Nanofluid is an advanced technology to enhance heat transportation. Additionally, the thermal conductivity of nanofluids is high therefore, they are more useful for heat transportation. Evaluation of entropy generation has been a helpful technique for tackling improvements in thermal features because it provides information that cannot be obtained via energy analysis. For thermodynamic irreversibilities, a good approximation is the rate of entropy generation. As a result of a reduction of entropy production, energy transport infrastructure has become more efficient. This study aims to analyse the bioconvective flow of nanofluid flow through a stretching sheet in the occurence of gyrotactic motile microorganisms. A magnetised nanomaterial model with thermophoretic and Brownian diffusion properties is analysed. The impacts of activation energy, temperature dependent and exponential base heat source are investigated in this analysis. The entropy generation of the system is also observed for nanofluid flow. The mathematical model is developed as partial differential equations. The governing equations are reduced to a dimensionless system of ordinary differential equations by applying similarity transformations. The ODEs are tacked numerically with the aid of shooting scheme in commercial software MATLAB. For graphical and numerical results of flow controlling parameters versus subjective fields, the commercial software MATLAB tool bvp4 is used with the shooting scheme. The novelty of this analysis computes numerical computation of bioconvective nanofluid flow with temperature-dependent and exponential base heat source investigated. Furthermore, the consequence of thermal radiation and entropy of the system is considered. The porous medium with activation energy is also taken into consideration. The results show that the velocity field is reduced with increased bioconvection Rayleigh number. The thermal field is increased via an exponential space-based heat source. The concentration is reduced via Lewis number. the microorganisms profile declines for larger bioconvection Lewis number. The Brinkman number Br, magnetic and permeability characteristics all showed a rising trend when plotted against the entropy production rate.
Citation: Madeeha Tahir, Ayesha Naz, Muhammad Imran, Hasan Waqas, Ali Akgül, Hussein Shanak, Rabab Jarrar, Jihad Asad. Activation energy impact on unsteady Bio-convection nanomaterial flow over porous surface[J]. AIMS Mathematics, 2022, 7(11): 19822-19845. doi: 10.3934/math.20221086
Nanofluid is an advanced technology to enhance heat transportation. Additionally, the thermal conductivity of nanofluids is high therefore, they are more useful for heat transportation. Evaluation of entropy generation has been a helpful technique for tackling improvements in thermal features because it provides information that cannot be obtained via energy analysis. For thermodynamic irreversibilities, a good approximation is the rate of entropy generation. As a result of a reduction of entropy production, energy transport infrastructure has become more efficient. This study aims to analyse the bioconvective flow of nanofluid flow through a stretching sheet in the occurence of gyrotactic motile microorganisms. A magnetised nanomaterial model with thermophoretic and Brownian diffusion properties is analysed. The impacts of activation energy, temperature dependent and exponential base heat source are investigated in this analysis. The entropy generation of the system is also observed for nanofluid flow. The mathematical model is developed as partial differential equations. The governing equations are reduced to a dimensionless system of ordinary differential equations by applying similarity transformations. The ODEs are tacked numerically with the aid of shooting scheme in commercial software MATLAB. For graphical and numerical results of flow controlling parameters versus subjective fields, the commercial software MATLAB tool bvp4 is used with the shooting scheme. The novelty of this analysis computes numerical computation of bioconvective nanofluid flow with temperature-dependent and exponential base heat source investigated. Furthermore, the consequence of thermal radiation and entropy of the system is considered. The porous medium with activation energy is also taken into consideration. The results show that the velocity field is reduced with increased bioconvection Rayleigh number. The thermal field is increased via an exponential space-based heat source. The concentration is reduced via Lewis number. the microorganisms profile declines for larger bioconvection Lewis number. The Brinkman number Br, magnetic and permeability characteristics all showed a rising trend when plotted against the entropy production rate.
