Examination of the nanofluid convective instability of vertical constant throughflow in a porous medium layer with variable gravity


In the current article, the impact of varying gravitational field and flow on the beginning of nanofluid convective instability in a permeable medium layer is studied numerically utilizing Galerkin technique. The permeable layer is directed to a regular vertical throughflow and irregular descendent gravitational force which changes with the length from the layer. The influences of three types of gravitational force inconsistency: (a) linear, (b) parabolic, and (c) exponential are examined on the formation of nanofluid convective instability with vanish nanoparticle flux condition at the plates. Results proved that the throughflow factor \(Q\) and gravity inconsistency factor \(\delta\) suspend the start of convective instability, while the nanoparticle Rayleigh-Darcy number \(R_{{{\text{np}}}}\) and the altered diffusivity ratio \({\text{NA}}_{{{\text{nf}}}}\) quick the start of nanofluid convection. The measurement of the convective cells diminishes with \(R_{{{\text{np}}}}\) and \({\text{NA}}_{{{\text{nf}}}}\), while \(Q\), \(\delta\) and the altered nanofluid Lewis number \({\text{Le}}_{{{\text{nf}}}}\) have duel effects on the measurement of convective cells.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


\(a\) :

Non-dimensional wave number

\(C\) :

Volumetric fraction of nanoparticles

\(D_{B}\) :

Brownian diffusion coefficient

\(D_{\theta }\) :

Thermophoresis diffusion coefficient

\({\hat{\mathbf{e}}}\) :

Unit vector

\(g\left( z \right)\) :

Variable gravity

\(g_{0}\) :

Reference gravity

\(K\) :

Permeability of the porous matrix

\(k_{m}\) :

Aggregate thermal conductivity of the porous matrix

\(L\) :

Height of nanofluid layer

\({\text{Le}}_{{{\text{nf}}}}\) :

Altered nanofluid Lewis number

\({\text{NA}}_{{{\text{nf}}}}\) :

Altered diffusivity ratio

\(P\) :


\(Q\) :

Throughflow factor

\(R_{m}\) :

Density Rayleigh–Darcy number

\(R_{{{\text{nf}}}}\) :

Nanofluid Rayleigh–Darcy number

\(R_{{{\text{np}}}}\) :

Nanoparticle Rayleigh–Darcy number

\({\mathbf{v}}\) :

Velocity vector

\(\left( {x,y,z} \right)\) :

Space coordinates

\(\alpha_{m}\) :

Thermal diffusivity

\(\beta\) :

Expansion coefficient

\(\beta_{C}\) :

Nanoparticles volumetric fraction extension coefficient

\(\beta_{\theta }\) :

Thermal extension coefficient

\(\mu\) :


\(\rho\) :


\(\varphi\) :

Volumetric portion of nanoparticle

\(\varepsilon\) :

Porosity of the porous lattice

\(\sigma\) :

Heat capacity ratio

\(\omega\) :

Progress rate of instability

\(\delta\) :

Gravity inconsistency factor

\(\tau\) :



Reference estimate

b :

Basic flow

c :


m :

Aggregate porous medium






  1. Akbarzadeh P, Mahian O (2018) The onset of nanofluid natural convection inside a porous layer with rough boundaries. J Mol Liq 272:344–352

    CAS  Article  Google Scholar 

  2. Alex SM, Patil PR, Venkatakrishnan K (2001) Variable gravity effects on thermal instability in a porous medium with internal heat source and inclined temperature gradient. Fluid Dyn Res 29(1):1

    Article  Google Scholar 

  3. Bao J, Li S, Zhang P, Ding X, Xue S, Cui Y et al (2020) Influence of the incorporation of recycled coarse aggregate on water absorption and chloride penetration into concrete. Constr Build Mater 239:117845. https://doi.org/10.1016/j.conbuildmat.2019.117845

