, Volume 54, Issue 3, pp 451–469 | Cite as

Effect of trigonometric sine, square and triangular wave-type time-periodic gravity-aligned oscillations on Rayleigh–Bénard convection in Newtonian liquids and Newtonian nanoliquids

  • P. G. Siddheshwar
  • C. KanchanaEmail author


The influence of trigonometric sine, square and triangular wave-types of time-periodic gravity-aligned oscillations on Rayleigh–Bénard convection in Newtonian liquids and in Newtonian nanoliquids is studied in the paper using the generalized Buongiorno two-phase model. The five-mode Lorenz model is derived under the assumptions of Boussinesq approximation, small-scale convective motion and some slip mechanisms. Using the method of multiscales, the Lorenz model is transformed to a Ginzburg–Landau equation the solution of which helps in quantifying the heat transport through the Nusselt number. Enhancement of heat transport in Newtonian liquids due to the presence of nanoparticles/nanotubes is clearly explained. The study reveals that all the three wave types of gravity modulation delay the onset of convection and thereby to a diminishment of heat transport. It is also found that in the case of trigonometric sine type of gravity modulation heat transport is intermediate to that of the cases of triangular and square types. The paper is the first such work that attempts to theoretically explain the effect of three different wave-types of gravity modulation on onset of convection and heat transport in the presence/absence of nanoparticles/nanotubes. Comparing the heat transport by the single-phase and by the generalized two-phase models, the conclusion is that the single-phase model under-predicts heat transport in nanoliquids irrespective of the type of gravity modulation being effected on the system. The results of the present study reiterate the findings of related experimental and numerical studies.


Nanoliquid Two-phase model Rayleigh–Bénard convection Lorenz model Ginzburg–Landau equation Gravity modulation Triangular Sinusoidal Square Wave forms Onset Heat transfer 



