {C - C_{\infty } } \right) + \frac{{D_{M} K_{T} }}{{T_{M} }}\frac{{\partial^{2} T}}{{\partial y^{2} }}$$

(5)

Fig. 1
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Flow geometry.

Using the Rosseland approximation, the radiative heat flux is expressed as:

$$q_{r} = \frac{{ - 4\sigma^{*} }}{{3k^{*} }}\frac{{\partial T^{4} }}{\partial y}$$

(6)

Here, refers to the absorption coefficient, and denotes the Stefan–Boltzmann constant. When the temperature variation within the flow is small, T4 can be expanded in a Taylor series about the ambient temperature T as:

$$T^{4} \cong 4T_{\infty }^{3} T - 3T_{\infty }^{4}$$

(7)

Also, ensure that the stretching velocity \({u}_{w}\)(x)= \({a}_{x}\) is applied consistently in both the formulation and the boundary conditions.

Using the relation from Eq. (4), the energy equation becomes:

$$u\frac{\partial T}{{\partial x}} + v\frac{\partial T}{{\partial y}} = \frac{{K_{f} }}{{\rho C_{p} }}\frac{{\partial^{2} T}}{{\partial y^{2} }} + \frac{{\left( {\mu + k} \right)}}{{\rho C_{p} }}\left( {\frac{\partial u}{{\partial y}}} \right)^{2} + \frac{{\sigma B_{0}^{2} }}{{\rho C_{p} }}u^{2} + \frac{1}{{\rho C_{p} }}\frac{{16\sigma^{*} T_{\infty }^{3} }}{{3K^{*} }}\frac{{\partial^{2} T}}{{\partial y^{2} }} + \frac{{D_{M} K_{T} }}{{C_{S} C_{P} }}\frac{{\partial^{2} T}}{{\partial y^{2} }}$$

(8)

Non-dimensional parameters reveal that the bounding conditions and similarity transformation cannot be separated.

$$\begin{aligned} K & = \frac{k}{\mu },M = \frac{{\sigma B_{0}^{2} }}{\rho b},K_{P} = \frac{v}{{6K_{p}^{*} }},Gr = \frac{{g\beta_{T} \left( {T_{w} - T_{\infty } } \right)x^{3} }}{{v^{2} }},G_{C} = \frac{{g\beta_{c} \left( {C_{w} - C_{\infty } } \right)x^{3} }}{{v^{2} }}, \\ \lambda & = \frac{{G_{r} }}{{\left( {{\text{Re}}_{x} } \right)^{2} }},\delta = \frac{{G_{c} }}{{\left( {{\text{Re}}_{x} } \right)^{2} }},P_{r} = \frac{{\rho vC_{p} }}{{K_{f} }},E_{c} = \frac{{u_{w}^{2} }}{{C_{p} \left( {T_{w} - T_{\infty } } \right)}},R = \frac{{4\sigma^{*} T_{\infty }^{3} }}{{3K^{*} k_{f} }}, \\ D_{u} & = \frac{{D_{M} K_{T} \left( {C_{w} - C_{\infty } } \right)}}{{vC_{s} C_{p} \left( {T_{w} - T_{\infty } } \right)}},Sc = \frac{v}{D},Kr = \frac{{K_{r}{\prime} }}{b},{\text{Re}}_{x} = \frac{{u_{w} x}}{v},Sr = \frac{{D_{M} K_{T} \left( {T_{w} - T_{\infty } } \right)}}{{vT_{M} \left( {C_{w} - C\infty } \right)}} \\ \end{aligned}$$

(9)

The way to handle boundaries with me:

$$\begin{aligned} & v = 0,u = u_{w} = b_{x} ,T = T_{w} ,N = - s\frac{\partial u}{{\partial y}},C = C_{w} \;at\;y = 0 \\ & N = 0,u = 0,C = C_{\infty } ,T = T_{\infty } \;as\;y \to \infty \\ \end{aligned}$$

(10)

We plan to convert Eqs. (2, 3, 4, 5) into ODE by using the dimensionless variables and similarity transformations described in the next section.