    CAS  Article  Google Scholar 

  4. Buongiorno J (2006) Convective transport in nanofluids. J Heat Transf 128(3):240–250

    Article  Google Scholar 

  5. Cai C, Wu X, Liu W, Zhu W, Chen H, Qiu JCD et al (2020) Selective laser melting of near-α titanium alloy Ti-6Al-2Zr-1Mo-1V: parameter optimization, heat treatment and mechanical performance. J Mater Sci Technol 57:51–64. https://doi.org/10.1016/j.jmst.2020.05.004

    Article  Google Scholar 

  6. Chand R, Yadav D, Rana GC (2015) Electrothermo convection in a horizontal layer of rotating nanofluid. Int J Nanoparticles 8:241–261

  7. Chand R, Rana GC, Yadav D (2016) Electrothermo convection in a porous medium saturated by nanofluid. J Appl Fluid Mech 9(3):1081–1088

  8. Chand R, Rana GC, Yadav D (2017) Thermal instability in a layer of couple stress nanofluid saturated porous medium. J Theor Appl Mech 47:69–84

  9. Chen H, Fan D, Huang J, Huang W, Zhang G et al (2020) Finite element analysis model on ultrasonic phased array technique for material defect time of flight diffraction detection. Sci Adv Mater 12(5):665–675. https://doi.org/10.1166/sam.2020.3689

    CAS  Article  Google Scholar 

  10. Choi SU, Eastman JA (1995) Enhancing thermal conductivity of fluids with nanoparticles.In: Argonne National Lab., IL (United States)

  11. Chu Y-M, Kumar R, Bach Q-V (2020a) Water-based nanofluid flow with various shapes of Al2O3 nanoparticles owing to MHD inside a permeable tank with heat transfer. Appl Nanosci. https://doi.org/10.1007/s13204-020-01609-2

    Article  Google Scholar 

  12. Chu Y-M, Abohamzeh E, Bach Q-V (2020b) Thermal two-phase analysis of nanomaterial in a pipe with turbulent flow. Appl Nanosci. https://doi.org/10.1007/s13204-020-01576-8

    Article  Google Scholar 

  13. Chu YM, Yadav D, Shafee A, Li Z, Bach QV (2020c) Influence of wavy enclosure and nanoparticles on heat release rate of PCM considering numerical study. J Mol Liq 319:114121. https://doi.org/10.1016/j.molliq.2020.114121

  14. Chu YM, Li Z, Bach QV (2020d) Application of nanomaterial for thermal unit including tube fitted with turbulator. Appl Nanosci. https://doi.org/10.1007/s13204-020-01587-5

    Article  Google Scholar 

  15. Chu Y-M, Hajizadeh MR, Li Z, Bach QV (2020e) Investigation of nano powders influence on melting process within a storage unit. J Mol Liq 318:114321. https://doi.org/10.1016/j.molliq.2020.114321

    CAS  Article  Google Scholar 

  16. Dehghan M, Rahmani Y, Ganji DD, Saedodin S, Valipour MS, Rashidi S (2015) Convection–radiation heat transfer in solar heat exchangers filled with a porous medium: Homotopy perturbation method versus numerical analysis. Renew Energy 74:448–455

  17. Eastman JA, Choi S, Li S, Yu W, Thompson L (2001) Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Appl Phys Lett 78(6):718–720

    CAS  Article  Google Scholar 

  18. Feldman MR, Anderson JD, Schubert G, Trimble V, Kopeikin SM, Lämmerzahl C (2016) Deep space experiment to measure G. Class Quantum Gravity 33(12):125013

    Article  Google Scholar 

  19. Feng Q, Li Y, Wang N, Hao Y, Chang J, Wang Z et al (2020) A biomimetic nanogenerator of reactive nitrogen species based on battlefield transfer strategy for enhanced immunotherapy. Small. https://doi.org/10.1002/smll.202002138

    Article  Google Scholar 

  20. Gangadharaiah YH, Suma SP, Ananda K (2013) Variable gravity field and throughflow effects on penetrative convection in a porous layer. Int J Comput Technol 5(3):172–191

