Combined Effects of Anisotropic Permeability, Chemical Reaction, and Dual Stratifications on Unsteady Free Convection Around a Vertical Circular Cylinder
Keywords:
Chemical reaction, Anisotropic porous medium, Free convection, Circular cylinder, Dual stratification.
Abstract
The unsteady free convection boundary layer flow past a vertically translating circular cylinder embedded in a saturated anisotropic porous medium is investigated under simultaneous thermal and solutal stratifications with a first-order homogeneous chemical reaction. The porous resistance is modelled with a tensor-based Darcy-Brinkman formulation in which the permeability is represented by a symmetric positive-definite second-order tensor, and wall suction/injection is permitted. Using boundary-layer scaling and similarity transformations, the governing PDE system is reduced to a seventh-order nonlinear two-point boundary value problem. The reduced system is solved by an adaptive Lobatto IIIa collocation scheme, selected for its A-stability and reliable performance on stiff boundary-layer equations. The solver reproduces the Crane flat-plate exact solution to six decimal places and matches published stretching-cylinder data to within 2%. The results show that porous drag and transverse curvature govern wall shear and heat transfer, the Schmidt number and chemical reaction strength govern mass transfer, and thermal stratification enhances the Nusselt number through boundary layer thinning. Global sensitivity analysis using Latin Hypercube sampling identifies the Prandtl number as most influential for heat transfer, the Schmidt number for mass transfer, and the porous drag parameter for skin friction. The main contribution is the unified treatment of anisotropic tensor permeability, chemical reaction, and dual stratification for an unsteady translating cylinder, together with a quantitative ranking of parameter influence under a single consistent modelling framework. The results are relevant to geothermal energy extraction, contaminant transport in stratified aquifers, and catalytic processes in enhanced oil recovery.
References
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G. Degan, M. Gibigaye, C. Akowanou, and N. C. Awanou. The similarity regime for natural convection in a vertical cylindrical well filled with an anisotropic porous medium. Journal of Engineering Mathematics, 62(3):277-287, 2008.
K. Vajravelu and K. V. Prasad. Mixed convection heat transfer in an anisotropic porous medium with oblique principal axes. Journal of Mechanics, 30(4):327-338, 2014.
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K. Slimi, A. Mhimid, M. Ben Salah, S. Ben Nasrallah, A. A. Mohamad, and L. Storesletten. Anisotropy effects on heat and fluid flow by unsteady natural convection and radiation in saturated porous media. Numerical Heat Transfer, Part A: Applications, 48(8):763-790, 2005.
B. S. Bhadauria, A. Kumar, J. Kumar, N. C. Sacheti, and P. Chandran. Natural convection in a rotating anisotropic porous layer with internal heat generation. Transport in Porous Media, 90(2):687-705, 2011.
A. M. Aly and S. E. Ahmed. An incompressible smoothed particle hydrodynamics method for natural/mixed convection in a non-Darcy anisotropic porous medium. International Journal of Heat and Mass Transfer, 77:1155-1168, 2014.
M. Iasiello, N. Bianco, W. K. S. Chiu, and V. Naso. Anisotropic convective heat transfer in open-cell metal foams: Assessment and correlations. International Journal of Heat and Mass Transfer, 154:119682, 2020.
S. A. Ali, M. Rudziva, and H. Sithole Mthethwa. A numerical study of doublediffusive convection in the anisotropic porous layer under rotational modulation with internal heat generation. International Communications in Heat and Mass Transfer, 137:106266, 2022.
A. Garg, Y. D. Sharma, and S. K. Jain. Onset of triply diffusive thermo-bio-convection in the presence of gyrotactic microorganisms and internal heating into an anisotropic porous medium: Oscillatory convection. Chinese Journal of Physics, 84:66-82, 2023.
B. Keshavarzian and H.-O. Sayehvand. Validity of the boundary layer assumptions for natural convection around a cylinder in a porous medium. Results in Engineering, 18:101069, 2023.
D. A. Nield and A. Bejan. Convection in Porous Media. Springer, Cham, 5 edition, 2017.
C. Y. Cheng. Double-diffusive natural convection along a vertical wavy truncated cone in non-Newtonian fluid saturated porous media with thermal and mass stratification. International Communications in Heat and Mass Transfer, 35(8):985-990, 2008.
C. Y. Cheng. Combined heat and mass transfer in natural convection flow from a vertical wavy surface in a power-law fluid saturated porous medium with thermal and mass stratification. International Communications in Heat and Mass Transfer, 36(4):351-356, 2009.