$$\begin{aligned} \eta & = \left( {\sqrt{\frac{b}{v}} } \right)y,\,u = bxf{\prime} \left( \eta \right),v = - \sqrt {bv} f\left( \eta \right),N = \left( {\sqrt {\frac{{b^{3} }}{v}} } \right)xG\left( \eta \right), \\ \varphi \left( \eta \right) & = \frac{{C - C_{\infty } }}{{C_{w} - C_{\infty } }},\;\theta \left( \eta \right) = \frac{{T - T_{\infty } }}{{T_{w} - T_{\infty } }} \\ \end{aligned}$$

(11)

In that case, the enduring equations are shortened:

$$ff^{\prime \prime } - \left( {1 + K} \right)f^{\prime \prime \prime } + \lambda \theta + \left( {f^{\prime } } \right)^{2} + \delta \phi - \left( {M + K_{p} } \right)f^{\prime } = 0$$

(12)

$$fG{\prime} - fG{\prime} + \left( {1 + \frac{K}{2}} \right)G^{^{\prime\prime}} - K\left( {2G + f^{^{\prime\prime}} } \right) = 0$$

(13)

$$\left( {1 + \frac{4}{3}R} \right)\theta^{^{\prime\prime}} + \Pr f\theta{\prime} + \left( {1 + K} \right)\Pr Ecf^{{^{{\prime\prime}{2}} }} + \Pr Du\phi^{^{\prime\prime}} = 0$$

(14)

$$\phi^{^{\prime\prime}} + Scf\phi{\prime} - ScKr\phi + ScSr\theta^{^{\prime\prime}} = 0$$

(15)

The exchanging boundary conditions are

$$\begin{aligned} & f{\prime} \left( \eta \right) = 1,G\left( \eta \right) = - sf^{^{\prime\prime}} \left( \eta \right),\,f\left( \eta \right) = 0,\theta \left( \eta \right) = 1,\,\varphi \left( \eta \right) = 1\;at\;\eta = 0 \\ & G\left( \eta \right) = 0,\,\theta \left( \eta \right) = 0,f{\prime} \left( \eta \right) = 0,\,\varphi \left( \eta \right) = 0\;as\;\eta \to \infty \\ \end{aligned}$$

(16)

The shear stress can be calculated here as

$$\tau_{w} = \left[ {\left( {\mu + k} \right)\left( {\frac{\partial u}{{\partial \zeta }}} \right) + kN} \right]_{\zeta = 0} = \left( {\mu + k} \right)bx\sqrt{\frac{b}{v}} f^{\prime\prime\prime}\left( 0 \right)$$

(17)

Coefficient of skin friction.

\(C_{f} = \frac{{\tau_{w} }}{{\rho u_{w}^{2} }} = \frac{{\left( {1 + K} \right)f^{^{\prime\prime}} \left( 0 \right)}}{{\sqrt {{\text{Re}}_{w} } }}\). Here \({\text{Re}}_{w} = \frac{{u_{w} x}}{v}\) means the local Re. The couple stress is denoted here at the surfaces \(M_{w} = \left( {\gamma \frac{\partial N}{{\partial \zeta }}} \right)_{\zeta = 0} = \mu u_{w} \left( {1 + \frac{k}{2}} \right)G{\prime} \left( 0 \right)\).

The surface of energy flow should be stated in terms of Fourier’s law as trails:

$$q_{x} \left( x \right) = - k_{f} \left( {\frac{\partial N}{{\partial \zeta }}} \right)_{\zeta = 0} = - k_{f} \left( {T_{w} - T_{\infty } } \right)\sqrt{\frac{b}{v}} \theta^{\prime}\left( 0 \right).$$

The formula for influencing the local surface energy flux transfer coefficient is

$$h\left( x \right) = \frac{{q_{w} \left( x \right)}}{{\left( {T_{w} - T_{\infty } } \right)}} = - k_{f} \sqrt{\frac{b}{v}} \theta^{\prime}\left( 0 \right).$$

And the Nusselt number is represented as \(Nu_{x} = \frac{xh\left( x \right)}{{k_{f} }} = - \sqrt{\frac{b}{v}} \,x\theta{\prime} \left( 0 \right)\) and write as \(\frac{{Nu_{x} }}{{\sqrt {{\text{Re}}_{w} } }} = - \theta{\prime} \left( 0 \right)\) Likewise, local solute flux is defined as \(J_{w} = - D\left( {\frac{\partial C}{{\partial \zeta }}} \right)_{\zeta = 0}\) and \(Sh_{x} = \frac{{J_{w} \left( x \right)}}{{D\left( {C_{w} - C_{\infty } } \right)}} = - \sqrt{\frac{b}{v}} x\varphi{\prime} \left( 0 \right)\) presents \(\frac{{Sh_{x} }}{{\sqrt {{\text{Re}}_{w} } }} = - \varphi{\prime} \left( 0 \right)\).

