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A theoretical study of enhanced heat transfer in nanoliquids with volumetric heat source

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Abstract

Rayleigh–Bénard convection in nanoliquids is studied in the presence of volumetric heat source. The present analytical work concerns twenty nanoliquids. Carrier liquids considered are water, ethylene glycol, engine oil and glycerine and with them five different nanoparticles considered are copper, copper oxide, silver, alumina and titania. Expression for the thermophysical properties of the nanoliquids is chosen from phenomenological laws or mixture theory. Heat source is characterized by an internal nanoliquid Rayleigh number \(R_{I_{nl}}\). Heat source adds to the energy of the system and hence an advanced onset is observed in this case compared to the problem with no heat source. In the case of heat sink, however, heat is drawn from the system leading to delay in onset. The individual effect of all the nanoparticles is to advance convection. Enhanced heat transport situation is observed in each of the nanoliquids with engine-oil-silver transporting maximum heat and water-titania the least. Additional Fourier modes are found not to have any profound effect on the results predicted by minimal modes. The connection between the Lorenz model and the Ginzburg–Landau model is clearly shown in the paper.

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References

  1. Adomian, G.: A review of the decomposition method and some recent results for nonlinear equations. Comput. Math. Appl. 21, 101–127 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method, vol. 60. Springer, Berlin (2013)

    MATH  Google Scholar 

  3. Agarwal, S., Bhadauria, B.S.: Convective heat transport by longitudinal rolls in dilute nanoliquids. J. Nanofluids 3, 380–390 (2014)

    Article  Google Scholar 

  4. Aghighi, S., Ammar, A., Metivier, C., Chinesta, F.: Parametric solution of the Rayleigh–Bénard convection model by using the PGD: Application to nanofluids. Int. J. Numer. Methods Heat Fluid Flow 25, 1252–1281 (2015)

    Article  MATH  Google Scholar 

  5. Bergman, T.L., Incropera, F.P., Lavine, A.S.: Fundamentals of Heat and Mass Transfer. John Wiley and Sons, New York (2011)

    Google Scholar 

  6. Bhadauria, B., Siddheshwar, P.G., Singh, A.K., Gupta, V.K.: A local nonlinear stability analysis of modulated double diffusive stationary convection in a couple stress liquid. J. Appl. Fluid Mech. 9, 1255–1264 (2016)

    Article  Google Scholar 

  7. Bhadauria, B.S., Kiran, P.: Chaotic and oscillatory magneto-convection in a binary viscoelastic fluid under g-jitter. Int. J. Heat Mass Transf. 84, 610–624 (2015)

    Article  Google Scholar 

  8. Bhatia, P.K., Steiner, J.M.: Thermal instability in a viscoelastic fluid layer in hydromagnetics. J. Math. Anal. Appl. 41(2), 271–283 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bianco, V., Manca, O., Nardini, S., Kambiz, V.: Heat Tansfer Enhancement with Nanofluids. CRC Press, Canada (2015)

    Book  Google Scholar 

  10. Brinkman, H.C.: The viscosity of concentrated suspensions and solutions. J. Chem. Phys. 20, 571–571 (1952)

    Article  Google Scholar 

  11. Buongiorno, J.: Convective transport in nanofluids. J. Heat Transf. 128, 240–250 (2005)

    Article  Google Scholar 

  12. Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford (1961)

    MATH  Google Scholar 

  13. Choi, S.U.S.: Enhancing thermal conductivity of fluids with nanoparticles. ASME Publ. Fed 231, 99–106 (1995)

    Google Scholar 

  14. Clever, R.M.: Heat transfer and stability properties of convection rolls in an internally heated fluid layer. Z. Angew. Math. Phys. 28, 585–597 (1977)

    Article  MATH  Google Scholar 

  15. Corcione, M.: Rayleigh–Bénard convection heat transfer in nanoparticle suspensions. Int. J. Heat Fluid Flow 32, 65–77 (2011)

    Article  Google Scholar 

  16. Das, S.K., Putra, N., Thiesen, P., Roetzel, W.: Temperature dependence of thermal conductivity enhancement for nanofluids. J. Heat Transf. 125, 567–574 (2003)

    Article  Google Scholar 

  17. Devi, R., Mahajan, A., et al.: Global stability for thermal convection in a couple-stress fluid. Int. Commun. Heat Mass Transf. 38, 938–942 (2011)