| [1] |
E. Tombácz, D. Bica, A. Hajdú, E. Illés, A. Majzik, L. Vékás, Surfactant double layer stabilized magnetic nanofluids for biomedical application, J. Phys.: Condens. Matter, 20 (2008), 204103. https://doi.org/10.1088/0953-8984/20/20/204103 doi: 10.1088/0953-8984/20/20/204103
|
| [2] |
T. Sharmaa, A. Mohana Reddy, T. Chandra, S. Ramaprabhu, Development of carbon nanotubes and nanofluids based microbial fuel cell, Int. J. Hydrogen Energ., 33 (2008), 749–6754. https://doi.org/10.1016/j.ijhydene.2008.05.112 doi: 10.1016/j.ijhydene.2008.05.112
|
| [3] |
M. Shaijumon, S. Ramaprabhu, N. Rajalakshmi, Platinum/multi walled carbon nanotubes-platinum/carbon composites as electro catalysts for oxygen reduction reaction in proton exchange membrane fuel cell, Appl. Phys. Lett., 88 (2006), 253105. https://doi.org/10.1063/1.2214139 doi: 10.1063/1.2214139
|
| [4] | S. Choi, J. Eastman, Enhancing thermal conductivity of fluids with nanoparticles, Proceedings of International mechanical engineering congress and exhibition, 1995, 1–7. |
| [5] |
D. Wen, Y. Ding, Formulation of nanofluids for natural convective heat transfer applications, Int. J. Heat Fluid Fl., 26 (2005), 855–864. https://doi.org/10.1016/j.ijheatfluidflow.2005.10.005 doi: 10.1016/j.ijheatfluidflow.2005.10.005
|
| [6] |
J. Buongiorno, Convective transport in nanofluids, J. Heat Transfer, 128 (2006), 240–250. https://doi.org/10.1115/1.2150834 doi: 10.1115/1.2150834
|
| [7] |
S. Murshed, K. Leong, C. Yang, Investigations of thermal conductivity and viscosity of nanofluids, Int. J. Therm. Sci., 47 (2008), 560–568. https://doi.org/10.1016/j.ijthermalsci.2007.05.004 doi: 10.1016/j.ijthermalsci.2007.05.004
|
| [8] |
S. Ganguly, S. Sikdar, S. Basu, Experimental investigation of the effective electrical conductivity of aluminum oxide nanofluids, Powder Technol., 196 (2009), 326–330. https://doi.org/10.1016/j.powtec.2009.08.010 doi: 10.1016/j.powtec.2009.08.010
|
| [9] |
A. Kuznetsov, D. Nield, Natural convective boundary-layer flow of a nanofluid past a vertical plate, Int. J. Therm. Sci., 49 (2010), 243–247. https://doi.org/10.1016/j.ijthermalsci.2009.07.015 doi: 10.1016/j.ijthermalsci.2009.07.015
|
| [10] |
M. Mustafa, T. Hayat, I. Pop, S. Asghar, S. Obaidat, Stagnation-point flow of a nanofluid towards a stretching sheet, Int. J. Heat Mass Tran., 54 (2011), 5588–5594. https://doi.org/10.1016/j.ijheatmasstransfer.2011.07.021 doi: 10.1016/j.ijheatmasstransfer.2011.07.021
|
| [11] |
F. Asadzadeh, M. Nasr Esfahany, N. Etesami, Natural convective heat transfer of Fe3O4/ethylene glycol nanofluid in electric field, Int. J. Therm. Sci., 62 (2012), 114–119. https://doi.org/10.1016/j.ijthermalsci.2011.11.010 doi: 10.1016/j.ijthermalsci.2011.11.010
|
| [12] |
T. Hayat, M. Waqas, S. Shehzad, A. Alsaedi, A model of solar radiation and Joule heating in magnetohydrodynamic (MHD) convective flow of thixotropic nanofluid, J. Mol. Liq., 215 (2016), 704–710. https://doi.org/10.1016/j.molliq.2016.01.005 doi: 10.1016/j.molliq.2016.01.005
|
| [13] |
M. Farooq, M. Ijaz Khan, M. Waqas, T. Hayatb, A. Alsaedi, M. Imran Khan, MHD stagnation point flow of viscoelastic nanofluid with non-linear radiation effects, J. Mol. Liq., 221 (2016), 1097–1103. https://doi.org/10.1016/j.molliq.2016.06.077 doi: 10.1016/j.molliq.2016.06.077
|
| [14] |
