Comparison of the Results of Steady Darcy-Be ́nard Convection Problems of the Classical and the Barletta Types
DOI:
https://doi.org/10.37934/cfdl.17.9.163177Keywords:
Porous medium, linear stability, weakly nonlinear stability, Newtonian fluid, Nusselt numberAbstract
The linear stability analysis of the Barletta-Darcy-Be ́nard convection problem in a horizontal fluid-saturated porous layer is extended to a weakly nonlinear stability analysis considering local thermal equilibrium (LTE) between the fluid and solid phases. The minimal Fourier-Galerkin expansion is used for the case of a free upper surface (Neumann boundary condition on the stream function) along with isothermal boundary condition for which heat transport is quantified in terms of the Nusselt number. The present article aims to fill the literature gap between the linear and non-linear stability analyses of classical Darcy-Be ́nard convection and of Barletta-Darcy-Be ́nard convection. Weakly non-linear stability analysis has not been performed in the case of the non-classical Darcy-Be ́nard convection problem. A comparison of results of the present problem with those of the classical Darcy-Be ́nard convection problem is made. It is found that the cell size is larger in the case of the former problem compared to the latter. The critical Darcy-Rayleigh number, however is smaller in the former one. The Nusselt number varies inversely as the Rayleigh number, R and hence the Nusselt number increases with decrease in R which implies that more heat is transported in Barletta-Darcy-Be ́nard convection compared to classical Darcy-Be ́nard convection.
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[1] Banu, Nurzahan and D. A. S. Rees. "Onset of Darcy–Benard convection using a thermal non-equilibrium model." International Journal of Heat and Mass Transfer 45, no. 11 (2002): 2221-2228. https://doi.org/10.1016/S0017-9310(01)00331-3 DOI: https://doi.org/10.1016/S0017-9310(01)00331-3
[2] Barletta, Antonio. "Thermal instabilities in a fluid saturated porous medium." In Heat transfer in multi-phase materials, pp. 381-414. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. https://doi.org/10.1007/8611_2010_8 DOI: https://doi.org/10.1007/8611_2010_8
[3] Barletta, Antonio, Michele Celli and H. Lagziri. "Instability of a horizontal porous layer with local thermal non-equilibrium: effects of free surface and convective boundary conditions." International Journal of Heat and Mass Transfer 89 (2015): 75-89. https://doi.org/10.1016/j.ijheatmasstransfer.2015.05.026 DOI: https://doi.org/10.1016/j.ijheatmasstransfer.2015.05.026
[4] Horton, C. W. and F. T. Rogers Jr. "Convection currents in a porous medium." Journal of Applied Physics 16, no. 6 (1945): 367-370. https://doi.org/10.1063/1.1707601 DOI: https://doi.org/10.1063/1.1707601
[5] Katto, Y. and T. Masuoka. "Criterion for the onset of convective flow in a fluid in a porous medium." International Journal of Heat and Mass Transfer 10, no. 3 (1967): 297-309. https://doi.org/10.1016/0017-9310(67)90147-0 DOI: https://doi.org/10.1016/0017-9310(67)90147-0
[6] Kuznetsov, A. V. and D. A. Nield. "Effect of local thermal non-equilibrium on the onset of convection in a porous medium layer saturated by a nanofluid." Transport in Porous Media 83 (2010): 425-436. https://doi.org/10.1007/s11242-009-9452-8 DOI: https://doi.org/10.1007/s11242-009-9452-8
[7] Lapwood, ER303660032. "Convection of a fluid in a porous medium." In Mathematical proceedings of the cambridge philosophical society, vol. 44, no. 4, pp. 508-521. Cambridge University Press, 1948. https://doi.org/10.1017/S030500410002452X DOI: https://doi.org/10.1017/S030500410002452X
[8] Nield, Donald A. and Adrian Bejan. Convection in porous media. Vol. 3. New York: springer, 2006.