The authors are grateful to the Bangalore University for support. The authors are grateful to the Reviewers of the paper and the Editor for their valuable comments.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Abu-Nada E, Masoud Z, Hijazi A (2008) Natural convection heat transfer enhancement in horizontal concentric annuli using nanofluids. Int Commun Heat Mass Transf 35(5):657–665CrossRefGoogle Scholar
  2. 2.
    Agarwal S, Bhadauria B (2014) Convective heat transport by longitudinal rolls in dilute nanoliquids. J Nanofluids 3(4):380–390CrossRefGoogle Scholar
  3. 3.
    Agarwal S, Bhadauria BS, Siddheshwar PG (2011) Thermal instability of a nanofluid saturating a rotating anisotropic porous medium. Spec Top Rev Porous Media: Int J 2(1):53–64CrossRefGoogle Scholar
  4. 4.
    Angayarkanni SA, Philip J (2015) Review on thermal properties of nanofluids: recent developments. Adv Colloid Interface Sci 225(Supplement C):146–176CrossRefGoogle Scholar
  5. 5.
    Azmi WH, Sharma KV, Mamat R, Najafi G, Mohamad MS (2016) The enhancement of effective thermal conductivity and effective dynamic viscosity of nanofluids: a review. Renew Sustain Energy Rev 53(Supplement C):1046–1058CrossRefGoogle Scholar
  6. 6.
    Bhadauria BS (2006) Time-periodic heating of Rayleigh–Bénard convection in a vertical magnetic field. Physica Scripta 73(3):296–302ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Biringen S, Peltier LJ (1990a) Computational study of 3-D Bénard convection with gravitational modulation. Phys Fluids A 2:279–283CrossRefGoogle Scholar
  8. 8.
    Biringen S, Peltier LJ (1990b) Numerical simulation of 3-D Bénard convection with gravitational modulation. Phys Fluids A: Fluid Dyn (1989–1993) 2(5):754–764Google Scholar
  9. 9.
    Boulal T, Aniss S, Belhaq M, Rand R (2007) Effect of quasiperiodic gravitational modulation on the stability of a heated fluid layer. Phys Rev E 76(5):056320ADSCrossRefGoogle Scholar
  10. 10.
    Brinkman HC (1952) The viscosity of concentrated suspensions and solutions. J Chem Phys 20:571–571ADSCrossRefGoogle Scholar
  11. 11.
    Buongiorno J (2006) Convective transport in nanofluids. ASME J Heat Transf 128(3):240–250CrossRefGoogle Scholar
  12. 12.
    Chandrasekhar S (1961) Hydrodynamic and hydromagnetic stability. Clarendon Press, OxfordzbMATHGoogle Scholar
  13. 13.
    Choi CK (1995) Enhancing thermal conductivity of fluids with nanoparticles. ASME-Publ-Fed 231:99–106Google Scholar
  14. 14.
    Colangelo G, Favale E, Milanese M, de Risi A, Laforgia D (2017) Cooling of electronic devices: nanofluids contribution. Appl Therm Eng 127(Supplement C):421–435CrossRefGoogle Scholar
  15. 15.
    Elhajjar B, Bachir G, Mojtabi A, Fakih C, Charrier-Mojtabi MC (2010) Modeling of Rayleigh–Bénard natural convection heat transfer in nanofluids. Comptes Rendus Méc 338(6):350–354ADSCrossRefzbMATHGoogle Scholar
  16. 16.
    Gershuni GZ, Zhukhovitskii EM (1963) On parametric excitation of convective instability. J Appl Math Mech 27(5):1197–1204MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gershuni GZ, Zhukhovitskii EM, Iurkov IS (1970) On convective stability in the presence of periodically varying parameter. J Appl Math Mech 34(3):442–452CrossRefGoogle Scholar
  18. 18.
    Ghasemi B, Aminossadati S (2009) Natural convection heat transfer in an inclined enclosure filled with a water-cuo nanofluid. Numer Heat Transf Part A: Appl 55(8):807–823ADSCrossRefGoogle Scholar
  19. 19.
    Gresho PM, Sani RL (1970) The effects of gravity modulation on the stability of a heated fluid layer. J Fluid Mech 40:783–806ADSCrossRefzbMATHGoogle Scholar
  20. 20.
    Hamilton RL, Crosser OK (1962) Thermal conductivity of heterogeneous two-component systems. Ind Eng Chem Fundam 1:187–191CrossRefGoogle Scholar
  21. 21.
    Jou RY, Tzeng SC (2006) Numerical research of nature convective heat transfer enhancement filled with nanofluids in rectangular enclosures. Int Commun Heat Mass Transf 33(6):727–736CrossRefGoogle Scholar