    Article  Google Scholar 

  18. Drazin, P.G., Reid, W.H.: Hydrodynamic Stability. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  19. Elhajjar, B., Bachir, G., Mojtabi, A., Fakih, C., Charrier-Mojtabi, M.C.: Modeling of Rayleigh–Bénard natural convection heat transfer in nanofluids. Comptes Rendus Mcanique 338, 350–354 (2010)

    Article  MATH  Google Scholar 

  20. Eslamian, M., Ahmed, M., El-Dosoky, M.F., Saghir, M.Z.: Effect of thermophoresis on natural convection in a Rayleigh–Bénard cell filled with a nanofluid. Int. J. Heat Mass Transf. 81, 142–156 (2015)

    Article  Google Scholar 

  21. Fatoorehchi, H., Abolghasemi, H.: Investigation of nonlinear problems of heat conduction in tapered cooling fins via symbolic programming. Appl. Appl. Math. 7, 717–734 (2012)

    MathSciNet  MATH  Google Scholar 

  22. Fatoorehchi, H., Abolghasemi, H.: Approximating the minimum reflux ratio of multicomponent distillation columns based on the Adomian decomposition method. J. Taiwan Inst. Chem. Eng. 45, 880–886 (2014)

    Article  Google Scholar 

  23. Fatoorehchi, H., Abolghasemi, H., Zarghami, R.: Analytical approximate solutions for a general nonlinear resistor-nonlinear capacitor circuit model. Appl. Math. Model. 39, 6021–6031 (2015)

    Article  MathSciNet  Google Scholar 

  24. Gaikwad, S.N., Malashetty, M.S., Prasad, K.R.: An analytical study of linear and non-linear double diffusive convection with soret and dufour effects in couple stress fluid. Int. J. Nonlinear Mech. 42, 903–913 (2007)

    Article  Google Scholar 

  25. Garoosi, F., Bagheri, G., Talebi, F.: Numerical simulation of natural convection of nanofluids in a square cavity with several pairs of heaters and coolers (HACs) inside. Int. J. Heat Mass Transf. 67, 362–376 (2013)

    Article  Google Scholar 

  26. Garoosi, F., Garoosi, S., Hooman, K.: Numerical simulation of natural convection and mixed convection of the nanofluid in a square cavity using Buongiorno model. Powder Tech. 268, 279–292 (2014)

    Article  Google Scholar 

  27. Getling, A.V.: Rayleigh–Bénard Convection: Structures and Dynamics. World Scientific, Singapore (1998)

    Book  MATH  Google Scholar 

  28. Gupta, U., Sharma, G.: On Rivlin–Erickson elastico-viscous fluid heated and soluted from below in the presence of compressibility, rotation and hall currents. J. Appl. Math. Comput. 25, 51–66 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hamilton, R.L., Crosser, O.K.: Thermal conductivity of heterogeneous two-component systems. Ind. Eng. Chem. Fund. 1, 187–191 (1962)

    Article  Google Scholar 

  30. Jawdat, J.M., Hashim, I., Bhadauria, B.S., Momani, S.: On onset of chaotic convection in couple-stress fluids. Math. Model. Anal. 19(3), 359–370 (2014)

    Article  MathSciNet  Google Scholar 

  31. Jawdat, J.M., Hashim, I., Momani, S.: Dynamical system analysis of thermal convection in a horizontal layer of nanofluids heated from below. Math. Probl. Eng. 2012, 1–13 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kakac, S., Pramuanjaroenkij, A.: Review of convective heat transfer enhancement with nanofluids. Int. J. Heat Mass Transf. 52, 3187–3196 (2009)

    Article  MATH  Google Scholar 

  33. Khanafer, K., Vafai, K., Lightstone, M.: Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. Int. J. Heat Mass Transf. 46, 3639–3653 (2003)

    Article  MATH  Google Scholar 

  34. Kim, J., Choi, C.K., Kang, Y.T., Kim, M.G.: Effects of thermodiffusion and nanoparticles on convective instabilities in binary nanofluids. Nanoscale Microscale Thermophys. Eng. 10, 29–39 (2006)

    Article  Google Scholar 

  35. Kim, J., Kang, Y.T., Choi, C.K.: Analysis of convective instability and heat transfer characteristics of nanofluids. Phys. Fluids 16, 2395–2401 (2004)