A. K. Kulkarni, H. R. Jacobs, and J. J. Hwang. Similarity solution for natural convection flow over an isothermal vertical wall immersed in thermally stratified medium. International Journal of Heat and Mass Transfer, 30(4):691-698, 1987.
D. Angirasa and J. Srinivasan. Natural convection flows due to the combined buoyancy of heat and mass diffusion in a thermally stratified medium. Journal of Heat Transfer, 111(3):657-663, 1989.
R. K. Deka and A. Paul. Convectively driven flow past an infinite moving vertical cylinder with thermal and mass stratification. Pramana, 81(4):641-665, 2013.
B. K. Jha and M. K. Musa. The combined effects of anisotropic porous medium and stably stratified fluid on free convective flow through an annulus. Journal of Taibah University for Science, 12(5):678-686, 2018.
B. K. Jha, M. K. Musa, and A. O. Ajibade. The role of stably stratified fluid and anisotropy porous medium in modeling fluid flow through an asymmetrically heated/cooled vertical annulus. Heat Transfer, 52(4):2972-2990, 2023.
B. K. Jha, M. K. Musa, and A. O. Ajibade. Buoyancy force distribution driven Couette flow of stably stratified fluid in a vertical channel filled with anisotropic porous material. International Journal of Applied and Computational Mathematics, 9:148, 2023.
J. A. Adigun, A. Adeniyan, and I. O. Abiala. Stagnation point MHD slip-flow of viscoelastic nanomaterial over a stretched inclined cylindrical surface in a porous medium with dual stratification. International Communications in Heat and Mass Transfer, 126:105479, 2021.
M. Thebault, S. Giroux-Julien, V. Timchenko, C. Menezo, and J. Reizes. Transitional natural convection flow in a vertical channel: Impact of the external thermal stratification. International Journal of Heat and Mass Transfer, 151:119476, 2020.
D. Srinivasacharya and Ch. RamReddy. Heat and mass transfer by natural convection in a doubly stratified non-Darcy micropolar fluid. International Communications in Heat and Mass Transfer, 37(7):873-880, 2010.
M. A. Mansour, N. F. El-Ansary, and A. M. Aly. Effects of chemical reaction and thermal stratification on MHD free convective heat and mass transfer over a vertical stretching surface embedded in a porous media considering Soret and Dufour numbers. Chemical Engineering Journal, 145(2):340-345, 2008.
R. Kandasamy, K. Periasamy, and K. K. S. Prabhu. Chemical reaction, heat and mass transfer on MHD flow over a vertical stretching surface with heat source and thermal stratification effects. International Journal of Heat and Mass Transfer, 48(21-22):4557-4561, 2005.
F. S. Ibrahim, A. M. Elaiw, and A. A. Bakr. Effects of chemical reaction and radiation absorption on the unsteady MHD free convection flow past a semi-infinite vertical permeable moving plate with heat source and suction. Communications in Nonlinear Science and Numerical Simulation, 13(6):1056-1066, 2008.
A. J. Chamkha, A. M. Aly, and M. A. Mansour. Chemical reaction effects on natural convection flow in a porous medium. Engineering Journal of Qatar University, 15:43-58, 2002.
A. Mahdy. Effect of chemical reaction and heat generation or absorption on double diffusive convection from a vertical truncated cone in porous media with variable viscosity. International Communications in Heat and Mass Transfer, 37(5):548-554, 2010.
P. Sudarsana Reddy and P. Sreedevi. Impact of chemical reaction and double stratification on heat and mass transfer characteristics of nanofluid flow over porous stretching sheet with thermal radiation. International Journal of Ambient Energy, 43(1):1626-1636, 2022.
C. Zhang, L. Zheng, X. Zhang, and G. Chen. MHD flow and radiation heat transfer of nanofluids in porous media with variable surface heat flux and chemical reaction. Applied Mathematical Modelling, 39(1):165-181, 2015.
K. Singh and P. Chauhan. Influence of heat source on radiative-convective heat and mass transfer flow of water-based copper nanofluid from a permeable stretchable cylinder in porous medium with chemical reaction and convective boundary condition. Journal of Thermal Analysis and Calorimetry, 150:1951-1968, 2025.
A. J. Badday and A. J. Harfash. Double-diffusive convection in bidispersive porous medium with chemical reaction and magnetic field effects. Transport in Porous Media, 139:45-66, 2021.
G. B. Zegeye, E. Haile, and G. Awgichew. Combined effects of Joule heating and binary chemical reaction of MHD Williamson nano fluid on Darcy-Forchheimer porous medium past unsteady stretching cylinder. International Journal of Thermofluids, 20:100474, 2023.