Method of results

Solving boundary value problems (BVPs) using the fourth-order Lobatto IIIa method on Arnoldi’s solver still works for different applications, including tough nonlinear and stiff equations, singular BVPs, and problems with known or unknown parameters. Even though the software hasn’t been updated by the creator, people in the community have found clever methods to make the app better. As an example, by employing dynamic boundary conditions, researchers have improved how models capture changes within the system during computation. Interest has also grown in connecting bvp4c with tools like Chebfun, to make it better at solving difficult equations such as the Fisher-KPP equation. Insights from these projects suggest that combining different approaches can result in faster solutions to some Boundary Value Problems. These additions work best for users who can choose to use special solutions or external tools.

Start with the basic IVP and end up with the boundary conditions (16) and the system of two nonlinear and coupled ODEs (12) to (15). Correctly build the sets.

\(y_{7} = \theta{\prime} ,\,y_{2} = f{\prime} ,\,y_{4} = G,\,y_{3} = f^{^{\prime\prime}} ,y_{1} = f,\,y_{5} = G{\prime} ,\,y_{8} = \varphi ,y_{6} = \theta ,\,\,y_{9} = \varphi{\prime}\) then the nonlinear ODEs are changed as:

$$\begin{gathered} y_{3}{\prime} = \left( {{{\left( { - y_{1} y_{3} + y_{2}^{2} - \lambda y_{6} - \delta y_{8} + (M + K_{p} )y_{2} } \right)} \mathord{\left/ {\vphantom {{\left( { - y_{1} y_{3} + y_{2}^{2} - \lambda y_{6} - \delta y_{8} + (M + K_{p} )y_{2} } \right)} {\left( {1 + K} \right)}}} \right. \kern-0pt} {\left( {1 + K} \right)}}} \right) \hfill \\ y_{5}{\prime} = \left( {{{ - y_{1} y_{5} + y_{2} y_{4} + K(2y_{4} + y_{3} )} \mathord{\left/ {\vphantom {{ - y_{1} y_{5} + y_{2} y_{4} + K(2y_{4} + y_{3} )} {\left( {1 + K/2} \right)}}} \right. \kern-0pt} {\left( {1 + K/2} \right)}}} \right) \hfill \\ y_{7}{\prime} = \left( {{{ - \Pr y_{1} y_{7} - \left( {1 + K} \right)\Pr Ecy_{3}^{2} - \Pr EcMy_{2}^{2} + \Pr DuScy_{1} y_{9} - \Pr DuScKry_{8} } \mathord{\left/ {\vphantom {{ - \Pr y_{1} y_{7} - \left( {1 + K} \right)\Pr Ecy_{3}^{2} - \Pr EcMy_{2}^{2} + \Pr DuScy_{1} y_{9} - \Pr DuScKry_{8} } {1 + }}} \right. \kern-0pt} {1 + }}\frac{4}{3}R - \Pr DuScSr} \right) \hfill \\ y_{9}{\prime} = - Scy_{1} y_{9} + ScKry_{8} - ScSr\left( {{{ - \Pr y_{1} y_{7} - (1 + K)\Pr Ecy_{3}^{2} - \Pr MEcy_{2}^{2} + \Pr DuScy_{1} y_{9} - \Pr DuScKry_{8} } \mathord{\left/ {\vphantom {{ - \Pr y_{1} y_{7} - (1 + K)\Pr Ecy_{3}^{2} - \Pr MEcy_{2}^{2} + \Pr DuScy_{1} y_{9} - \Pr DuScKry_{8} } {\left( {1 + \frac{4}{3}R - \Pr DuScSr} \right)}}} \right. \kern-0pt} {\left( {1 + \frac{4}{3}R - \Pr DuScSr} \right)}}} \right) \hfill \\ \end{gathered}$$