M. Bhatti, A. Zeeshan, R. Ellahi, Simultaneous effects of coagulation and variable magnetic field on peristaltically induced motion of Jeffrey nanofluid containing gyrotactic microorganism, Microvasc Res., 110 (2017), 32–42. https://doi.org/10.1016/j.mvr.2016.11.007 doi: 10.1016/j.mvr.2016.11.007
|
| [15] |
M. Ijaz Khan, T. Hayat, M. Imran Khan, A. Alsaedi, Activation energy impact in nonlinear radiative stagnation point flow of Cross nanofluid, Int. Commun. Heat Mass, 91 (2018), 216–224. https://doi.org/10.1016/j.icheatmasstransfer.2017.11.001 doi: 10.1016/j.icheatmasstransfer.2017.11.001
|
| [16] |
S. Nadeem, N. Abbas, A. Khan, Characteristics of three dimensional stagnation point flow of Hybrid nanofluid past a circular cylinder, Results Phys., 8 (2018), 829–835. https://doi.org/10.1016/j.rinp.2018.01.024 doi: 10.1016/j.rinp.2018.01.024
|
| [17] |
M. Bhatti, H. Öztop, R. Ellahi, I. Sarris, M. Doranehgard, Insight into the investigation of diamond (C) and Silica (SiO2) nanoparticles suspended in water-based hybrid nanofluid with application in solar collector, J. Mol. Liq., 357 (2022), 119134. https://doi.org/10.1016/j.molliq.2022.119134 doi: 10.1016/j.molliq.2022.119134
|
| [18] |
M. Bhatti, O. Bég, S. Abdelsalam, Computational framework of magnetized MgO-Ni/water-based stagnation nanoflow past an elastic stretching surface: application in solar energy coatings, Nanomaterials, 12 (2022), 1049. https://doi.org/10.3390/nano12071049 doi: 10.3390/nano12071049
|
| [19] | M. Rashidi, M. Alhuyi Nazari, I. Mahariq, N. Ali, Modeling and sensitivity analysis of thermal conductivity of ethylene glycol-water based nanofluids with Alumina nanoparticles, Exp. Tech., in press. https://doi.org/10.1007/s40799-022-00567-4 |
| [20] |
S. Rawat, H. Upreti, M. Kumar, Numerical study of activation energy and thermal radiation effects on Oldroyd-B nanofluid flow using the Cattaneo-Christov double diffusion model over a convectively heated stretching sheet, Heat Transf., 50 (2021), 5304–5331. https://doi.org/10.1002/htj.22125 doi: 10.1002/htj.22125
|
| [21] |
A. Mishra, H. Upreti, A comparative study of Ag-MgO/water and Fe3O4-CoFe2O4/EG-water hybrid nanofluid flow over a curved surface with chemical reaction using Buongiorno model, Partial Differential Equations in Applied Mathematics, 5 (2022), 100322. https://doi.org/10.1016/j.padiff.2022.100322 doi: 10.1016/j.padiff.2022.100322
|
| [22] |
H. Upreti, A. Pandey, S. Rawat, M. Kumar, Modified Arrhenius and thermal radiation effects on three-dimensional magnetohydrodynamic flow of carbon nanotubes nanofluids over bi-directional stretchable surface, Journal of Nanofluids, 10 (2021), 538–551. https://doi.org/10.1166/jon.2021.1804 doi: 10.1166/jon.2021.1804
|
| [23] |
H. Upreti, M. Kumar, Influence of non-linear radiation, Joule heating and viscous dissipation on the boundary layer flow of MHD nanofluid flow over a thin moving needle, Multidiscip. Model. Ma., 16 (2020), 208–224. https://doi.org/10.1108/MMMS-05-2019-0097 doi: 10.1108/MMMS-05-2019-0097
|
| [24] |
A. Pandey, H. Upreti, Mixed convective flow of Ag-H2O magnetic nanofluid over a curved surface with volumetric heat generation and temperature-dependent viscosity, Heat Transf., 50 (2021), 7251–7270. https://doi.org/10.1002/htj.22227 doi: 10.1002/htj.22227
|
| [25] |
H. Upreti, N. Joshi, A. Pandey, S. Rawat, Assessment of convective heat transfer in Sisko fluid flow via stretching surface due to viscous dissipation and suction, Nanosci. Technol., 13 (2022), 31–44. https://doi.org/10.1615/NanoSciTechnolIntJ.2022039531 doi: 10.1615/NanoSciTechnolIntJ.2022039531
