Exploring assisting and opposing flow characteristics in a saturated darcy medium with non-uniform heat source or sink and suspended microbes

Khalid Masood; Khalid Masood

doi:10.3934/math.2025666

AIMS Mathematics

2025, Volume 10, Issue 6: 14804-14839. doi: 10.3934/math.2025666

Previous Article Next Article

Research article Special Issues

Exploring assisting and opposing flow characteristics in a saturated darcy medium with non-uniform heat source or sink and suspended microbes

Khalid Masood ^,

Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia

Received: 26 April 2025 Revised: 02 June 2025 Accepted: 16 June 2025 Published: 27 June 2025

Aim of the study
In this study, I investigated buoyancy-driven convection involving suspended microorganisms in a saturated Darcy porous medium, considering the effects of thermal radiation, Brownian motion, thermophoresis, and non-uniform heat source/sink. I focused on slender geometries, specifically a cone and a cylinder, under both assisting and opposing flow conditions, to better understand their impact on heat and mass transfer behavior.

Significance and novelty:
Bio-convection has wide-ranging applications in drug delivery, cancer therapy, electronics cooling, microbial fuel cells, water purification, and cosmetic formulations. Unlike other researchers, I uniquely integrated the effects of non-uniform thermal conditions, gyrotactic microorganism dynamics, and slender body geometry. The inclusion of a multilinear regression model added an analytical layer for interpreting parameter sensitivity, which enhanced the model's utility in engineering design.

Methodology
The governing nonlinear partial differential equations were transformed into a system of ordinary differential equations using similarity transformations. These equations were solved numerically using the fourth-order Runge-Kutta method coupled with a shooting technique. To further analyze parametric trends, multilinear regression was applied to the simulation data.

Results and insights
The presence of gyrotactic microorganisms significantly enhanced mass transfer rates compared to microbe-free scenarios. An increase in thermophoresis led to higher temperature distributions but reduced both concentration and microorganism density. Fluid velocity increased under assisting flows due to stronger buoyancy forces. These trends were demonstrated through detailed stream function and temperature plots and numerical findings for important flow and heat transfer parameters in tabular form for the flow dynamics.

Conclusion
This study provides meaningful insights into the role of microorganisms in enhancing mass and heat transfer in porous media. The model serves as a strong foundation for future extensions involving experimental validation, time-dependent simulations, turbulent bio-convection, and more complex geometrical configurations for industrial and biomedical applications.
- micro-organisms,
- multi-linear regression (MLR),
- non-uniform heat source or sink,
- Thermal radiation,
- porous channel,
- buoyancy,
- slender body of revolution
Citation: Khalid Masood. Exploring assisting and opposing flow characteristics in a saturated darcy medium with non-uniform heat source or sink and suspended microbes[J]. AIMS Mathematics, 2025, 10(6): 14804-14839. doi: 10.3934/math.2025666

Related Papers:

Abstract

Aim of the study

In this study, I investigated buoyancy-driven convection involving suspended microorganisms in a saturated Darcy porous medium, considering the effects of thermal radiation, Brownian motion, thermophoresis, and non-uniform heat source/sink. I focused on slender geometries, specifically a cone and a cylinder, under both assisting and opposing flow conditions, to better understand their impact on heat and mass transfer behavior.

Significance and novelty:

Bio-convection has wide-ranging applications in drug delivery, cancer therapy, electronics cooling, microbial fuel cells, water purification, and cosmetic formulations. Unlike other researchers, I uniquely integrated the effects of non-uniform thermal conditions, gyrotactic microorganism dynamics, and slender body geometry. The inclusion of a multilinear regression model added an analytical layer for interpreting parameter sensitivity, which enhanced the model's utility in engineering design.

Methodology

The governing nonlinear partial differential equations were transformed into a system of ordinary differential equations using similarity transformations. These equations were solved numerically using the fourth-order Runge-Kutta method coupled with a shooting technique. To further analyze parametric trends, multilinear regression was applied to the simulation data.

Results and insights

The presence of gyrotactic microorganisms significantly enhanced mass transfer rates compared to microbe-free scenarios. An increase in thermophoresis led to higher temperature distributions but reduced both concentration and microorganism density. Fluid velocity increased under assisting flows due to stronger buoyancy forces. These trends were demonstrated through detailed stream function and temperature plots and numerical findings for important flow and heat transfer parameters in tabular form for the flow dynamics.