[9] Rees, D. Andrew S. and A. P. Bassom. "The onset of Darcy-Bénard convection in an inclined layer heated from below." Acta Mechanica 144 (2000): 103-118. https://doi.org/10.1007/BF01181831 DOI: https://doi.org/10.1007/BF01181831
[10] Rees, D. A. S. and I. Pop. "Local thermal non-equilibrium in porous medium convection." In Transport phenomena in porous media III, pp. 147-173. Pergamon, 2005. https://doi.org/10.1016/B978-008044490-1/50010-7 DOI: https://doi.org/10.1016/B978-008044490-1/50010-7
[11] Siddheshwar, P. G. and K. M. Lakshmi. "Darcy-Bénard convection of Newtonian liquids and Newtonian nanoliquids in cylindrical enclosures and cylindrical annuli." Physics of Fluids 31, no. 8 (2019). https://doi.org/10.1063/1.5109183 DOI: https://doi.org/10.1063/1.5109183
[12] Straughan, Brian. "Convection in porous media." In Stability and Wave Motion in Porous Media, pp. 1-45. New York, NY: Springer New York, 2008. https://doi.org/10.1007/978-0-387-76543-3_4 DOI: https://doi.org/10.1007/978-0-387-76543-3_4
[13] Suthar, Om P., P. G. Siddheshwar and B. S. Bhadauria. "A study on the onset of thermally modulated Darcy–Bénard convection." Journal of Engineering Mathematics 101 (2016): 175-188. https://doi.org/10.1007/s10665-016-9853-y DOI: https://doi.org/10.1007/s10665-016-9853-y
[14] Rees, D. Andrew S. and Abdelkader Mojtabi. "The effect of conducting boundaries on weakly nonlinear Darcy–Bénard convection." Transport in porous media 88 (2011): 45-63. https://doi.org/10.1007/s11242-011-9722-0 DOI: https://doi.org/10.1007/s11242-011-9722-0
[15] Saleh, H., N. H. Saeid, Ishak Hashim and Zainol Mustafa. "Effect of conduction in bottom wall on Darcy–Bénard convection in a porous enclosure." Transport in porous media 88 (2011): 357-368. https://doi.org/10.1007/s11242-011-9743-8 DOI: https://doi.org/10.1007/s11242-011-9743-8
[16] Capone, Florinda, JACOPO ALFONSO Gianfrani, Giuliana Massa and D. Andrew S. Rees. "A weakly nonlinear analysis of the effect of vertical throughflow on Darcy–Bénard convection." Physics of Fluids 35, no. 1 (2023). https://doi.org/10.1063/5.0135258 DOI: https://doi.org/10.1063/5.0135258
[17] Turkyilmazoglu, Mustafa. "Darcy–Bénard convection through a uniformly permeable porous slab." Communications in Nonlinear Science and Numerical Simulation 125 (2023): 107392. https://doi.org/10.1016/j.cnsns.2023.107392 DOI: https://doi.org/10.1016/j.cnsns.2023.107392
[18] Dubey, Rashmi and P. V. S. N. Murthy. "The onset of convective instability of horizontal throughflow in a porous layer with inclined thermal and solutal gradients." Physics of Fluids 30, no. 7 (2018). https://doi.org/10.1063/1.5040901 DOI: https://doi.org/10.1063/1.5040901
[19] Swain, Sradharam, Golam Mortuja Sarkar and Bikash Sahoo. "Stability analysis of MHD stagnation point flow of Casson fluid past a shrinking sheet in porous medium considering heat sink or source, thermal radiation and suction effects." Journal of Computational Science 75 (2024): 102207. https://doi.org/10.1016/j.jocs.2023.102207 DOI: https://doi.org/10.1016/j.jocs.2023.102207
[20] Swain, Sradharam, Golam Mortuja Sarkar and Bikash Sahoo. "Dual solutions and linear temporal stability analysis of mixed convection flow of non-Newtonian special third grade fluid with thermal radiation." International Journal of Thermal Sciences 189 (2023): 108262. https://doi.org/10.1016/j.ijthermalsci.2023.108262 DOI: https://doi.org/10.1016/j.ijthermalsci.2023.108262