|
| [26] |
N. Joshi, H. Upreti, A. Pandey, MHD Darcy-Forchheimer Cu-Ag/H2O-C2H6O2 hybrid nanofluid flow via a porous stretching sheet with suction/blowing and viscous dissipation, Int. J. Comput. Meth. En., 23 (2022), 527–535. https://doi.org/10.1080/15502287.2022.2030426 doi: 10.1080/15502287.2022.2030426
|
| [27] |
H. Upreti, S. Rawat, M. Kumar, Radiation and non-uniform heat sink/source effects on 2D MHD flow of CNTs-H2O nanofluid over a flat porous plate, Multidiscip. Model. Ma., 16 (2020), 791–809. https://doi.org/10.1108/MMMS-08-2019-0153 doi: 10.1108/MMMS-08-2019-0153
|
| [28] |
N. Joshi, A. Pandey, H. Upreti, M. Kumar, Mixed convection flow of magnetic hybrid nanofluid over a bidirectional porous surface with internal heat generation and a higher-order chemical reaction, Heat Transf., 50 (2021), 3661–3682. https://doi.org/10.1002/htj.22046 doi: 10.1002/htj.22046
|
| [29] |
S. Rawat, H. Upreti, M. Kumar, Thermally stratified nanofluid flow over porous surface cone with Cattaneo-Christov heat flux approach and heat generation (or) absorption, SN Appl. Sci., 2 (2020), 302. https://doi.org/10.1007/s42452-020-2099-3 doi: 10.1007/s42452-020-2099-3
|
| [30] |
S. Rawat, S. Negi, H. Upreti, M. Kumar, A non-Fourier's and non-Fick's approach to study MHD mixed convective copper water nanofluid flow over flat plate subjected to convective heating and zero wall mass flux condition, Int. J. Appl. Comput. Math., 7 (2021), 246. https://doi.org/10.1007/s40819-021-01190-4 doi: 10.1007/s40819-021-01190-4
|
| [31] |
A. Bejan, A study of entropy generation in fundamental convective heat transfer, J. Heat Transfer, 101 (1979), 718–725. https://doi.org/10.1115/1.3451063 doi: 10.1115/1.3451063
|
| [32] |
A. Bejan, Second law analysis in heat transfer, Energy, 5 (1980), 720–732. https://doi.org/10.1016/0360-5442(80)90091-2 doi: 10.1016/0360-5442(80)90091-2
|
| [33] |
A. Bejan, Method of entropy generation minimization, or modeling and optimization based on combined heat transfer and thermodynamics, Revue Générale de Thermique, 35 (1996), 637–646. https://doi.org/10.1016/S0035-3159(96)80059-6 doi: 10.1016/S0035-3159(96)80059-6
|
| [34] |
S. Tasnim, M. Shohel, M. Mamun, Entropy generation in a porous channel with hydromagnetic effect, Exergy, 2 (2002), 300–308. https://doi.org/10.1016/S1164-0235(02)00065-1 doi: 10.1016/S1164-0235(02)00065-1
|
| [35] |
S. Mahmud, R. Fraser, The second law analysis in fundamental convective heat transfer problems, Int. J. Therm. Sci., 42 (2003), 177–186. https://doi.org/10.1016/S1290-0729(02)00017-0 doi: 10.1016/S1290-0729(02)00017-0
|
| [36] |
R. Ellahi, M. Hassan, A. Zeeshan, Shape effects of nanosize particles in nanofluid on entropy generation, Int. J. Heat Mass, 81 (2015), 449–456. https://doi.org/10.1016/j.ijheatmasstransfer.2014.10.041 doi: 10.1016/j.ijheatmasstransfer.2014.10.041
|
| [37] |
G. Shit, R. Haldar, S. Mandal, Entropy generation on MHD flow and convective heat transfer in a porous medium of exponentially stretching surface saturated by nanofluids, Adv. Powder Technol., 28 (2017), 1519–1530. https://doi.org/10.1016/j.apt.2017.03.023 doi: 10.1016/j.apt.2017.03.023
|
| [38] |
M. Sheikholeslami, M. Jafaryar, A. Shafee, Z. Li, R. Ul Haq, Heat transfer of nanoparticles employing innovative turbulator considering entropy generation, Int. J. Heat Mass, 136 (2019), 1233–1240. https://doi.org/10.1016/j.ijheatmasstransfer.2019.03.091 doi: 10.1016/j.ijheatmasstransfer.2019.03.091