Conclusion

This study provides meaningful insights into the role of microorganisms in enhancing mass and heat transfer in porous media. The model serves as a strong foundation for future extensions involving experimental validation, time-dependent simulations, turbulent bio-convection, and more complex geometrical configurations for industrial and biomedical applications.

References

[1]	A. V. Kuznetsov, A. A. Avramenko, Effect of Small Particles on the Stability of Bioconvection in a Suspension of Gyrotactic Microorganisms in a Layer of Finite Depth, Int. Commun. Heat Mass, 31 (2004), 1–10. https://doi.org/10.1016/S0735-1933(03)00196-9 doi: 10.1016/S0735-1933(03)00196-9
[2]	P. Geng, A. V. Kuznetsov, Effect of Small Solid Particles on the Development of Bioconvection Plumes, Int. Commun. Heat Mass, 31 (2004), 629–638. https://doi.org/10.1016/S0735-1933(04)00050-8 doi: 10.1016/S0735-1933(04)00050-8
[3]	P. Geng, A. V. Kuznetsov, Settling of Bidispersed Small Solid Particles in a Dilute Suspension Containing Gyrotactic Micro-Organisms, Int. J. Eng. Sci., 43 (2005), 992–1010. https://doi.org/10.1016/j.ijengsci.2005.03.002 doi: 10.1016/j.ijengsci.2005.03.002
[4]	P. Geng, A. V. Kuznetsov, Introducing the Concept of Effective Diffusivity to Evaluate the Effect of Bioconvection on Small Solid Particles, Int. J. Transp. Phenom., 7 (2005), 321–338.
[5]	A. V. Kuznetsov, The Onset of Bioconvection in a Suspension of Gyrotactic Microorganisms in a Fluid Layer of Finite Depth Heated from Below, Int. Commun. Heat Mass, 32 (2005), 574–582. https://doi.org/10.1016/j.icheatmasstransfer.2004.10.021 doi: 10.1016/j.icheatmasstransfer.2004.10.021
[6]	M. V. S. Rao, K. Gangadhar, A. J. Chamkha, et al., Bioconvection in a Convectional Nanofluid Flow Containing Gyrotactic Microorganisms over an Isothermal Vertical Cone Embedded in a Porous Surface with Chemical Reactive Species, Arab. J. Sci. Eng., 46 (2021), 2493–2503. https://doi.org/10.1007/s13369-020-05132-y doi: 10.1007/s13369-020-05132-y
[7]	A. V. Kuznetsov, Thermo-Bioconvection in a Suspension of Oxytactic Bacteria, Int. Commun. Heat Mass, 32 (2005), 991–999. https://doi.org/10.1016/j.icheatmasstransfer.2004.11.005 doi: 10.1016/j.icheatmasstransfer.2004.11.005
[8]	A. V. Kuznetsov, Investigation of the Onset of Thermo-Bioconvection in a Suspension of Oxytactic Microorganisms in a Shallow Fluid Layer Heated from Below, Theor. Comput. Fluid Dyn., 19 (2005), 287–299. https://doi.org/10.1007/s00162-005-0167-3 doi: 10.1007/s00162-005-0167-3
[9]	A. V. Kuznetsov, The Onset of Thermo-Bioconvection in a Shallow Fluid Saturated Porous Layer Heated from Below in a Suspension of Oxytactic Microorganisms, Eur. J. Mech. B/Fluids, 25 (2006), 223–233. https://doi.org/10.1016/j.euromechflu.2005.06.003 doi: 10.1016/j.euromechflu.2005.06.003
[10]	A. V. Kuznetsov, Non-Oscillatory and Oscillatory Nanofluid Biothermal Convection in a Horizontal Layer of Finite Depth, Eur. J Mech. B/ Fluids, 30 (2011), 156–165. https://doi.org/10.1016/j.euromechflu.2010.10.007 doi: 10.1016/j.euromechflu.2010.10.007
[11]	A. V. Kuznetsov, Nanofluid Bioconvection in Water-Based Suspensions Containing Nanoparticles and Oxytactic microorganisms: Oscillatory instability, Nanoscale Res. Lett., 6 (2011), 100. https://doi.org/10.1186/1556-276X-6-100 doi: 10.1186/1556-276X-6-100