  22. 22.
    Kanchana C, Zhao Y (2018) Effect of internal heat generation/absorption on Rayleigh–Bénard convection in water well-dispersed with nanoparticles or carbon nanotubes. Int J Heat Mass Transf 127:1031–1047CrossRefGoogle Scholar
  23. 23.
    Kanchana C, Zhao Y, Siddheshwar PG (2018) A comparative study of individual influences of suspended multiwalled carbon nanotubes and alumina nanoparticles on Rayleigh–Bénard convection in water. Phys Fluids 30:084101–114ADSCrossRefGoogle Scholar
  24. 24.
    Khanafer K, Vafai K, Lightstone M (2003) Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. Int J Heat Mass Transf 46(19):3639–3653CrossRefzbMATHGoogle Scholar
  25. 25.
    Kim J, Kang YT, Choi CK (2004) Analysis of convective instability and heat transfer characteristics of nanofluids. Phys Fluids 16(7):2395–2401ADSCrossRefzbMATHGoogle Scholar
  26. 26.
    Kim J, Choi CK, Kang YT, Kim MG (2006) Effects of thermodiffusion and nanoparticles on convective instabilities in binary nanofluids. Nanoscale Microscale Thermophys Eng 10(1):29–39CrossRefGoogle Scholar
  27. 27.
    Maheshwary PB, Handa CC, Nemade KR (2017) A comprehensive study of effect of concentration, particle size and particle shape on thermal conductivity of titania/water based nanofluid. Appl Therm Eng 119(Supplement C):79–88CrossRefGoogle Scholar
  28. 28.
    Malashetty MS, Padmavathi V (1997) Effect of gravity modulation on the onset of convection in a fluid and porous layer. Int J Eng Sci 35(9):829–840MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Meenakshi N, Siddheshwar PG (2017) A theoretical study of enhanced heat transfer in nanoliquids with volumetric heat source. J Appl Math Comput.
  30. 30.
    Murshed SMS, Leong KC, Yang C (2005) Enhanced thermal conductivity of \({T}i O_2\) water based nanofluids. Int J Therm Sci 44(4):367–373CrossRefGoogle Scholar
  31. 31.
    Nield DA, Kuznetsov AV (2009) Thermal instability in a porous medium layer saturated by a nanofluid. Int J Heat Mass Transf 52(25):5796–5801CrossRefzbMATHGoogle Scholar
  32. 32.
    Noghrehabadi A, Samimi A (2012) Natural convection heat transfer of nanofluids due to thermophoresis and Brownian diffusion in a square enclosure. Int J Eng Adv Technol 1:81–93Google Scholar
  33. 33.
    Pinto RV, Fiorelli FAS (2016) Review of the mechanisms responsible for heat transfer enhancement using nanofluids. Appl Therm Eng 108(Supplement C):720–739CrossRefGoogle Scholar
  34. 34.
    Roberts NA, Walker DG (2010) Convective performance of nanofluids in commercial electronics cooling systems. Appl Therm Eng 30(16):2499–2504CrossRefGoogle Scholar
  35. 35.
    Sheremet MA, Pop I, Nazar R (2015) Natural convection in a square cavity filled with a porous medium saturated with a nanofluid using the thermal nonequilibrium model with a Tiwari and Das nanofluid model. Int J Mech Sci 100:312–321CrossRefGoogle Scholar
  36. 36.
    Shima PD, Philip J (2014) Role of thermal conductivity of dispersed nanoparticles on heat transfer properties of nanofluid. Ind Eng Chem Res 3(2):980–988CrossRefGoogle Scholar
  37. 37.
    Shu Y, Li BQ, Groh DHC (2002) Magnetic damping of g-jitter induced double-diffusive convection. Numer Heat Transf: Part A: Appl 42(4):345–364ADSCrossRefGoogle Scholar
  38. 38.
    Siddheshwar PG (2010) A series solution for the Ginzburg–Landau equation with a time-periodic coefficient. Appl Math 1(06):542–554CrossRefGoogle Scholar
  39. 39.
    Siddheshwar PG, Abraham A (2007) Rayleigh–Bénard convection in a dielectric liquid: time-periodic body force. Proc Appl Math Mech 7(1):2100083–2100084CrossRefGoogle Scholar
  40. 40.
    Siddheshwar PG, Kanchana C (2017) Unicellular unsteady Rayleigh–Bénard convection in Newtonian liquids and Newtonian nanoliquids occupying enclosures: new findings. Int J Mech Sci 131–132:1061–1072CrossRefGoogle Scholar
  41. 41.