    Article  MATH  Google Scholar 

  36. Kim, J., Kang, Y.T., Choi, C.K.: Soret and dufour effects on convective instabilities in binary nanofluids for absorption application. Int. J. Refrig. 30, 323–328 (2007)

    Article  Google Scholar 

  37. Krishnamurti, R.: Convection induced by selective absorption of radiation: A laboratory model of conditional instability. Dyn. Atmos. Oceans 27, 367–382 (1998)

    Article  Google Scholar 

  38. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)

    Article  MATH  Google Scholar 

  39. Magyari, E.: Comment on the homogeneous nanofluid models applied to convective heat transfer problems. Acta Mech. 222, 381–385 (2011)

    Article  MATH  Google Scholar 

  40. Malashetty, M.S., Gaikwad, S.N., Swamy, M.: An analytical study of linear and non-linear double diffusive convection with soret effect in couple stress liquids. Int. J. Therm. Sci. 45, 897–907 (2006)

    Article  Google Scholar 

  41. McKenzie, D.P., Roberts, J.M., Weiss, N.O.: Convection in the Earth’s mantle: Towards a numerical simulation. J. Fluid Mech. 62, 465–538 (1974)

    Article  MATH  Google Scholar 

  42. Narayana, M., Sibanda, P., Siddheshwar, P.G., Jayalatha, G.: Linear and nonlinear stability analysis of binary viscoelastic fluid convection. Appl. Math. Model. 37, 8162–8178 (2013)

    Article  MathSciNet  Google Scholar 

  43. Nield, D.A., Kuznetsov, A.V.: The onset of convection in a horizontal nanofluid layer of finite depth. Eur. J. Mech. B/Fluids 29, 217–223 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Nield, D.A., Kuznetsov, A.V.: The onset of convection in an internally heated nanofluid layer. J. Heat Transf. 136(1), 014501 (2014)

    Article  Google Scholar 

  45. Park, H.M.: Rayleigh–Bénard convection of nanofluids based on the pseudo-single-phase continuum model. Int. J. Therm. Sci. 90, 267–278 (2015)

    Article  Google Scholar 

  46. Payne, L.E., Straughan, B.: Critical Rayleigh numbers for oscillatory and nonlinear convection in an isotropic thermomicropolar fluid. Int. J. Eng. Sci. 27, 827–836 (1989)

    Article  MATH  Google Scholar 

  47. Platten, J.K., Legros, J.C.: Convection in Liquids. Springer, Berlin (2012)

    MATH  Google Scholar 

  48. Putra, N., Roetzel, W., Das, S.K.: Natural convection of nano-fluids. Heat Mass Transf. 39, 775–784 (2003)

    Article  MATH  Google Scholar 

  49. Riahi, N., Hsui, A.T.: Nonlinear double-diffusive convection with local heat and solute sources. Int. J. Eng. Sci. 24, 529–544 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  50. Roberts, P.H.: Convection in horizontal layers with internal heat generation. J. Fluid Mech. 30, 33–49 (1967)

    Article  Google Scholar 

  51. Rudraiah, N., Siddheshwar, P.G.: Effect of non-uniform basic temperature gradient on the onset of Marangoni convection in a fluid with suspended particles. Aerosp. Sci. Tech. 4, 517–523 (2000)

    Article  MATH  Google Scholar 

  52. Sharma, A., Shandil, R.G.: Effect of magnetic field dependent viscosity on ferroconvection in the presence of dust particles. J. Appl. Math. Comput. 27, 7–22 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  53. Shivakumara, I.S., Kumar, S.B.N.: Linear and weakly nonlinear triple diffusive convection in a couple stress fluid layer. Int. J. Heat Mass Transf. 68, 542–553 (2014)

    Article  Google Scholar 

  54. Siddheshwar, P.G., Kanchana, C.: Unicellular, unsteady Rayleigh–Bénard convection in Newtonian liqenclosure Newtonain nanloquids occupying enclosures. Int. J. Mech. Sci. (2017) (In press)

  55. Siddheshwar, P.G., Kanchana, C., Kakimoto, Y., Nakayama, A.: 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, 012402 (2017)

