Weakly Nonlinear Bio-Thermal Convection in a Porous Media Layer Under Rotation, Gravity Modulation, and Heat Source

  • Michael I. Kopp Institute for Single Cristals, Nat. Academy of Science Ukraine, Kharkiv, Ukraine https://orcid.org/0000-0001-7457-3272
  • Volodymyr V. Yanovsky Institute for Single Cristals, Nat. Academy of Science Ukraine, Kharkiv, Ukraine; V.N. Karazin Kharkiv National University, Kharkiv, Ukraine https://orcid.org/0000-0003-0461-749X
Keywords: Darcy-Brinkman model, Bio-thermal convection, Gravity modulation, Porous rotating medium, Gyrotactic microorganism


In this paper, the influence of gravitational modulation on weakly nonlinear biothermal convection in a porous rotating layer is investigated. We consider a layer of porous medium saturated with Newtonian fluid, containing gyrotactic microorganisms, and subject to gravitational modulation, rotation, and internal heating. To analyze linear stability, it is sufficient to represent disturbances in the form of normal modes, while nonlinear analysis includes a truncated Fourier series containing a harmonic of the nonlinear interaction. A six-dimensional nonlinear Lorentz-type model is constructed, exhibiting both reflection symmetry and dissipation. We determined heat and mass transfer using a weakly nonlinear theory based on the representation of a truncated Fourier series. Additionally, the behavior of nonstationary Nusselt and Sherwood numbers was investigated by numerically solving finite amplitude equations. Applying the expansion of regular perturbations in a small parameter to a six-dimensional model of Lorentz equations with periodic coefficients, we obtained the Ginzburg-Landau (GL) equation. This equation describes the evolution of the finite amplitude of the onset of convection. The amplitude of convection in the unmodulated case is determined analytically and serves as a standard for comparison. The study examines the effect of various parameters on the system, including the Vadasz number, modified Rayleigh-Darcy number, Taylor number, cell eccentricity, and modulation parameters such as amplitude and frequency. By varying these parameters, in different cases, we analyzed heat and mass transfer, quantitatively expressed by the Nusselt and Sherwood numbers. It has been established that the modulation amplitude has a significant effect on the enhancement of heat and mass transfer, while the modulation frequency has a decreasing effect.


Download data is not yet available.


D. Ingham and L. Pop, Transport Phenomena in Porous Media (Elsevier, Oxford, 2005).

D.A. Nield and A. Bejan, “Internal Natural Convection: Heating from Below,” in: Convection in Porous Media, (Springer, Cham, 2017). https://doi.org/10.1007/978-3-319-49562-0

P. Vadasz, “Instability and convection in rotating porous media: A review,” Fluids 4, 147 (2019). http://dx.doi.org/10.3390/fluids4030147

A.K. Agarwal, and A. Verma, “The effect of compressibility, rotation and magnetic field on thermal instability of Walters' fluid permeated with suspended particles in porous medium,” Thermal Science 18, 539-550 (2014). https://doi.org/10.2298/TSCI110805087A

G. Padma, and S.V. Suneetha, “Hall effects on MHD Flow through Porous Medium in a Rotating Parallel Plate Channel,” Int. J. Appl. Eng. Res. 13, 9772-9789 (2018). https://www.ripublication.com

P. Vasseur, and L. Robillard, “Natural convection in enclosures filled with anisotropic porous media,” Trans. Phenom. Porous Media, 331-356 (1998). https://doi.org/10.1016/B978-008042843-7/50014-3

M. Fahs, A. Younes, and A. Makradi, “A reference benchmark solution for free convection in a square cavity filled with a heterogeneous porous medium,” Numer. Heat Transfer Part B Fundam. 67, 437-462 (2015). https://doi.org/10.1080/10407790.2014.977183

M. Zhao, S. Wang, S.C. Li, Q.Y. Zhang, and U.S. Mahabaleshwar, “Chaotic Darcy-Brinkman convection in a fluid saturated porous layer subjected to gravity modulation,” Results in Physics, 9, 1468-1480 (2018). https://doi.org/10.1016/j.rinp.2018.04.047

T.S. Lundgren, “Slow Flow through Stationary Random Beds and Suspensions of Spheres,” J. Fluid Mech. 51, 273-299 (1972). http://dx.doi.org/10.1017/S002211207200120X

D. Yadav, R. Bhargava, and G.S. Agrawal, “Boundary and internal heat source effects on the onset of Darcy-Brinkman convection in a porous layer saturated by nanofluid,” Int. J. Therm. Sci. 60, 244-254 (2012). https://doi.org/10.1016/j.ijthermalsci.2012.05.011

D.A. Nield, and A.V. Kuznetsov, “The Onset of Convection in an Internally Heated Nanofluid Layer,” J. Heat Transfer, 136, 014501 (2014). https://doi.org/10.1115/1.4025048

