Heat and Mass Transfer Effect of Micropolar Nanofluid Flow over a Horizontal Moving Plate with Non-Uniform Heat Source/Sink

Abstract

This study investigates the stagnation-point flow and heat transfer in mixed convection of a micropolar fluid over an exponentially expanding vertical sheet, including the effects of buoyancy, magnetic fields, radiation, and viscous dissipation. The governing partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) using similarity transformations and solved with the bvp4c method. The results show that increasing buoyancy, magnetic field, radiation, and viscous dissipation enhance the velocity profile near the surface. Buoyancy suppresses particle rotation, while magnetic fields enhance it, with micropolar fluids reversing this trend under higher radiation. Temperature decreases with stronger magnetic fields and buoyancy, indicating improved heat dissipation. Differences in fluid behaviour between micropolar and Newtonian fluids highlight the unique characteristics of micropolarity. These findings are relevant for applications such as polymer extrusion, thin-film manufacturing, microfluidic devices, lubrication systems, and cooling technologies, where precise control of flow and heat transfer is essential.

Introduction

This study investigates the micropolar fluid flow and heat transfer over exponentially stretching/shrinking vertical surfaces, which are relevant in industrial applications such as polymer extrusion, thin-film manufacturing, and cooling systems.

Background & Literature Review

Research on Newtonian and non-Newtonian boundary layer flow has advanced significantly, driven by applications in manufacturing processes involving stretching surfaces. Numerous studies have analyzed the effects of heat sources, magnetic fields, viscous dissipation, radiation, and micropolar effects in these contexts. Notably:

• Sakiadis and Crane laid the foundation for boundary-layer flow on moving and stretching surfaces.

• Subsequent works explored MHD flows, micropolar fluids, nanofluids, activation energy effects, and hybrid thermal systems.

• Recent studies address complex fluid behavior, emphasizing entropy generation, heat and mass transfer, and flow optimization.

Problem Definition

The study examines micropolar fluid flow toward a stagnation point on an exponentially stretching/shrinking sheet, subject to:

• External magnetic fields

• Thermal radiation

• Buoyancy forces

• Viscous and ohmic dissipation

Governing Equations

The system is governed by parabolic PDEs for:

1. Continuity

2. Linear momentum

3. Angular (rotational) momentum

4. Energy (heat)

These are reduced via similarity transformations into a set of ODEs involving:

• Velocity f(η)

• Microrotation h(η)

• Temperature θ(η)

Key dimensionless parameters include:

• Magnetic field strength (M)

• Micropolar parameter (K)

• Buoyancy (λ)

• Radiation (R)

• Eckert number (Ec)

• Prandtl number (Pr)

Numerical Method

The bvp4c solver in MATLAB is used to solve the resulting boundary value problem (BVP), employing shooting methods and adaptive mesh refinement to ensure accuracy. The equations are converted into first-order ODEs for computational efficiency.

Validation

Numerical results show excellent agreement with previous studies (e.g., Waini et al., Bakar & Soid), confirming the method’s reliability. Tables I & II validate the skin friction coefficient and local Nusselt number for various stretching/shrinking parameters.

Results & Discussion

The simulation results reveal how key parameters affect flow and heat transfer:

• Stretching rate (ε): Increases heat transfer (Nusselt number) but decreases skin friction.

• Magnetic parameter (M): Inhibits velocity but increases temperature due to Lorentz force.

• Micropolar parameter (K): Affects both angular velocity and shear stress, showing the impact of fluid microstructure.

• Radiation parameter (R) and buoyancy (λ): Enhance temperature profiles and convective effects.

• Eckert number (Ec): Higher viscous dissipation leads to more thermal energy generation.

Conclusion

This investigation focuses on the thermal study and magnetohydrodynamic (MHD) micropolar fluid of stagnation-point flow across a vertically aligned surface undergoing exponential stretching or shrinking, incorporating the effects of buoyancy. The analysis confirmed the existence of solutions under both stretching and shrinking scenarios. The study further elaborates on the influence of magnetic forces, buoyancy-induced flow, thermal radiation, energy dissipation due to viscosity, and the rotational characteristics of micropolar fluids over the exponentially varying surface. From this comprehensive analysis, the study has led to the derivation of several important findings. The velocity declines with increasing micropolar effects due to rotational inertia, the magnetic fields suppress the velocity through Lorentz forces, but increase under the consequence of magnetic fields, radiation and buoyancy forces enhancing the fluid’s motion. The angular velocity decreases with both buoyancy forces and magnetic fields, showing that these parameters suppress micro-rotational effects within the fluid. The temperature drops with growing magnetic fields and buoyancy forces due to enhanced convective heat transfer but increases with higher micropolar effects, which retain heat. Radiation enhances heat dissipation, lowering the temperature near the surface. Higher Prandtl numbers accelerate heat dissipation, resultant in a supplementary rapid decline in temperature within boundary layer, while higher Eckert numbers increase the temperature due to viscous dissipation. As the fluid temperature rises because of increased viscous dissipation, the thermal boundary layer next to the surface expands. Buoyancy forces promote velocity and improve heat transfer by enhancing fluid motion by providing additional control over the flow. Radiation and micropolar parameters influence both energy transfer and rotational behaviour, with radiation enhancing heat transfer and micropolarity slowing down fluid motion and retaining heat. It is evident from the findings that higher values of the heat source/sink parameters A^* and B^* contribute to an increase in the thermal energy of the fluid. In contrast, negative values enhance the rate of temperature reduction. Therefore, these parameters are crucial in regulating the thermal behaviour across the boundary layer.