    Article  Google Scholar 

  21. Gao N, Cheng B, Hou H, Zhang R (2018) Mesophase pitch based carbon foams as sound absorbers. Mater Lett 212:243–246. https://doi.org/10.1016/j.matlet.2017.10.074

    CAS  Article  Google Scholar 

  22. Hayakawa Y,  Kumar VN, Arivanandhan M, Rajesh G, Koyama T, Momose Y, Sakata K, Ozawa T, Okano Y, Inatomi Y (2017) Effects of gravity and crystal orientation on the growth of InGaSb ternary alloy semiconductors-experiments at the international space station and on Earth. Int J Microgravity Sci Appl 34:340111-12

  23. He L, Liu J, Liu Y, Cui B, Hu B, Wang M et al (2019) Titanium dioxide encapsulated carbon-nitride nanosheets derived from MXene and melamine-cyanuric acid composite as a multifunctional electrocatalyst for hydrogen and oxygen evolution reaction and oxygen reduction reaction. Appl Catal B Environ 248:366–379. https://doi.org/10.1016/j.apcatb.2019.02.033

    CAS  Article  Google Scholar 

  24. Herron IH (2001) Onset of convection in a porous medium with internal heat source and variable gravity. Int J Eng Sci 39(2):201–208

    Article  Google Scholar 

  25. Hirt C, Claessens S, Fecher T, Kuhn M, Pail R,  Rexer M (2013) New ultrahigh‐resolution picture of Earth's gravity field. Geophys Res Lett 40:4279–4283

  26. Hu J, Lin J, Zhang Y, Lin Z, Qiao Z, Liu Z et al (2019) A new anti-biofilm strategy of enabling arbitrary surfaces of materials and devices with robust bacterial anti-adhesion via a spraying modified microsphere method. J Mater Chem A Mater Energy Sustain 7(45):26039–26052. https://doi.org/10.1039/c9ta07236e

    CAS  Article  Google Scholar 

  27. Hu X, Ma P, Wang J, Tan G (2020) A hybrid cascaded DC-DC boost converter with ripple reduction and large conversion ratio. IEEE J Emerg Sel Top Power Electron 8(1):761–770. https://doi.org/10.1109/JESTPE.2019.2895673

    Article  Google Scholar 

  28. Lee S, Choi S-S, Li S, Eastman J (1999) Measuring thermal conductivity of fluids containing oxide nanoparticles. J Heat Transf 121(2):280–289

    CAS  Article  Google Scholar 

  29. Li Q, Wang J, Wang J, Baleta J, Min C, Sundén B (2018) Effects of gravity and variable thermal properties on nanofluid convective heat transfer using connected and unconnected walls. Energy Convers Manag 171:1440–1448

    CAS  Article  Google Scholar 

  30. Liao Q, Wei W, Zuo H, Li X, Yang Z, Xiao S et al (2020) Interfacial bonding enhancement and properties improvement of carbon/copper composites based on nickel doping. Compos Interfaces. https://doi.org/10.1080/09276440.2020.1798681

    Article  Google Scholar 

  31. Lin J, Hu J, Wang W, Liu K, Zhou C, Liu Z et al (2020a) Thermo and light-responsive strategies of smart titanium-containing composite material surface for enhancing bacterially anti-adhesive property. Chem Eng J. https://doi.org/10.1016/j.cej.2020.125783

    Article  Google Scholar 

  32. Lin J, Cai X, Liu Z, Liu N, Xie M, Zhou B et al (2020b) Anti-liquid-interfering and bacterially antiadhesive strategy for highly stretchable and ultrasensitive strain sensors based on cassie-baxter wetting state. Adv Funct Mater 30(23):2000398. https://doi.org/10.1002/adfm.202000398

    CAS  Article  Google Scholar 

  33. Liu X, Pu J, Wang L, Xue Q (2013) Novel DLC/ionic liquid/graphene nanocomposite coatings towards high-vacuum related space applications. J Mater Chem A 1(11):3797–3809