The boundary limitations are

$$\begin{gathered} y_{8} \left( 0 \right) = 1,\,y_{1} \left( 0 \right) = 0,\,y_{4} \left( 0 \right) + sy_{3} \left( 0 \right),y_{2} \left( 0 \right) = 1,y_{6} \left( 0 \right) = 1 \hfill \\ y_{8} \left( \infty \right) = 0,y_{2} \left( \infty \right) = 0,y_{6} \left( \infty \right) = 0,y_{4} \left( \infty \right) = 0 \hfill \\ \end{gathered}$$

figure a

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Results and discussion

This part show how change in number make line in graph up or down for speed, spin, hot, and mix in layer. Solve in MATLAB BVP4C and get line and table for skin friction, Nusselt, Sherwood. Magnetic M make speed small, spin more near wall, and hot more because electric hot. Micro N change spin and speed near sheet. Light heat R make hot layer thick and more hot in flow. Soret Sr make mix from hot change, Dufour Du make hot from mix change and less near wall. Chemical Kr eat mix near sheet. Ohmic hot from M and σ put more hot inside, viscous warm Ec make more hot from speed change. When hole in medium big (φ), speed go small, hot and mix go more. Permeability K change how easy flow go in hole. All show in figure line go up down, use in filter, oil take, heat machine, and many place in work.

Physical mechanisms

According to Fig. 2, when M increases, the velocity of the fluid falls. The electrically conducting fluid in plasma is affected by Lorentz force, a back-up force caused by resistive effects in the magnetic field. As a result, a thinner boundary layer forms with a greater M, showing that the magnetic field improves the accuracy of cooling the fluid. The impact of the M on microrotation or the spin of fluid particles in micropolar fluids is shown in Fig. 3. Enhanced M increases microrotation as the B0 makes the fluid’s particles mix better and transmit heat. In the boundary region shown in Fig. 4, we see a rise in temperature because of Joule heating, caused by the fluid current set up by the B0. When the viscous part of the process meets the effects of the magnetic field, it results in a rise in temperature. From Fig. 5 that high magnetic fields help particles get stuck near the boundary layer. So, making the magnetic field stronger could prevent up-flowing matter from dispersion while filtration processes are going on.

Fig. 2
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Effects of \(M\) on f’(η).

Fig. 3
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Possessions of \(M\) on Microrotation outline.

Fig. 4
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Possessions of \(M\) on Temperature contour.

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Possessions of \(M\) on Concentration profile.

Through Figs. 6, 7, 8, and 9, we can see that the higher φ becomes, the less steep the slope in the boundary layer velocities. The reduction in flow velocity occurs because the boundary layer fluid enters the porous medium through its pores. By contrast, higher porosity is associated with a rise in both temperature and concentration. Such properties are important for applications in heat exchangers and catalytic processes since more heat and particles are contained within the region close to the surface. The next four figures examine what occurs when the micropolar parameter is considered. With an increased micropolar parameter, both viscosity and microrotation near the boundary decrease. When the micropolar parameter rises, the temperature profile decreases, suggesting that micropolar fluids have lower thermal conductivity than ordinary fluids. It is found that increasing the micropolar parameters leads to a slightly reduced concentration.

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Effects of \(K_{p}\) over Velocity.

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Effects of \(K_{p}\) over Microrotation outline.

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Effects of \(K_{p}\) on θ(η).

Fig. 9
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Effects of \(K_{p}\) on ϕ(η).

Figure 10 shows the way the permeability of the porous medium changes the speed profile of the micropolar fluid. Additional fluid passing through the porous matrix increases permeability, so the boundary layer may thin and the velocity gradient near the sheet may differ from before. How freely the fluid seeps into the medium are called the mechanism that determines the total resistance to flow. This matters greatly in EOR operations and filtering, where lowering permeability achieves the best results in injecting or filtering fluids. Figure 11 illustrates different speeds of microrotation for the Soret Effect case. When the Soret number rises, the angular motion of fluid within the microstructure indicates that thermal diffusion has a growing effect on the fluid’s structure. Due to temperature differences, the Soret effect affects mass flux, which spreads microrotation unevenly within the boundary layer. Its role is important in fluids that contain particles, such as colloidal suspensions or biological fluids, since moving particles due to temperature change the overall properties. In Fig. 12, we see how the Soret Effect affects the temperature in the fluid. It is seen that as the Soret number rises, the temperature gradient in the boundary layer becomes smaller. As a result, increased thermal diffusion makes warmer fluids more likely to transfer outward, which thin out the boundary layer where heat is transferred. Such a phenomenon matters in chemical reactors and separation processes, as the goal is to have similar temperatures and concentrations for better results and safety. Figure 13 shows the change in concentration profile due to the Soret effect. The local density of solute moves away from the hot zone as the rate of thermal diffusion rises.