References

[1] Sakiadis, B. C. (1961). Boundary?layer behaviour on continuous solid surfaces: I. Boundary?layer equations for two?dimensional and axisymmetric flow. AIChE Journal, 7(1), 26-28. [2] Sikiadis, B. C. (1961). Boundary-Layer behaviour on continuous solid surfaces. AI Ch. EJ, 7, 26-28. [3] Crane, L. J. (1970). Flow past a stretching plate. Zeitschrift für angewandte Mathematik und Physik ZAMP, 21, 645-647. [4] Gupta, P. S., & Gupta, A. S. (1977). Heat and mass transfer on a stretching sheet with suction or blowing. The Canadian journal of chemical engineering, 55(6), 744-746. [5] Sandeep, N., & Sulochana, C. (2015). Dual solutions for unsteady mixed convection flow of MHD micropolar fluid over a stretching/shrinking sheet with non-uniform heat source/sink. Engineering Science and Technology, an International Journal, 18(4), 738-745. [6] Prasannakumara, B. C., Gireesha, B. J., & Manjunatha, P. T. (2015). Melting phenomenon in MHD stagnation point flow of dusty fluid over a stretching sheet in the presence of thermal radiation and non-uniform heat source/sink. International Journal for Computational Methods in Engineering Science and Mechanics, 16(5), 265-274. [7] Mabood, F., Ibrahim, S. M., Rashidi, M. M., Shadloo, M. S., & Lorenzini, G. (2016). Non-uniform heat source/sink and Soret effects on MHD non-Darcian convective flow past a stretching sheet in a micropolar fluid with radiation. International Journal of Heat and Mass Transfer, 93, 674-682. [8] Seth, G. S., Sharma, R., Kumbhakar, B., & Tripathi, R. (2017). MHD stagnation point flow over exponentially stretching sheet with exponentially moving free-stream, viscous dissipation, thermal radiation and non-uniform heat source/sink. In Diffusion Foundations (Vol. 11, pp. 182-190). Trans Tech Publications Ltd. [9] Ram, M. S., Shamshuddin, M. D., & Spandana, K. (2021). Numerical simulation of stagnation point flow in magneto micropolar fluid over a stretchable surface under influence of activation energy and bilateral reaction. International Communications in Heat and Mass Transfer, 129, 105679. [10] Murugan, R. D., Sivakumar, N., Tarakaramu, N., Sarhan, N., & Awwad, E. M. (2024). Mixed convection hybrid nanofluid flow over a rotating cone in a rotating fluid environment with interfacial nanolayer effect. Numerical Heat Transfer, Part B: Fundamentals, 1-20. [11] Murugan, R. D., Sivakumar, N., Tarakaramu, N., Ahmad, H., & Askar, S. (2024). Entropy generation on MHD motion of hybrid nanofluid with porous medium in presence of thermo-radiation and ohmic viscous dissipation. Discover Applied Sciences, 6(4), 199. [12] Murugan, R. D., Sivakumar, N., Tarakaramu, N., Alhazmi, H., & Abdullaev, S. (2024). Entropy and energy transfer analysis of a Maxwell thin?film fluid over an inclined surface with viscous dissipation effect. ZAMM?Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, e202300381. [13] Tarakaramu, N., Sivakumar, N., Tamam, N., Satya Narayana, P. V., & Ramalingam, S. (2024). Theoretical analysis of Arrhenius activation energy on 3D MHD nanofluid flow with convective boundary condition. Modern Physics Letters B, 38(16), 2341009. [14] Tarakaramu, N., Reddappa, B., Radha, G., Abduvalieva, D., Sivakumar, N., Awwad, F. A., ... & Reddy, K. A. (2023). Thermal radiation and heat generation on three-dimensional Casson fluid motion via porous stretching surface with variable thermal conductivity. Open Physics, 21(1), 20230137. [15] Masthanaiah, Y., Tarakaramu, N., Khan, M. I., Rushikesava, A., Moussa, S. B., Fadhl, B. M., ... & Eldin, S. M. (2023). Impact of viscous dissipation and entropy generation on cold liquid via channel with porous medium by analytical analysis. Case Studies in Thermal Engineering, 47, 103059. [16] Jagadeesh, S., Chenna Krishna Reddy, M., Tarakaramu, N., Ahmad, H., Askar, S., & Shukhratovich Abdullaev, S. (2023). Convective heat and mass transfer rate on 3D Williamson nanofluid flow via linear stretching sheet with thermal radiation and heat absorption. Scientific Reports, 13(1), 9889. [17] Reddy, A. B., & Kumar, R. Y. R. (2022, December). Performance and security analysis in cloud using drops and T-coloring methods. In 2022 Fourth International Conference on