    CAS  Article  Google Scholar 

  34. Liu Y, Yang C, Sun Q, Wu S, Lin S et al (2019a) Enhanced embedding capacity for the SMSD-based data-hiding method. Signal Process Image Commun 78:216–222. https://doi.org/10.1016/j.image.2019.07.013

    Article  Google Scholar 

  35. Liu Y, Zhang Q, Xu M, Yuan H, Chen Y, Zhang J et al (2019b) Novel and efficient synthesis of Ag-ZnO nanoparticles for the sunlight-induced photocatalytic degradation. Appl Surf Sci 476:632–640. https://doi.org/10.1016/j.apsusc.2019.01.137

    CAS  Article  Google Scholar 

  36. Liu Q, Song Z, Han H, Donkor S, Jiang L, Wang W et al (2020) A novel green reinforcement corrosion inhibitor extracted from waste Platanus acerifolia leaves. Constr Build Mater 260:119695. https://doi.org/10.1016/j.conbuildmat.2020.119695

    CAS  Article  Google Scholar 

  37. Mahabaleshwar US, Basavaraja D, Wang S, Lorenzini G, Lorenzini E (2017) Convection in a porous medium with variable internal heat source and variable gravity. Int J Heat Mass Transf 111:651–656

    Article  Google Scholar 

  38. Mahajan A, Sharma MK (2018) The onset of convection in a magnetic nanofluid layer with variable gravity effects. Appl Math Comput 339:622–635

    Article  Google Scholar 

  39. Nield DA,  Bejan A (2006) Convection in porous media. Springer, New York

  40. Nield D, Kuznetsov A (2009) Thermal instability in a porous medium layer saturated by a nanofluid. Int J Heat Mass Transf 52(25–26):5796–5801

    CAS  Article  Google Scholar 

  41. Nield D, Kuznetsov A (2011) The effect of vertical throughflow on thermal instability in a porous medium layer saturated by a nanofluid. Transp Porous Media 87(3):765–775

    CAS  Article  Google Scholar 

  42. Nield D, Kuznetsov A (2015) The effect of vertical throughflow on thermal instability in a porous medium layer saturated by a nanofluid: a revised model. J Heat Transf 137(5):052601

    Article  Google Scholar 

  43. Rao CV, Chakravarthi V, Raju ML (1994) Forward modeling: gravity anomalies of two-dimensional bodies of arbitrary shape with hyperbolic and parabolic density functions. Comput Geosci 20(5):873–880

  44. Rionero S, Straughan B (1990) Convection in a porous medium with internal heat source and variable gravity effects. Int J Eng Sci 28(6):497–503

    CAS  Article  Google Scholar 

  45. Savino R, di Francescantonio N, Fortezza R, Abe Y (2007) Heat pipes with binary mixtures and inverse Marangoni effects for microgravity applications. Acta Astronaut 61(1–6):16–26

    Article  Google Scholar 

  46. Sheikholeslami M, Jafaryar M, Said Z, Alsabery AI, Babazadeh H, Shafee A (2021a) Modification for helical turbulator to augment heat transfer behavior of nanomaterial via numerical approach. Appl Therm Eng 182:115935

    Article  Google Scholar 

  47. Sheikholeslami M, Farshad SA, Shafee A, Babazadeh H (2021b) Performance of solar collector with turbulator involving nanomaterial turbulent regime. Renew Energy 163:1222–1237

    Article  Google Scholar 

  48. Shi M, Narayanasamy M, Yang C, Zhao L, Jiang J, Angaiah S et al (2020) 3D interpenetrating assembly of partially oxidized MXene confined Mn–Fe bimetallic oxide for superior energy storage in ionic liquid. Electrochim Acta 334:135546. https://doi.org/10.1016/j.electacta.2019.135546