    Siddheshwar PG, Kanchana C (2018) A study of unsteady, unicellular Rayleigh–Bénard convection of nanoliquids in enclosures using additional modes. J Nanofluids 7:791–800CrossRefGoogle Scholar
  42. 42.
    Siddheshwar PG, Meenakshi N (2017) Amplitude equation and heat transport for Rayleigh–Bénard convection in Newtonian liquids with nanoparticles. Int J Appl Comput Math 3(1):271–291MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Siddheshwar PG, Revathi BR (2013) Effect of gravity modulation on weakly non-linear stability of stationary convection in a dielectric liquid. World Acad Sci Eng Technol 7(1):119–124Google Scholar
  44. 44.
    Siddheshwar PG, Veena BN (2018) A theoretical study of natural convection of water-based nanoliquids in low-porosity enclosures using single-phase model. J Nanofluids 7:163–174CrossRefGoogle Scholar
  45. 45.
    Siddheshwar PG, Sekhar GN, Jayalatha G (2010) Effect of time-periodic vertical oscillations of the Rayleigh–Bénard system on nonlinear convection in viscoelastic liquids. J Non-Newtonian Fluid Mech 165(19):1412–1418CrossRefzbMATHGoogle Scholar
  46. 46.
    Siddheshwar PG, Bhadauria BS, Mishra P, Srivastava AK (2012) Study of heat transport by stationary magneto-convection in a Newtonian liquid under temperature or gravity modulation using Ginzburg–Landau model. Int J Non-Linear Mech 47(5):418–425CrossRefGoogle Scholar
  47. 47.
    Siddheshwar PG, Kanchana C, Kakimoto Y, Nakayama A (2016a) Steady finite-amplitude Rayleigh–Bénard convection in nanoliquids using a two-phase model: theoretical answer to the phenomenon of enhanced heat transfer. ASME J Heat Transf 139(1):012402–18CrossRefGoogle Scholar
  48. 48.
    Siddheshwar PG, Kanchana C, Kakimoto Y, Nakayama A (2016b) Study of heat transport in Newtonian water-based nanoliquids using two-phase model and Ginzburg-Landau approach. In: Proceedings of Vignana Bharathi Golden Jubilee Volume, Bangalore University, India 0971–6882(1), 85–101Google Scholar
  49. 49.
    Tiwari RK, Das MK (2007) Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int J Heat Mass Transf 50(9–10):2002–2018CrossRefzbMATHGoogle Scholar
  50. 50.
    Tzou D (2008a) Instability of nanofluids in natural convection. ASME J Heat Transf 130(7):072401–19CrossRefGoogle Scholar
  51. 51.
    Tzou D (2008b) Thermal instability of nanofluids in natural convection. Int J Heat Mass Transf 51(11):2967–2979CrossRefzbMATHGoogle Scholar
  52. 52.
    Umavathi JC (2015) Rayleigh–Bénard convection subject to time dependent wall temperature in a porous medium layer saturated by a nanofluid. Meccanica 50(4):981–994MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Usri NA, Azmi WH, Mamat R, Hamid KA, Najafi G (2015) Thermal conductivity enhancement of \(Al_2O_3\) nanofluid in ethylene glycol and water mixture. Energy Procedia 79(Supplement C):397–402CrossRefGoogle Scholar
  54. 54.
    Venezian G (1969) Effect of modulation on the onset of thermal convection. J Fluid Mech 35:243–254ADSCrossRefzbMATHGoogle Scholar
  55. 55.
    Wang X-Q, Mujumdar AS (2007) Heat transfer characteristics of nanofluids: a review. Int J Therm Sci 46(1):1–19CrossRefGoogle Scholar
  56. 56.
    Wheeler AA, Mc Fadden GB, Murray BT, Coriell SR (1991) Convective stability in the Rayleigh–Bénard and directional solidification problems: high-frequency gravity modulation. Phys Fluids A: Fluid Dyn 3(12):2847–2858ADSCrossRefzbMATHGoogle Scholar
  57. 57.
    Yadav D, Agrawal S, Bhargava R (2011) Thermal instability of rotating nanofluid layer. Int J Eng Sci 49(11):1171–1184MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Yu Y, Chan CL, Chen CF (2007) Effect of gravity modulation on the stability of a horizontal double diffusive layer. J Fluid Mech 589:183–213ADSCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019
corrected publication March 2019

Authors and Affiliations

  1. 1.Department of MathematicsBangalore UniversityBangaloreIndia
  2. 2.Harbin Institute of Technology, ShenzhenShenzhenChina

Personalised recommendations