[12]	A. V. Kuznetsov, Bio-Thermal Convection Induced by Two Different Species of Microorganisms, Int. Commun. Heat Mass, 38 (2011), 548–553. https://doi.org/10.1016/j.icheatmasstransfer.2011.02.006 doi: 10.1016/j.icheatmasstransfer.2011.02.006
[13]	S. Nadeem, S. Saleem, Analytical treatment of unsteady mixed convection MHD flow on a rotating cone in a rotating frame, J. Taiwan Inst., Chem. Eng., 44 (2013), 596–604. https://doi.org/10.1016/j.jtice.2013.01.007 doi: 10.1016/j.jtice.2013.01.007
[14]	L. Tham, R. Nazar, I. Pop, Steady mixed convection flow on a horizontal circular cylinder embedded in a porous medium filled by a nano fluid containing gyro tactic micro-organisms, J. Heat Transfer., 135 (2013), 102601. https://doi.org/10.1115/1.4024387 doi: 10.1115/1.4024387
[15]	K. Zaimi, A. Ishak, I. Pop, Stagnation point flow toward a stretching/shrinking sheet in a nanofluid containing both nano particles and gyrotactic micro-organisms, J. Heat Transfer., 136 (2014), 041705. https://doi.org/10.1115/1.4026011 doi: 10.1115/1.4026011
[16]	C. S. K. Raju, N. Sandeep, Dual solutions for unsteady heat and mass transfer in Bio-convection flow towards a rotating cone/plate in a rotating fluid, Int. J. Eng. Res. Afr., 20 (2016), 161–176. https://doi.org/10.4028/www.scientific.net/JERA.20.161 doi: 10.4028/www.scientific.net/JERA.20.161
[17]	W. J. Minkowycx, P. Cheng, Free convection about a vertical cylinder embedded in a porous medium, Int. J. Heat Mass, 19 (1976), 805–813. https://doi.org/10.1016/0017-9310(76)90135-6 doi: 10.1016/0017-9310(76)90135-6
[18]	P. Cheng, W. J. Minkowycz, Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike, J. Geophys. Res, 82 (1977), 2040–2044. https://doi.org/10.1029/JB082i014p02040 doi: 10.1029/JB082i014p02040
[19]	M. Kumari, I. Pop, G. Nath, Finite-difference and improved perturbation solutions for free convection on a vertical cylinder embedded in a saturated porous medium, Int. J. Heat Mass, 28 (1985), 2171–2174. https://doi.org/10.1016/0017-9310(85)90112-7 doi: 10.1016/0017-9310(85)90112-7
[20]	J. H. Merkin, Free convection from a vertical cylinder embedded in a saturated porous medium, Acta Mechanica, 62 (1986), 19–28. https://doi.org/10.1007/bf01175850 doi: 10.1007/bf01175850
[21]	J. H. Merkin, I. Pop, Mixed convection boundarylayer on a vertical cylinder embedded in a saturated porous medium, Acta Mechanica, 66 (1987), 251–262. https://doi.org/10.1007/BF01184297 doi: 10.1007/BF01184297
[22]	F. C. Lai, I. Pop, F. A. Kulacki, Free and mixed convection from slender bodies of revolution in porous media, Int. J. Heat Mass, 33 (1990), 1767–1769.
[23]	F. C. Lai, C. Y. Choi, F. A. Kulacki, Coupled heat and mass transfer by natural convection from slender bodies of revolution in porous media, Int. Commun. Heat Mass, 17 (1990), 609–620. https://doi.org/10.1016/0735-1933(90)90009-9 doi: 10.1016/0735-1933(90)90009-9
[24]	S. M. Hussain, M. K. Mishra, I. Alraddadi, I. A. Al-Luhaybi, Thermal analysis of trihybrid (Co+ ZrO2+ Au/H2O) nanofluid flow through the exponentially deformable surface with Lorentz force and Navier's slippage constraint, Int. J. Numer. Method. H., 35 (2025), 1680–1712. https://doi.org/10.1108/HFF-03-2025-0140 doi: 10.1108/HFF-03-2025-0140