H. A. Isede, A. Adeniyan, and O. Oladosu. Entropy optimization and heat transfer analysis of MHD heat generating fluid flow through an anisotropic porous parallel wall channel: An analytic solution. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 9(1):81-103, July 2023.
J. A. Gbadeyan, T. M. Asiru, and H. A. Isede. Influence of Dufour and Soret on unsteady, magneto-hydrodynamics (MHD), convective heat and mass transfer flow in non-Darcy porous medium. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 2019(1):468-482, 2019.
H. D. Nguyen, S. Paik, R. W. Douglass, and I. Pop. Unsteady non-Darcy reactiondriven flow from an anisotropic cylinder in porous media. Chemical Engineering Science, 51(22):4963-4977, 1996.
A. M. Rashad. Unsteady nanofluid flow over an inclined stretching surface with convective boundary condition and anisotropic slip impact. International Journal of Heat and Technology, 35(1):82-90, 2017.
L. J. Crane. Flow past a stretching plate. Zeitschrift für angewandte Mathematik und Physik ZAMP, 21(4):645-647, 1970.
C. Y. Wang. Fluid flow due to a stretching cylinder. The Physics of Fluids, 31(3):466-468, 1988.
S. Znaidia, S. Rehman, N. Drissi, and F. Asiri. Mixed convection flow and heat transfer of Williamson drilling nanofluid over a stretching cylinder using Brinkman-Maxwell-Garnett model and temperature-dependent viscosity. Journal of Nonlinear Mathematical Physics, 32:18, 2025.
A. Ishak, R. Nazar, and I. Pop. Uniform suction/blowing effect on flow and heat transfer due to a stretching cylinder. Applied Mathematical Modelling, 32(10):2059-2066, 2008.
N. Bachok and A. Ishak. Flow and heat transfer over a stretching cylinder with prescribed surface heat flux. Malaysian Journal of Mathematical Sciences, 4(2):159-169, 2010.
E. M. A. Elbashbeshy. Heat transfer over a stretching surface with variable surface heat flux. Journal of Physics D: Applied Physics, 31(16):1951-1956, 1998.
T. S. Chen and C. F. Yuh. Combined heat and mass transfer in natural convection along a vertical cylinder. International Journal of Heat and Mass Transfer, 23(4):451-461, 1980.
I. Khan, N. Ali Shah, A. Tassaddiq, N. Mustapha, and S. A. Kechil. Natural convection heat transfer in an oscillating vertical cylinder. PLoS ONE, 13(1):e0188656, 2018.
P. Virtanen, R. Gommers, T. E. Oliphant, et al. SciPy 1.0: fundamental algorithms for scientific computing in Python. Nature Methods, 17(3):261-272, 2020.
G. Degan, M. Gibigaye, C. Akowanou, and N. C. Awanou. The similarity regime for natural convection in a vertical cylindrical well filled with an anisotropic porous medium. Journal of Engineering Mathematics, 62(3):277-287, 2008.
K. Vajravelu and K. V. Prasad. Mixed convection heat transfer in an anisotropic porous medium with oblique principal axes. Journal of Mechanics, 30(4):327-338, 2014.
A. Nakayama, F. Kuwahara, T. Umemoto, and T. Hayashi. Heat and fluid flow within an anisotropic porous medium. Journal of Heat Transfer, 124(4):746-753, 2002.
C. S. Roper, K. D. Fink, S. T. Lee, J. A. Kolodziejska, and A. J. Jacobsen. Anisotropic convective heat transfer in microlattice materials. AIChE Journal, 59(2):622-629, 2013.
K. Slimi, A. Mhimid, M. Ben Salah, S. Ben Nasrallah, A. A. Mohamad, and L. Storesletten. Anisotropy effects on heat and fluid flow by unsteady natural convection and radiation in saturated porous media. Numerical Heat Transfer, Part A: Applications, 48(8):763-790, 2005.
B. S. Bhadauria, A. Kumar, J. Kumar, N. C. Sacheti, and P. Chandran. Natural convection in a rotating anisotropic porous layer with internal heat generation. Transport in Porous Media, 90(2):687-705, 2011.
A. M. Aly and S. E. Ahmed. An incompressible smoothed particle hydrodynamics method for natural/mixed convection in a non-Darcy anisotropic porous medium. International Journal of Heat and Mass Transfer, 77:1155-1168, 2014.
M. Iasiello, N. Bianco, W. K. S. Chiu, and V. Naso. Anisotropic convective heat transfer in open-cell metal foams: Assessment and correlations. International Journal of Heat and Mass Transfer, 154:119682, 2020.