Emerging Research in Electronics, Computer Science and Technology (ICERECT) (pp. 1-7). IEEE. [18] Li, S., Tarakaramu, N., Khan, M. I., Sivakumar, N., Satya Narayana, P. V., Abdullaev, S., ... & Eldin, S. M. (2024). Enhanced heat transfer and fluid motion in 3D nanofluid with anisotropic slip and magnetic field. Open Physics, 22(1), 20230131. [19] Sedki, A. M., & Qahiti, R. (2023). Unsteady magnetohydrodynamic radiative Casson nanofluid within chemically reactive flow over a stretchable surface with variable thickness through a porous medium. Energies, 16(23), 7776. [20] Rehman, Fiaz Ur, Sohail Nadeem, Hafeez Ur Rehman, and Rizwan Ul Haq. \"Thermophysical analysis for three-dimensional MHD stagnation-point flow of nano-material influenced by an exponential stretching surface.\" Results in physics 8 (2018): 316-323. https://doi.org/10.1016/j.rinp.2017.12.026 [21] Bejawada, S. G., & Nandeppanavar, M. M. (2023). Effect of thermal radiation on magnetohydrodynamics heat transfer micropolar fluid flow over a vertical moving porous plate. Experimental and Computational Multiphase Flow, 5(2), 149-158. [22] Bakar, F. N. A., & Soid, S. K. (2023). MHD Stagnation-Point Flow and Heat Transfer in a Micropolar Fluid over an Exponentially Vertical Sheet. CFD Letters, 15(3), 81-96. [23] Khan, M. N., Ahmad, S., Wang, Z., Ahammad, N. A., & Elkotb, M. A. (2023). Bioconvective surface-catalyzed Casson hybrid nanofluid flow analysis by using thermodynamics heat transfer law on a vertical cone. Tribology International, 188, 108859. [24] Khan, M. N., Aldosari, F. M., Wang, Z., Yasir, M., Afikuzzaman, M., & Elseesy, I. E. (2024). Overview of solar thermal applications of heat exchangers with thermophysical features of hybrid nanomaterials. Nanoscale Advances, 6(1), 136-145. [25] Khan, M. N., Ahmad, S., Wang, Z., Hussien, M., Alhuthali, A. M., & Ghazwani, H. A. (2024). Flow and heat transfer insights into a chemically reactive micropolar Williamson ternary hybrid nanofluid with cross-diffusion theory. Nanotechnology Reviews, 13(1), 20240081. [26] Khan, M. N., Wang, Z., Ahammad, N. A., Rezapour, S., Shutaywi, M., Ali, N. B., & Elkotb, M. A. (2024). Mixed convective flow analysis of a Maxwell fluid with double diffusion theory on a vertically exponentially stretching surface. Applied Water Science, 14(8), 172. [27] Naveed Khan, M., Alhowaity, A., Wang, Z., Gepreel, K. A., & Hussien, M. (2024). Numerical analysis of the heat transfer application on a convective tangent hyperbolic nanofluid flow over a porous stretching cylinder with stratification effects. Numerical Heat Transfer, Part A: Applications, 1-17. [28] Waini, Iskandar, Anuar Ishak, and Ioan Pop. \"Hybrid nanofluid flow towards a stagnation point on an exponentially stretching/shrinking vertical sheet with buoyancy effects.\" International Journal of Numerical Methods for Heat & Fluid Flow 31, no. 1 (2020): 216-235. [29] Bakar, F. N. A., & Soid, S. K. (2023). MHD Stagnation-Point Flow and Heat Transfer in a Micropolar Fluid over an Exponentially Vertical Sheet. CFD Letters, 15(3), 81-96. [30] Yasir, M., Khan, M., & Malik, Z. U. (2023). Analysis of thermophoretic particle deposition with Soret-Dufour in a flow of fluid exhibit relaxation/retardation times effect. International Communications in Heat and Mass Transfer, 141, 106577. [31] Khan, M. N., Haider, J. A., Wang, Z., Gul, S., Lone, S. A., & Elkotb, M. A. (2024). Mathematical modelling of the partial differential equations in microelectromechanical systems (MEMS) and its applications. Modern Physics Letters B, 38(05), 2350207. [32] Yasir, M., Bilal, S., Ahammad, N. A., & Elseesy, I. E. (2024). Thermal irregular generation and absorption of nanoscale energy transportation of thermodynamic material of a micropolar fluid. Ain Shams Engineering Journal, 15(9), 102948. [33] Yasir, M., Khan, M., Al-Zubaidi, A., & Saleem, S. (2023). Arrhenius activation energy effect in thermally viscous dissipative flow of micropolar material with gyrotactic microorganisms. Alexandria Engineering Journal, 84, 204-214.

Copyright

Copyright © 2025 Machindranath Diwate, Jagadish V. Tawade, Pradeep G Janthe, Nitiraj V. Kulkarni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.