[25]	I. Alraddadi, S. M. Hussain, A. G. Almutairi, M. K. Mishra, Implications of slippage constraint and aligned Lorentz force on the ternary hybrid nanofluid flow over an exponentially deformable surface, Multiscale and Multidiscip. Model. Exp. and Des., 8 (2025), 269. https://doi.org/10.1007/s41939-025-00864-6 doi: 10.1007/s41939-025-00864-6
[26]	S. M. Mabrouk, E. H. Nasr, A. E. Kabeel, A. S. Rashed, Enhanced Bioconvection Adjacent to Permeable Cylinder in Hybrid Nanofluids: Bacteria Distribution and Physical Features under Magnetic Field Influence, J. Eng. Phys. Thermophy., 98 (2025), 454–467. https://doi.org/10.1007/s10891-025-03119-w doi: 10.1007/s10891-025-03119-w
[27]	A. S. Rashed, E. Nasr, S. M. Mabrouk, Bioconvective Flow Surrounding a Thin Surgical Needle in Blood Incorporating Ternary Hybrid Nanoparticles, Comput. Methods Differe., 2024.
[28]	M. Sohail, E. R. El-Zahar, A. A. A. Mousa, S. Althobaiti, N. A. Shah, J. D. Chung, Finite element analysis for ternary hybrid nanoparticles on thermal enhancement in pseudo-plastic liquid through porous stretching sheet, Sci. Rep., 12 (2022), 9219. https://doi.org/10.1038/s41598-022-12857-3 doi: 10.1038/s41598-022-12857-3
[29]	G. K. Ramesh, J. K. Madhukesh, R. Das, N. A. Shah, S. J. Yook, Thermodynamic activity of a ternary nanofluid flow passing through a permeable slipped surface with heat source and sink, Waves Random Complex, 35 (2022), 3499–3519. https://doi.org/10.1080/17455030.2022.2053237 doi: 10.1080/17455030.2022.2053237
[30]	A. Rauf, Faisal, N. A. Shah, T. Botmart, Hall current and morphological effects on MHD micropolar non-Newtonian trihybrid nanofluid flow between two parallel surfaces, Sci Rep, 12 (2022), 16608. https://doi.org/10.1038/s41598-022-19625-3 doi: 10.1038/s41598-022-19625-3
[31]	C. S. K. Raju, S. Saleem, S. U. Mamatha, I, Hussain. Heat and mass transport phenomena of radiated slender body of three revolutions with saturated porous: Buongiorno's model. Int. J. Therm. Sci., 132 (2018), 309–315. https://doi.org/10.1016/j.ijthermalsci.2018.06.016 doi: 10.1016/j.ijthermalsci.2018.06.016
[32]	M. D. Kumar, C. S. K. Raju, H. Ashraf, N. A. Shah, A. Ali, A. Mennouni, et al., Analysis of dynamical assisting and opposing flow characteristics of darcy surface-filled ternary nanoparticles and Fourier flux: Artificial neural network and levenberg method, J. Circuit., Syst. Comp., 33 (2024), 2440001. https://doi.org/10.1142/S0218126624400012 doi: 10.1142/S0218126624400012
[33]	B. Manvi, J. Tawade, M. Biradar, S. Noeiaghdam, U. F. Gamiz, V. Govindan, The effects of MHD radiating and non-uniform heat source/sink with heating on the momentum and heat transfer of Eyring-Powell fluid over a stretching, Results Eng., 14 (2022), 100435. https://doi.org/10.1016/j.rineng.2022.100435 doi: 10.1016/j.rineng.2022.100435
[34]	S. M. Hussain, M. R. Mishra, G. S. Seth, A. J. Chamkha, Dynamics of heat absorbing and radiative hydromagnetic nanofluids through a stretching surface with chemical reaction and viscous dissipation, In: Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering, 238 (2022), 101–111. https://doi.org/10.1177/095440892210961
[35]	F. C. Lai, C. Y. Choi, F. A. Kulacki, Coupled heat and mass transfer by natural convection from slender bodies of revolution in porous media, Int. Commun. Heat Mass, 17 (1990), 609–620. https://doi.org/10.1016/0735-1933(90)90009-9 doi: 10.1016/0735-1933(90)90009-9

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)