S. A. Ali, M. Rudziva, and H. Sithole Mthethwa. A numerical study of doublediffusive convection in the anisotropic porous layer under rotational modulation with internal heat generation. International Communications in Heat and Mass Transfer, 137:106266, 2022.
A. Garg, Y. D. Sharma, and S. K. Jain. Onset of triply diffusive thermo-bio-convection in the presence of gyrotactic microorganisms and internal heating into an anisotropic porous medium: Oscillatory convection. Chinese Journal of Physics, 84:66-82, 2023.
B. Keshavarzian and H.-O. Sayehvand. Validity of the boundary layer assumptions for natural convection around a cylinder in a porous medium. Results in Engineering, 18:101069, 2023.
D. A. Nield and A. Bejan. Convection in Porous Media. Springer, Cham, 5 edition, 2017.
C. Y. Cheng. Double-diffusive natural convection along a vertical wavy truncated cone in non-Newtonian fluid saturated porous media with thermal and mass stratification. International Communications in Heat and Mass Transfer, 35(8):985-990, 2008.
C. Y. Cheng. Combined heat and mass transfer in natural convection flow from a vertical wavy surface in a power-law fluid saturated porous medium with thermal and mass stratification. International Communications in Heat and Mass Transfer, 36(4):351-356, 2009.
A. K. Kulkarni, H. R. Jacobs, and J. J. Hwang. Similarity solution for natural convection flow over an isothermal vertical wall immersed in thermally stratified medium. International Journal of Heat and Mass Transfer, 30(4):691-698, 1987.
D. Angirasa and J. Srinivasan. Natural convection flows due to the combined buoyancy of heat and mass diffusion in a thermally stratified medium. Journal of Heat Transfer, 111(3):657-663, 1989.
R. K. Deka and A. Paul. Convectively driven flow past an infinite moving vertical cylinder with thermal and mass stratification. Pramana, 81(4):641-665, 2013.
B. K. Jha and M. K. Musa. The combined effects of anisotropic porous medium and stably stratified fluid on free convective flow through an annulus. Journal of Taibah University for Science, 12(5):678-686, 2018.
B. K. Jha, M. K. Musa, and A. O. Ajibade. The role of stably stratified fluid and anisotropy porous medium in modeling fluid flow through an asymmetrically heated/cooled vertical annulus. Heat Transfer, 52(4):2972-2990, 2023.
B. K. Jha, M. K. Musa, and A. O. Ajibade. Buoyancy force distribution driven Couette flow of stably stratified fluid in a vertical channel filled with anisotropic porous material. International Journal of Applied and Computational Mathematics, 9:148, 2023.
J. A. Adigun, A. Adeniyan, and I. O. Abiala. Stagnation point MHD slip-flow of viscoelastic nanomaterial over a stretched inclined cylindrical surface in a porous medium with dual stratification. International Communications in Heat and Mass Transfer, 126:105479, 2021.
M. Thebault, S. Giroux-Julien, V. Timchenko, C. Menezo, and J. Reizes. Transitional natural convection flow in a vertical channel: Impact of the external thermal stratification. International Journal of Heat and Mass Transfer, 151:119476, 2020.
D. Srinivasacharya and Ch. RamReddy. Heat and mass transfer by natural convection in a doubly stratified non-Darcy micropolar fluid. International Communications in Heat and Mass Transfer, 37(7):873-880, 2010.
M. A. Mansour, N. F. El-Ansary, and A. M. Aly. Effects of chemical reaction and thermal stratification on MHD free convective heat and mass transfer over a vertical stretching surface embedded in a porous media considering Soret and Dufour numbers. Chemical Engineering Journal, 145(2):340-345, 2008.
R. Kandasamy, K. Periasamy, and K. K. S. Prabhu. Chemical reaction, heat and mass transfer on MHD flow over a vertical stretching surface with heat source and thermal stratification effects. International Journal of Heat and Mass Transfer, 48(21-22):4557-4561, 2005.
F. S. Ibrahim, A. M. Elaiw, and A. A. Bakr. Effects of chemical reaction and radiation absorption on the unsteady MHD free convection flow past a semi-infinite vertical permeable moving plate with heat source and suction. Communications in Nonlinear Science and Numerical Simulation, 13(6):1056-1066, 2008.
A. J. Chamkha, A. M. Aly, and M. A. Mansour. Chemical reaction effects on natural convection flow in a porous medium. Engineering Journal of Qatar University, 15:43-58, 2002.