|
| [39] |
P. Biswal, T. Basak, Entropy generation vs energy efficiency for natural convection based energy flow in enclosures and various applications: a review, Renew. Sust. Energ. Rev., 80 (2017), 1412–1457. https://doi.org/10.1016/j.rser.2017.04.070 doi: 10.1016/j.rser.2017.04.070
|
| [40] |
T. Hayat, S. Nawaz, A. Alsaedi, Entropy generation in peristalsis with different shapes of nanomaterial, J. Mol. Liq., 248 (2017), 447–458. https://doi.org/10.1016/j.molliq.2017.10.058 doi: 10.1016/j.molliq.2017.10.058
|
| [41] |
T. Hayat, S. Farooq, B. Ahmad, A. Alsaedi, Effectiveness of entropy generation and energy transfer on peristaltic flow of Jeffrey material with Darcy resistance, Int. J. Heat Mass, 106 (2017), 244–252. https://doi.org/10.1016/j.ijheatmasstransfer.2016.10.017 doi: 10.1016/j.ijheatmasstransfer.2016.10.017
|
| [42] |
M. Ijaz Khan, S. Qayyum, T. Hayata, M. Waqas, M. Imran Khan, A. Alsaedi, Entropy generation minimization and binary chemical reaction with Arrhenius activation energy in MHD radiative flow of nanomaterial, J. Mol. Liq., 259 (2018), 274–283. https://doi.org/10.1016/j.molliq.2018.03.049 doi: 10.1016/j.molliq.2018.03.049
|
| [43] |
M. Bhatti, M. Sheikholeslami, A. Shahid, M. Hassan, T. Abbas, Entropy generation on the interaction of nanoparticles over a stretched surface with thermal radiation, Colloid. Surfaces A, 570 (2019), 368–376. https://doi.org/10.1016/j.colsurfa.2019.03.058 doi: 10.1016/j.colsurfa.2019.03.058
|
| [44] |
A. Kumar, R. Tripathi, R. Singh, Entropy generation and regression analysis on stagnation point flow of Casson nanofluid with Arrhenius activation energy, J. Braz. Soc. Mech. Sci. Eng., 41 (2019), 306. https://doi.org/10.1007/s40430-019-1803-y doi: 10.1007/s40430-019-1803-y
|
| [45] |
F. Sultan, W. Khan, M. Ali, M. Shahzad, H. Sun, M. Irfan, Importance of entropy generation and infinite shear rate viscosity for non-Newtonian nanofluid, J. Braz. Soc. Mech. Sci. Eng., 41 (2019), 439. https://doi.org/10.1007/s40430-019-1950-1 doi: 10.1007/s40430-019-1950-1
|
| [46] |
A. Ullah, Z. Shah, P. Kumam, M. Ayaz, S. Islam, M. Jameel, Viscoelastic MHD nanofluid thin film flow over an unsteady vertical stretching sheet with entropy generation, Processes, 7 (2019), 262. https://doi.org/10.3390/pr7050262 doi: 10.3390/pr7050262
|
| [47] |
M. Nayak, A. Abdul Hakeem, B. Ganga, M. Ijaz Khan, M. Waqas, O. Makinde, Entropy optimized MHD 3D nanomaterial of non-Newtonian fluid: a combined approach to good absorber of solar energy and intensification of heat transport, Comput. Meth. Prog. Bio., 186 (2020), 105131. https://doi.org/10.1016/j.cmpb.2019.105131 doi: 10.1016/j.cmpb.2019.105131
|
| [48] |
N. Khan, I. Riaz, M. Hashmi, S. Musmar, S. Khan, Z. Abdelmalek, et al., Aspects of chemical entropy generation in flow of Casson nanofluid between radiative stretching disks, Entropy, 22 (2022), 495. https://doi.org/10.3390/e22050495 doi: 10.3390/e22050495
|
| [49] |
G. Shit, S. Mandal, Entropy analysis on unsteady MHD flow of Casson nanofluid over a stretching vertical plate with thermal radiation effect, Int. J. Appl. Comput. Math., 6 (2020), 2. https://doi.org/10.1007/s40819-019-0754-4 doi: 10.1007/s40819-019-0754-4
|
| [50] |
R. Vincent, N. Hill, Bioconvection in a suspension of phototactic algae, J. Fluid Mech., 327 (1996), 343–371. https://doi.org/10.1017/S0022112096008579 doi: 10.1017/S0022112096008579
|
| [51] |