I.K. Khalid, N.F.M. Mokhtar, I. Hashim, Z.B. Ibrahim, and S.S.A. Gani, “Effect of Internal Heat Source on the Onset of Double-Diffusive Convection in a Rotating Nanofluid Layer with Feedback Control Strategy,” Adv. Math. Phys. 2017, 2789024. https://doi.org/10.1155/2017/2789024

C. Jain, and V.S. Solomatov, “Onset of convection in internally heated fluids with strongly temperature-dependent viscosity,” Phys. Fluids, 34, 096604 (2022). https://doi.org/10.1063/5.0105170

M. Devi, J. Sharma, and U. Gupta, “Effect of internal heat source on Darcy-Brinkman convection in a non-newtonian casson nanofluid layer,” J. Porous Media, 25, 17-35 (2022). https://doi.org/10.1615/JPorMedia.2022039506

T.J. Pedley, N.A. Hill, and J.O. Kessler, “The growth of bioconvection patterns in a uniform suspension of gyrotactic microorganisms,” J. Fluid Mech. 195, 223-338 (1988). https://doi.org/10.1017/S0022112088002393

N.A. Hill, T.J. Pedley, and J.O. Kessler, “Growth of bioconvection patterns in a suspension of gyrotactic microorganisms in a layer of finite depth,” J. Fluid Mech. 208, 509-543 (1989). https://doi.org/10.1017/S0022112089002922

T.J. Pedley, and J.O. Kessler, “Hydrodynamic phenomena in suspensions of swimming microorganisms,” Ann. Rev. Fluid Mech. 24, 313-358 (1992). http://dx.doi.org/10.1146/ANNUREV.FL.24.010192.001525

S. Childress, M. Levandowsky, and E.A. Spiegel, “Pattern formation in a suspension of swimming microorganisms: equations and stability theory,” J. Fluid Mech. 69, 591-613 (1975). https://doi.org/10.1017/S0022112075001577

A V. Kuznetsov, and A. A. Avramenko, “Stability Analysis of Bioconvection of Gyrotactic Motile Microorganisms in a Fluid Saturated Porous Medium,” Transp. Porous Media, 53, 95-104 (2003). http://dx.doi.org/10.1023/A:1023582001592

D.A. Nield, A.V. Kuznetsov, and A.A. Avramenko, “The onset of bioconvection in a horizontal porous-medium layer,” Transp. Porous Media, 54, 335-344 (2004). http://dx.doi.org/10.1023/B:TIPM.0000003662.31212.5b

A.A. Avramenko, and A.V. Kuznetsov, “The Onset of Convection in a Suspension of Gyrotactic Microorganisms in Superimposed Fluid and Porous Layers: Effect of Vertical Throughflow,” Transp. Porous Media, 65, 159-176 (2006). http://dx.doi.org/10.1007/s11242-005-6086-3

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, 223-233 (2006). http://dx.doi.org/10.1016/j.euromechflu.2005.06.003

A.A. Avramenko, “Model of Lorenz instability for bioconvection,” Dopov. Nac. akad. nauk Ukr. 10, 68-76 (2010).

E. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci. 20, 130-141 (1963). https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2

Y. Hwang, and T.J. Pedley, “Bioconvection under uniform shear: linear stability analysis,” J. Fluid Mech. 738, 522-562 (2014). https://doi.org/10.1017/jfm.2013.604

N.P. Dmitrenko, “Main aspects of the process of bioconvection in nanofluids and porous media,” Industrial Heat Engineering 39(5), 19-25 (2017). https://doi.org/10.31472/ihe.5.2017.03

Y.D. Sharma, and V. Kumar, “The effect of high-frequency vertical vibration in a suspension of gyrotactic micro-organisms,” Mech. Res. Commun. 44, 40-46 (2012). https://doi.org/10.1016/j.mechrescom.2012.06.001

A.K. Kushwaha, Y.D. Sharma, and A. Sharma, “Stability analysis of Vibrational System of Shallow Layers repleted with Random Swimming Gyrotactic Microorganisms,” Research Square, https://doi.org/10.21203/rs.3.rs-1814108/v1

A. Garg, Y.D. Sharma, and S.K. Jain, “Stability analysis of thermo-bioconvection flow of Jeffrey fluid containing gravitactic microorganism into an anisotropic porous medium,” Forces in Mechanics, 10, 100152 (2023). https://doi.org/10.1016/j.finmec.2022.100152

M. Zhao, S. Wang, H. Wang, and U.S. Mahabaleshwar, “Darcy-Brinkman bio-thermal convection in a suspension of gyrotactic microorganisms in a porous medium,” Neural Comput. and Applic. 31, 1061-1067 (2019). https://doi.org/10.1007/s00521-017-3137-y

M.I. Kopp, V.V. Yanovsky, and U.S. Mahabaleshwar, “A Bio-Thermal Convection in a Porous Medium Saturated by Nanofluid Containing Gyrotactic Microorganisms Under an External Magnetic Field,” East European Journal of Physics 4, 23 47 (2022). https://doi.org/10.26565/2312-4334-2022-4-02