Fig. 10
Fig. 10

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Effects of \(K\) on f’(η).

Fig. 11
Fig. 11

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Possessions of \(K\) on Microrotation profile.

Fig. 12
Fig. 12

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Effects of \(K\) over a Temperature outline.

Fig. 13
Fig. 13

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Effects of \(K\) on Concentration.

In environmental engineering, this method is useful for controlling contaminants in soil or where pollutants might be dispersed, by using temperature gradients to move them between pores. Figure 14 explains that by increasing the R, we see that the fluid takes in more heat due to a thicker thermal boundary layer. It can be detected in Fig. 15 that boosting Ec raises the fluid temperature, meaning that friction creates more heating inside the system. The figures demonstrate that when Sr is higher, the velocity and concentration profiles show Barenblatt-Poiseuille profiles, meaning mass diffuses because of temperature gradients, causing thermo-diffusion. The Dufour number is shown in Fig. 16 to affect the flow velocity of the liquid. Because the Dufour effect relates concentration gradients to heat flux, it is responsible for changes in heat energy as caused by variation in solute concentration which in turn changes the motion of fluids. An augmented Dufour number generally enhances the exchange of energy which can change the speed of the fluids in the boundary layer. It matters in multi-component heat exchangers and energy systems where performance depends on coordinating heat and mass transfer. There is an impact of the Dufour Number on the degree of concentration (Fig. 17). Figure 17 demonstrates that increasing the Dufour number shifts the energy profile and usually causes a decline in local concentration close to the surface. A strengthened Dufour effect results in more contribution from concentration gradients to heat which may thin the layer of solute near the surface and distribute the solute within the fluid. In industrial systems for drying, desalination and making chemicals, accurate transfer of heat and mass is essential to meet the conditions for the final products or good results. Figure 18 demonstrates that the chemical reaction parameter which stands for higher reaction rates, leads to a reduction in concentration near the boundary layer, since the reactants are being burned by the reaction. Using these insights, we understand how several factors affect micropolar fluid flows, making it easier to take advantage of ideal heat and mass transfer when dealing with complex systems.

Fig. 14
Fig. 14

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Effects of \(R\) on θ (η).

Fig. 15
Fig. 15

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Effects of \(Ec\) on θ (η).

Fig. 16
Fig. 16

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Possessions of \(Sr\) on f’ (η).

Fig. 17
Fig. 17

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Influence of \(Sr\) on ϕ (η).

Fig. 18
Fig. 18

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Effects of \(Kr\) on f’ (η).

Physical interpretations

Magnetic Parameter (M) A rise in M generates Lorentz force, which suppresses the primary flow, reduces velocity, and increases fluid temperature due to Ohmic heating (Figs. 2 and 4).

Micropolar Parameter (N) Increased N enhances microrotation near the surface but lowers it away from the wall (Fig. 3), suggesting more rotational activity close to boundaries.

Radiation (R) Elevated R values lead to a thicker thermal boundary layer and increased fluid temperature due to radiative heat gain (Fig. 14).

Soret Number (Sr) As Sr increases, both concentration and microrotation profiles are affected due to stronger thermal diffusion effects (Figs. 11 and 13).

Dufour Number (Du) Higher Du increases temperature profiles and lowers concentration near the wall due to diffusion-thermo interaction (Figs. 16 and 17).

Chemical Reaction (Kr) A rise in the reaction rate reduces the concentration boundary layer thickness, indicating faster consumption of solute species (Fig 18).

Dgdg Table 1 comparison of Nusselt number \(- \theta^{\prime}\left( 0 \right)\) for different values of \(\Pr\) with \(M = K_{p} = \lambda = \delta = K = Ec = Du = Sr = s = 0\).

Table 1 Comparison of Nusselt number.