A. Kuznetsov, Nanofluid bioconvection: interaction of microorganisms oxytactic upswimming, nanoparticle distribution, and heating/cooling from below, Theor. Comput. Fluid Dyn., 26 (2012), 291–310. https://doi.org/10.1007/s00162-011-0230-1 doi: 10.1007/s00162-011-0230-1
|
| [52] |
M. Alqarni, S. Yasmin, H. Waqas, S. Khan, Recent progress in melting heat phenomenon for bioconvection transport of nanofluid through a lubricated surface with swimming microorganisms, Sci. Rep., 12 (2022), 8447. https://doi.org/10.1038/s41598-022-12230-4 doi: 10.1038/s41598-022-12230-4
|
| [53] | R. Alluguvelli, C. Balla, K. Naikoti, O. Makinde, Nanofluid bioconvection in porous enclosure with viscous dissipation, Indian J. Pure Ap. Phy., 60 (2022), 78–89. |
| [54] |
V. Puneeth, S. Manjunatha, O. Makinde, B. Gireesha, Bioconvection of a radiating hybrid nanofluid past a thin needle in the presence of heterogeneous-homogeneous chemical reaction, J. Heat Transfer, 143 (2021), 042502. https://doi.org/10.1115/1.4049844 doi: 10.1115/1.4049844
|
| [55] |
R. Alluguvelli, C. Balla, K. Naikoti, Bioconvection in porous square cavity containing oxytactic microorganisms in the presence of viscous dissipation, Discontinuity, Nonlinearity, and Complexity, 11 (2022), 301–313. https://doi.org/10.5890/DNC.2022.06.009 doi: 10.5890/DNC.2022.06.009
|
| [56] |
O. Makinde, I. Animasaun, Thermophoresis and Brownian motion effects on MHD bioconvection of nanofluid with nonlinear thermal radiation and quartic chemical reaction past an upper horizontal surface of a paraboloid of revolution, J. Mol. Liq., 221 (2016), 733–743. https://doi.org/10.1016/j.molliq.2016.06.047 doi: 10.1016/j.molliq.2016.06.047
|
| [57] |
K. Avinash, N. Sandeep, O. Makinde, I. Animasaun, Aligned magnetic field effect on radiative bioconvection flow past a vertical plate with thermophoresis and Brownian motion, Defect and Diffusion Forum, 377 (2017), 127–140. https://doi.org/10.4028/www.scientific.net/DDF.377.127 doi: 10.4028/www.scientific.net/DDF.377.127
|
| [58] |
M. Ijaz Khan, S. Ullah, T. Hayata, M. Waqas, M. Imran Khan, A. Alsaedi, Salient aspects of entropy generation optimization in mixed convection nanomaterial flow, Int. J. Heat Mass, 126 (2018), 1337–1346. https://doi.org/10.1016/j.ijheatmasstransfer.2018.05.168 doi: 10.1016/j.ijheatmasstransfer.2018.05.168
|
| [59] |
B. Sahoo, Y. Do, Effects of slip on sheet-driven flow and heat transfer of a third-grade fluid past a stretching sheet, Int. Commun. Heat Mass, 37 (2010), 1064–1071. https://doi.org/10.1016/j.icheatmasstransfer.2010.06.018 doi: 10.1016/j.icheatmasstransfer.2010.06.018
|
| [60] |
C. Wang, Flow due to a stretching boundary with partial slip-an exact solution of the Navier-Stokes equations, Chem. Eng. Sci., 57 (2002), 3745–3747. https://doi.org/10.1016/S0009-2509(02)00267-1 doi: 10.1016/S0009-2509(02)00267-1
|
| [61] |
A. Noghrehabadi, M. Saffarian, R. Pourrajab, M. Ghalambaz, Entropy analysis for nanofluid flow over a stretching sheet in the presence of heat generation/absorption and partial slip, J. Mech. Sci. Technol., 27 (2013), 927–937. https://doi.org/10.1007/s12206-013-0104-0 doi: 10.1007/s12206-013-0104-0
|
Figures(23) / Tables(1)
Madeeha Tahir, Ayesha Naz, Muhammad Imran, Hasan Waqas, Ali Akgül, Hussein Shanak, Rabab Jarrar, Jihad Asad. Activation energy impact on unsteady Bio-convection nanomaterial flow over porous surface[J]. AIMS Mathematics, 2022, 7(11): 19822-19845. doi: 10.3934/math.20221086
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