    CAS  Article  Google Scholar 

  49. Siddheshwar PG, Kanchana C (2017) Unicellular unsteady Rayleigh–Bénard convection in Newtonian liquids and Newtonian nanoliquids occupying enclosures: New findings. Int J Mech Sci 131–132:1061–1072

  50. Silk EA, Golliher EL, Selvam RP (2008) Spray cooling heat transfer: technology overview and assessment of future challenges for micro-gravity application. Energy Convers Manag 49(3):453–468

    CAS  Article  Google Scholar 

  51. Song Q, Zhao H, Jia J, Yang L, Lv W, Bao J et al (2020a) Pyrolysis of municipal solid waste with iron-based additives: a study on the kinetic, product distribution and catalytic mechanisms. J Clean Prod 258:120682. https://doi.org/10.1016/j.jclepro.2020.120682

    CAS  Article  Google Scholar 

  52. Song Q, Zhao H, Jia J, Yang L, Lv W, Gu Q et al (2020b) Effects of demineralization on the surface morphology, microcrystalline and thermal transformation characteristics of coal. J Anal Appl Pyrol 145:104716. https://doi.org/10.1016/j.jaap.2019.104716

    CAS  Article  Google Scholar 

  53. Sui D, Langåker VH, Yu Z (2017) Investigation of thermophysical properties of nanofluids for application in geothermal energy. Energy Proc 105:5055–5060

    CAS  Article  Google Scholar 

  54. Sun Q, Qu J, Wang Q, Yuan J (2017) Operational characteristics of oscillating heat pipes under micro-gravity condition. Int Commun Heat Mass Transf 88:28–36

    Article  Google Scholar 

  55. Tapley BD, Bettadpur S, Ries JC, Thompson, PF, Watkins MM (2004) GRACE measurements of mass variability in the Earth system. Science 305:503–505

  56. Umavathi J, Yadav D, Mohite MB (2015) Linear and nonlinear stability analyses of double-diffusive convection in a porous medium layer saturated in a Maxwell nanofluid with variable viscosity and conductivity. Elixir Mech Eng 79:30407–30426

  57. Vafai K (2010) Porous media: applications in biological systems and biotechnology. CRC Press, United States

  58. Wan C, Wang L-T, Sha J-Y, Ge H-H (2019) Effect of carbon nanoparticles on the crystallization of calcium carbonate in aqueous solution. Nanomaterials 9(2):179

    CAS  Article  Google Scholar 

  59. Wang W, Guo J, Long C, Li W, Guan J (2015) Flaky carbonyl iron particles with both small grain size and low internal strain for broadband microwave absorption. J Alloy Compd 637:106–111. https://doi.org/10.1016/j.jallcom.2015.02.220

    CAS  Article  Google Scholar 

  60. Wang G, Yao Y, Chen Z, Hu P (2019) Thermodynamic and optical analyses of a hybrid solar CPV/T system with high solar concentrating uniformity based on spectral beam splitting technology. Energy 166:256–266. https://doi.org/10.1016/j.energy.2018.10.089

    Article  Google Scholar 

  61. Wang M, Guo Y, Wang B, Luo H, Zhang X, Wang Q et al (2020) An engineered self-supported electrocatalytic cathode and dendrite-free composite anode based on 3D double-carbon hosts for advanced Li–SeS2 batteries. J Mater Chem A 8(6):2969–2983. https://doi.org/10.1039/C9TA11124G

    CAS  Article  Google Scholar 

  62. Won Y, Cho J, Agonafer D, Asheghi M, Goodson KE (2015) Fundamental cooling limits for high power density gallium nitride electronics. IEEE Trans Compon Packag Manuf Technol 5(6):737–744

    CAS  Article  Google Scholar 

  63. Wong KV, De Leon O (2017) Applications of nanofluids: current and future. In: Nanotechnology and energy. Jenny Stanford Publishing Singapore, pp 105–132