[21] Swain, Sradharam, Golam Mortuja Sarkar and Bikash Sahoo. "Flow and heat transfer analysis of a special third grade fluid over a stretchable surface in a parallel free stream." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 237, no. 1 (2023): 39-53. https://doi.org/10.1177/09544062221113417 DOI: https://doi.org/10.1177/09544062221113417
[22] Swain, Sradharam, Suman Sarkar and Bikash Sahoo. "Flow and heat transfer analysis of a special third grade fluid over a stretchable surface." Indian Journal of Physics 97, no. 9 (2023): 2745-2754. https://doi.org/10.1007/s12648-023-02638-7 DOI: https://doi.org/10.1007/s12648-023-02638-7
[23] Ramakrishna, Sumithra, Shivaraja Munikrishnappa and Arul Selvamary Thomaiyar. "The Effect of Linear, Parabolic and Inverted Parabolic Salinity Gradients on The Onset of Darcy Brinkman Rayleigh Benard Two-Component Convection in a Two-Layered System with Dufour Effect." Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 101, no. 2 (2023): 121-136. https://doi.org/10.37934/arfmts.101.2.121136 DOI: https://doi.org/10.37934/arfmts.101.2.121136
[24] Arasteh, Hossein, Ramin Mashayekhi, Davood Toghraie, Arash Karimipour, Mehdi Bahiraei and Alireza Rahbari. "Optimal arrangements of a heat sink partially filled with multilayered porous media employing hybrid nanofluid." Journal of Thermal Analysis and Calorimetry 137 (2019): 1045-1058. https://doi.org/10.1007/s10973-019-08007-z DOI: https://doi.org/10.1007/s10973-019-08007-z
[25] Barnoon, Pouya, Davood Toghraie, Reza Balali Dehkordi and Masoud Afrand. "Two phase natural convection and thermal radiation of Non-Newtonian nanofluid in a porous cavity considering inclined cavity and size of inside cylinders." International Communications in Heat and Mass Transfer 108 (2019): 104285. https://doi.org/10.1016/j.icheatmasstransfer.2019.104285 DOI: https://doi.org/10.1016/j.icheatmasstransfer.2019.104285
[26] Arasteh, Hossein, Ramin Mashayekhi, Milad Ghaneifar, Davood Toghraie and Masoud Afrand. "Heat transfer enhancement in a counter-flow sinusoidal parallel-plate heat exchanger partially filled with porous media using metal foam in the channels’ divergent sections." Journal of Thermal Analysis and Calorimetry 141, no. 5 (2020): 1669-1685. https://doi.org/10.1007/s10973-019-08870-w DOI: https://doi.org/10.1007/s10973-019-08870-w
[27] El-Shorbagy, M. A., Farshad Eslami, Muhammad Ibrahim, Pouya Barnoon, Wei-Feng Xia and Davood Toghraie. "Numerical investigation of mixed convection of nanofluid flow in a trapezoidal channel with different aspect ratios in the presence of porous medium." Case Studies in Thermal Engineering 25 (2021): 100977. https://doi.org/10.1016/j.csite.2021.100977 DOI: https://doi.org/10.1016/j.csite.2021.100977
[28] Liu, Xinglong, Davood Toghraie, Maboud Hekmatifar, Omid Ali Akbari, Arash Karimipour and Masoud Afrand. "Numerical investigation of nanofluid laminar forced convection heat transfer between two horizontal concentric cylinders in the presence of porous medium." Journal of Thermal Analysis and Calorimetry 141 (2020): 2095-2108. https://doi.org/10.1007/s10973-020-09406-3 DOI: https://doi.org/10.1007/s10973-020-09406-3