M.I. Kopp, and V.V. Yanovsky, “Darcy-Brinkman bio-thermal convection in a porous rotating layer saturated by Newtonian fluid containing gyrotactic microorganisms,” Ukr. J. Phys. 68, 30-37 (2023). https://doi.org/10.15407/ujpe68.1.30

P.M. Gresho, and R. Sani. “The effects of gravity modulation on the stability of a heated fluid layer,” J. Fluid Mech. 40, 783 806 (1970). https://doi.org/10.1017/S0022112070000447

M.S. Malashetty, and I. Begum. “Effect of Thermal/Gravity Modulation on the Onset of Convection in a Maxwell Fluid Saturated Porous Layer,” Transp. Porous Med. 90, 889-909 (2011). https://doi.org/10.1007/s11242-011-9822-x

P. Kiran, “Nonlinear thermal convection in a viscoelastic nanofluid saturated porous medium under gravity modulation,” Ain Shams Engineering Journal, 7, 639-651 (2016). https://doi.org/10.1016/j.asej.2015.06.005

P. Kiran, “Gravity modulation effect on weakly nonlinear thermal convection in a fluid layer bounded by rigid boundaries,” Int. J. Nonlinear Sci. Num. Simul. (2021). https://doi.org/10.1515/ijnsns-2021-0054.

P. Kiran, “Nonlinear throughflow and internal heating effects on vibrating porous medium,” Alex. Eng. J. 55, 757-767 (2016). http://dx.doi.org/10.1016/j.aej.2016.01.012

P. Kiran, “Throughflow and gravity modulation effects on heat transport in a porous medium,” J. Appl. Fluid Mech. 9, 1105 1113 (2016). https://doi.org/10.18869/acadpub.jafm.68.228.24682

P. Kiran, S.H. Manjula, and R. Roslan. “Weak nonlinear analysis of nanofluid convection with g-jitter using the Ginzburg-Landau model,” Open Physics, 20, 1283-1294 (2022). https://doi.org/10.1515/phys-2022-0217

S. H. Manjula, Palle Kiran, and S. N. Gaikwad. ''Study of Heat and Mass Transfer in a Rotating Nanofluid Layer Under Gravity Modulation,'' J. Nanofluids 12, 842-852 (2023). https://doi.org/10.1166/jon.2023.1971

M.I. Kopp, and V.V. Yanovsky, “Effect of gravity modulation on weakly nonlinear bio-thermal convection in a porous medium layer,” J. Appl. Phys. 134, 104702 (2023). http://dx.doi.org/10.1063/5.0165178

P. Kiran, and S.H. Manjula, “Weakly nonlinear bio-convection in a porous media under temperature modulation and internal heating,” Research Square, (2023). https://doi.org/10.21203/rs.3.rs-3313311/v1

R. Chand, and G.C. Rana, “On the onset of thermal convection in rotating nanofluid layer saturating a Darcy-Brinkman porous medium,” International Journal of Heat and Mass Transfer, 55, 5417-5424 (2012). https://doi.org/10.1016/j.ijheatmasstransfer.2012.04.043

G.C. Rana, and R. Chand, “On the onset of thermal convection in a rotating nanofluid layer saturating a Darcy-Brinkman porous medium: a more realistic model,” Journal of Porous Media, 18, 629-635 (2015). https://doi.org/10.1016/j.ijheatmasstransfer.2012.04.043

P.G. Siddheshwar, and C. Kanchana, “Unicellular unsteady Rayleigh-Benard convection in Newtonian liquids and Newtonian nanoliquids occupying enclosures: new findings,” Int. J. Mech. Sci. 131, 1061-1072 (2017). https://doi.org/10.1016/j.ijmecsci.2017.07.050

R. Haberman, Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 4th ed. (Pearson/Prentice Hall, N.J., 2004).

B.S. Bhadauria, and S. Agarwal, “Natural convection in a nanofluid saturated rotating porous layer: a nonlinear study,” Transp. Porous Med. 87, 585-602 (2011). https://doi.org/10.1007/s11242-010-9702-9

M.I. Kopp, A.V. Tur, and V.V. Yanovsky, “Weakly Nonlinear Magnetic Convection in a Nonuniformly Rotating Electrically Conductive Medium Under the Action of Modulation of External Fields,” East Eur. J. Phys. 2, 5-37 (2020). https://doi.org/10.26565/2312-4334-2020-2-01

How to Cite
Kopp, M. I., & Yanovsky, V. V. (2024). Weakly Nonlinear Bio-Thermal Convection in a Porous Media Layer Under Rotation, Gravity Modulation, and Heat Source. East European Journal of Physics, (1), 175-191. https://doi.org/10.26565/2312-4334-2024-1-15