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Furthermore, Table 2 shows the result of skin friction, couple stress, Nusselt and Sherwood numbers for different values of \(M\),\(K\) and \(s\) with another influence are \(\Pr = 0.733\),\(Sc = 0.5\),\(Kr = 0.5\), \(Ec = 0.22\).

Table 2 Skin friction, couple stress, Nusselt and Sherwood numbers for different values.

Full size table

Conclusions

This study provides a detailed analysis of MHD micropolar fluid flow over a stretching sheet within a porous medium, considering Soret and Dufour effects along with Ohmic heating, viscous dissipation, radiation, and chemical reaction impacts. By applying similarity transformations and using MATLAB’s BVP4C solver, the influence of various physical parameters on velocity, microrotation, temperature, and concentration profiles was examined systematically.

The inclusion of Ohmic heating demonstrated its strong influence in raising temperature levels within the thermal boundary layer, while viscous dissipation contributed additional internal heat, especially under high shear or viscous flow conditions. These effects are critical in applications involving electrically conducting fluids and high thermal gradients. The results showed that increasing the magnetic parameter suppresses fluid velocity due to the Lorentz force, whereas radiation and micropolar effects enhance energy distribution and rotational behavior near the wall. Soret and Dufour effects altered the coupling between heat and mass transport, highlighting their importance in non-equilibrium transport processes.

It would be valuable for future researchers to experience the Soret and Dufour effects experimentally in real porous materials such as those with changing textures and non-Newtonian fluids. In addition, studying the influence of electromagnetic fields and oscillatory flows on heat and mass transport in micropolar fluids may improve the use of these models in areas like microwave cooling and drug administration (e.g., controlled drug delivery). In addition, developing CFD simulations for multi-phase micropolar flows in porous media could increase our understanding of advanced materials and energy fields.

Data availability

The authors declare that the data supporting this study’s findings are available within the paper. More information is available from the corresponding author upon reasonable request.

Change history

Abbreviations

x,y:

Cartesian coordinates

u,v:

Velocity componentsalong x and y directions

T:

Temperature

T∞ :

Ambient/reference temperature

C:

Concentration

C∞ :

Ambient/reference concentration

ρ:

Fluid density

ԛr :

Radiative heat flex

DT :

Thermodiffusion (Soret effect) coefficient

k* :

Rosseland mean absorption coefficient

σ* :

Stefan Boltzmann constant

Pr :

Prandtl number

SC :

Schmidt number

N:

Micropolar parameter

Gr:

Thermal Grashof number

Gc:

Solutal Grashof number

K:

Permeability of Porosity parameter

Sr:

Soret number

Ec:

Eckart number

Du:

Dufour number

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Acknowledgements

This study is supported via funding from the Prince Sattam bin Abdulaziz University project number (PSAU/2025/R/1447).

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Authors and Affiliations

  1. Department of Mathematics, Thiruvalluvar University, Vellore, Tamil Nadu, 632 115, India

    Shakila Ahamedsheriff & Vijayaragavan Rajaram

  2. Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai, Tamil Nadu, 600062, India

    Gunaseelan Mani

  3. Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai, Tamil Nadu, 602 105, India

    Sabri T. M. Thabet

  4. Department of Mathematics, Radfan University College, University of Lahej, Lahej, Yemen

    Sabri T. M. Thabet

  5. Department of Mathematics, College of Science, Korea University, 145 Anam-Ro, Seongbuk-Gu, Seoul, 02814, Republic of Korea

    Sabri T. M. Thabet

  6. Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, 11942, Al-Kharj, Saudi Arabia

    Imed Kedim

Authors

  1. Shakila Ahamedsheriff
  2. Vijayaragavan Rajaram
  3. Gunaseelan Mani
  4. Sabri T. M. Thabet
  5. Imed Kedim

Contributions

Every author contributed equally to each part of the paper.

Corresponding author

Correspondence to Sabri T. M. Thabet.

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The authors declare no competing interests.

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This article has been retracted. Please see the retraction notice for more detail: https://doi.org/10.1038/s41598-026-48538-8

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Ahamedsheriff, S., Rajaram, V., Mani, G. et al. RETRACTED ARTICLE: Impacts of soret and dufour possessions on micropolar fluid past a stretching sheet in a porous medium. Sci Rep 15, 33059 (2025). https://doi.org/10.1038/s41598-025-18302-5

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