  64. Yadav D (2014) Hydrodynamic and hydromagnetic instability in nanofluids. Lambert Academic Publishing, Germany

  65. Yadav D (2019a) Numerical investigation of the combined impact of variable gravity field and throughflow on the onset of convective motion in a porous medium layer. Int Commun Heat Mass Transf 108:104274

    Article  Google Scholar 

  66. Yadav D (2019b) The effect of pulsating throughflow on the onset of magneto convection in a layer of nanofluid confined within a Hele-Shaw cell. Proc Inst Mech Eng Part E J Process Mech Eng 0(0):1–12

    Google Scholar 

  67. Yadav D (2019c) Impact of chemical reaction on the convective heat transport in nanofluid occupying in porous enclosures: a realistic approach. Int J Mech Sci 157–158:357–373

    Article  Google Scholar 

  68. Yadav D (2019d) The onset of longitudinal convective rolls in a porous medium saturated by a nanofluid with non-uniform internal heating and chemical reaction. J Therm Anal Calorim 135(2):1107–1117

    CAS  Article  Google Scholar 

  69. Yadav D (2020a) The density-driven nanofluid convection in an anisotropic porous medium layer with rotation and variable gravity field: a numerical investigation. J Appl Comput Mech 6(3):699–712

    Google Scholar 

  70. Yadav D (2020b) Numerical solution of the onset of Buoyancy-driven nanofluid convective motion in an anisotropic porous medium layer with variable gravity and internal heating. Heat Transf Asian Res 49(3):1170–1191

    Article  Google Scholar 

  71. Yadav D (2020c) The onset of Darcy-Brinkman convection in a porous medium layer with vertical throughflow and variable gravity field effects. Heat Transf Asian Res 49(5):3161–3173

    Article  Google Scholar 

  72. Yadav D, Wang J (2019) Convective heat transport in a heat generating porous layer saturated by a non-Newtonian nanofluid. Heat Transf Eng 40(16):1363–1382

    CAS  Article  Google Scholar 

  73. Yadav D, Bhargava R, Agrawal GS (2012a) 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

    Article  Google Scholar 

  74. Yadav D, Agrawal G, Bhargava R (2012b) The onset of convection in a binary nanofluid saturated porous layer. Int J Theor Appl Multiscale Mech 2(3):198–224

    CAS  Article  Google Scholar 

  75. Yadav D, Bhargava R, Agrawal GS, Yadav N, Lee J, Kim MC (2014a) Thermal instability in a rotating porous layer saturated by a non-Newtonian nanofluid with thermal conductivity and viscosity variation. Microfluid Nanofluid 16(1–2):425–440

    CAS  Article  Google Scholar 

  76. Yadav D, Bhargava R, Agrawal GS, Hwang GS, Lee J, Kim MC (2014b) Magneto-convection in a rotating layer of nanofluid. Asia Pac J Chem Eng 9(5):663–677

    CAS  Article  Google Scholar 

  77. Yadav D, Lee J, Cho HH (2016) Throughflow and quadratic drag effects on the onset of convection in a Forchheimer-extended Darcy porous medium layer saturated by a nanofluid. J Braz Soc Mech Sci Eng 38(8):2299–2309

    CAS  Article  Google Scholar 

  78. Yadav D, Mohamad A, Rana G (2020) Effect of throughflow on the convective instabilities in an anisotropic porous medium layer with inconstant gravity. J Appl Comput Mech. https://doi.org/10.22055/jacm.2020.32381.2006

  79. Yan H, Xue X, Chen W, Wu X, Dong J, Liu Y et al (2020) Reversible Na+ insertion/extraction in conductive polypyrrole-decorated NaTi2(PO4)3 nanocomposite with outstanding electrochemical property. Appl Surf Sci 530:147295. https://doi.org/10.1016/j.apsusc.2020.147295

    CAS  Article  Google Scholar 

  80. Yang Y, Liu J, Yao J, Kou J, Li Z, Wu T et al (2020a) Adsorption behaviors of shale oil in kerogen slit by molecular simulation. Chem Eng J 387:124054. https://doi.org/10.1016/j.cej.2020.124054