A. Mahdy. Effect of chemical reaction and heat generation or absorption on double diffusive convection from a vertical truncated cone in porous media with variable viscosity. International Communications in Heat and Mass Transfer, 37(5):548-554, 2010.
P. Sudarsana Reddy and P. Sreedevi. Impact of chemical reaction and double stratification on heat and mass transfer characteristics of nanofluid flow over porous stretching sheet with thermal radiation. International Journal of Ambient Energy, 43(1):1626-1636, 2022.
C. Zhang, L. Zheng, X. Zhang, and G. Chen. MHD flow and radiation heat transfer of nanofluids in porous media with variable surface heat flux and chemical reaction. Applied Mathematical Modelling, 39(1):165-181, 2015.
K. Singh and P. Chauhan. Influence of heat source on radiative-convective heat and mass transfer flow of water-based copper nanofluid from a permeable stretchable cylinder in porous medium with chemical reaction and convective boundary condition. Journal of Thermal Analysis and Calorimetry, 150:1951-1968, 2025.
A. J. Badday and A. J. Harfash. Double-diffusive convection in bidispersive porous medium with chemical reaction and magnetic field effects. Transport in Porous Media, 139:45-66, 2021.
G. B. Zegeye, E. Haile, and G. Awgichew. Combined effects of Joule heating and binary chemical reaction of MHD Williamson nano fluid on Darcy-Forchheimer porous medium past unsteady stretching cylinder. International Journal of Thermofluids, 20:100474, 2023.
H. A. Isede, A. Adeniyan, and O. Oladosu. Entropy optimization and heat transfer analysis of MHD heat generating fluid flow through an anisotropic porous parallel wall channel: An analytic solution. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 9(1):81-103, July 2023.
J. A. Gbadeyan, T. M. Asiru, and H. A. Isede. Influence of Dufour and Soret on unsteady, magneto-hydrodynamics (MHD), convective heat and mass transfer flow in non-Darcy porous medium. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 2019(1):468-482, 2019.
H. D. Nguyen, S. Paik, R. W. Douglass, and I. Pop. Unsteady non-Darcy reactiondriven flow from an anisotropic cylinder in porous media. Chemical Engineering Science, 51(22):4963-4977, 1996.
A. M. Rashad. Unsteady nanofluid flow over an inclined stretching surface with convective boundary condition and anisotropic slip impact. International Journal of Heat and Technology, 35(1):82-90, 2017.
L. J. Crane. Flow past a stretching plate. Zeitschrift für angewandte Mathematik und Physik ZAMP, 21(4):645-647, 1970.
C. Y. Wang. Fluid flow due to a stretching cylinder. The Physics of Fluids, 31(3):466-468, 1988.
S. Znaidia, S. Rehman, N. Drissi, and F. Asiri. Mixed convection flow and heat transfer of Williamson drilling nanofluid over a stretching cylinder using Brinkman-Maxwell-Garnett model and temperature-dependent viscosity. Journal of Nonlinear Mathematical Physics, 32:18, 2025.
A. Ishak, R. Nazar, and I. Pop. Uniform suction/blowing effect on flow and heat transfer due to a stretching cylinder. Applied Mathematical Modelling, 32(10):2059-2066, 2008.
N. Bachok and A. Ishak. Flow and heat transfer over a stretching cylinder with prescribed surface heat flux. Malaysian Journal of Mathematical Sciences, 4(2):159-169, 2010.
E. M. A. Elbashbeshy. Heat transfer over a stretching surface with variable surface heat flux. Journal of Physics D: Applied Physics, 31(16):1951-1956, 1998.
T. S. Chen and C. F. Yuh. Combined heat and mass transfer in natural convection along a vertical cylinder. International Journal of Heat and Mass Transfer, 23(4):451-461, 1980.
I. Khan, N. Ali Shah, A. Tassaddiq, N. Mustapha, and S. A. Kechil. Natural convection heat transfer in an oscillating vertical cylinder. PLoS ONE, 13(1):e0188656, 2018.
P. Virtanen, R. Gommers, T. E. Oliphant, et al. SciPy 1.0: fundamental algorithms for scientific computing in Python. Nature Methods, 17(3):261-272, 2020.
Published
2026-04-15
How to Cite
Adeniyan, A., Rauf, Q. O., & Tiamiyu, A. T. (2026). Combined Effects of Anisotropic Permeability, Chemical Reaction, and Dual Stratifications on Unsteady Free Convection Around a Vertical Circular Cylinder. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 12(1), 104 - 128. https://doi.org/10.5281/zenodo.20467246
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