    CAS  Article  Google Scholar 

  81. Yu X, Zhang J, Zhang J, Niu J, Zhao J, Wei Y et al (2019) Photocatalytic degradation of ciprofloxacin using Zn-doped Cu2O particles: analysis of degradation pathways and intermediates. Chem Eng J 374:316–327. https://doi.org/10.1016/j.cej.2019.05.177

    CAS  Article  Google Scholar 

  82. Yu H, He Z, Qian G, Gong X, Qu X (2020) Research on the anti-icing properties of silicone modified polyurea coatings (SMPC) for asphalt pavement. Constr Build Mater 242:117793. https://doi.org/10.1016/j.conbuildmat.2019.117793

    CAS  Article  Google Scholar 

  83. Zargartalebi H, Ghalambaz M, Sheremet MA, Pop I (2017) Unsteady free convection in a square porous cavity saturated with nanofluid: the case of local thermal nonequilibrium and Buongiorno’s mathematical models. J Porous Media 20(11):999–1016

  84. Zhang W (2020) Parameter adjustment strategy and experimental development of hydraulic system for wave energy power generation. Symmetry 12(5):711. https://doi.org/10.3390/sym12050711

    Article  Google Scholar 

  85. Zhang Y, Huang P (2019) Influence of mine shallow roadway on airflow temperature. Arab J Geosci. https://doi.org/10.1007/s12517-019-4934-7

    Article  Google Scholar 

  86. Zhang Z, Li W, Kan J, Xu D (2016) Theoretical and experimental analysis of a solar thermoelectric power generation device based on gravity-assisted heat pipes and solar irradiation. Energy Convers Manag 127:301–311

    Article  Google Scholar 

  87. Zhao H, Li Y, Song Q, Liu S, Yan J, Wang X et al (2019) Investigation on the physicochemical structure and gasification reactivity of nascent pyrolysis and gasification char prepared in the entrained flow reactor. Fuel 240:126–137. https://doi.org/10.1016/j.fuel.2018.11.145

    CAS  Article  Google Scholar 

  88. Zhong P, Lin H, Wang L, Mo Z, Meng X, Tang H et al (2020) Electrochemically enabled synthesis of sulfide imidazopyridines via a radical cyclization cascade. Green Chem Int J Green Chem Resour GC. https://doi.org/10.1039/D0GC02125C

    Article  Google Scholar 

  89. Zhu W, Zhang Z, Chen D, Chai W, Chen D, Zhang J et al (2020) Interfacial voids trigger carbon-based, all-inorganic CsPbIBr2 perovskite solar cells with photovoltage exceeding 1.33 V. Nanomicro Lett 12(1):1–14. https://doi.org/10.1007/s40820-020-00425-1

    CAS  Article  Google Scholar 

  90. Zuo H, Salahshoor Z, Yadav D, Hajizadeh MR, Vuong BX (2020) Investigation of thermal treatment of hybrid nanoparticles in a domain with different permeabilities. J Therm Anal Calorim (2020). https://doi.org/10.1007/s10973-020-09824-3

Download references


The authors would like to acknowledge the University of Nizwa for continuous support during this research. Also, the article partially was supported by National Natural Science Foundation of China (No. 71601072) and Key Scientific Research Project of Higher Education Institutions in Henan Province of China (No. 20B110006). Besides, this research was supported by the National Natural Science Foundation of China (Grant Nos. 11971142, 11701176, 11626101, 11871202, 61673169, 11601485).

Author information



Corresponding author

Correspondence to Yu-Ming Chu.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yadav, D., Chu, YM. & Li, Z. Examination of the nanofluid convective instability of vertical constant throughflow in a porous medium layer with variable gravity. Appl Nanosci (2021). https://doi.org/10.1007/s13204-021-01700-2

Download citation


  • Nanomaterial
  • Convective instability
  • Throughflow and inconsistency gravity
  • Permeable medium