    Article  Google Scholar 

  56. Siddheshwar, P.G., Meenakshi, N.: Amplitude equation and heat transport of Rayleigh–Bénard convection in Newtonian liquids with nanoparticles. Int. J. Appl. Comput. Math. 2, 1–22 (2015)

    Article  Google Scholar 

  57. Siddheshwar, P.G., Pranesh, S.: Effect of a non-uniform basic temperature gradient on Rayleigh–Bénard convection in a micropolar fluid. Int. J. Eng. Sci. 36, 1183–1196 (1998)

    Article  Google Scholar 

  58. Siddheshwar, P.G., Pranesh, S.: Magnetoconvection in a micropolar fluid. Int. J. Eng. Sci. 36, 1173–1181 (1998)

    Article  Google Scholar 

  59. Siddheshwar, P.G., Pranesh, S.: Suction-injection effects on the onset of Rayleigh–Bénard–Marangoni convection in a fluid with suspended particles. Acta Mech. 152, 241–252 (2001)

    Article  MATH  Google Scholar 

  60. Siddheshwar, P.G., Pranesh, S.: Magnetoconvection in fluids with suspended particles under 1g and \(\mu \)g. Aerosp. Sci. Tech. 6, 105–114 (2002)

    Article  MATH  Google Scholar 

  61. Siddheshwar, P.G., Pranesh, S.: An analytical study of linear and non-linear convection in Boussinesq–Stokes suspensions. Int. J. Nonlin. Mech. 39, 165–172 (2004)

    Article  MATH  Google Scholar 

  62. Siddheshwar, P.G., Sakshath, T.N.: Rayleigh–Bénard–Taylor convection of Newtonian nanoliquid. World Acad. Sci. Eng. Technol. Int. J. Mech. Aerosp. Ind. Mech. Manuf. Eng. 11, 1131–1135 (2017)

    Google Scholar 

  63. Siddheshwar, P.G., Sekhar, G.N., Jayalatha, G.: Effect of time-periodic vertical oscillations of the Rayleigh–Bénard system on nonlinear convection in viscoelastic liquids. J. Non Newtonian Fluid Mech. 165, 1412–1418 (2010)

    Article  MATH  Google Scholar 

  64. Siddheshwar, P.G., Sekhar, G.N., Jayalatha, G.: Surface tension driven convection in viscoelastic liquids with thermorheological effect. Int. Commun. Heat Mass Transf. 38, 468–473 (2011)

    Article  Google Scholar 

  65. Siddheshwar, P.G., Titus, S.P.: Nonlinear Rayleigh–Bénard convection with variable heat source. J. Heat Transf. 135, 1–12 (2013)

    Article  Google Scholar 

  66. Siddheshwar, P.G., Veena, B.N.: Unsteady Rayleigh–Bénard convection of nanoliquids in enclosures. World Acad. Sci. Eng. Technol. Int. J. Mech. Aerosp. Ind. Mech. Manuf. Eng. 11(6), 1051–1060 (2017)

    Google Scholar 

  67. Simo, C., Puigjaner, D., Herrero, J., Giralt, F.: Dynamics of particle trajectories in a Rayleigh–Bénard problem. Commun. Nonlin. Sci. Numer. Simul. 15, 24–39 (2010)

    Article  MATH  Google Scholar 

  68. Sofos, F., Karakasidis, T., Liakopoulos, A.: Transport properties of liquid argon in krypton nanochannels: anisotropy and non-homogeneity introduced by the solid walls. Int. J. Heat Mass Transf. 52(3), 735–743 (2009)

    Article  MATH  Google Scholar 

  69. Straughan, B.: The Energy Method, Stability, and Nonlinear Convection. Springer, Berlin (2013)

    MATH  Google Scholar 

  70. Thirlby, R.: Convection in an internally heated layer. J. Fluid Mech. 44, 673–693 (1970)

    Article  MATH  Google Scholar 

  71. Tiwari, R.K., Das, M.K.: Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int. J. Heat Mass Transf. 50, 2002–2018 (2007)

    Article  MATH  Google Scholar 

  72. Tritton, D.J., Zarraga, M.N.: Convection in horizontal layers with internal heat generation. Exp. J. Fluid Mech. 30, 21–31 (1967)

    Article  Google Scholar 

  73. Tveitereid, M., Palm, E.: Convection due to internal heat sources. J. Fluid Mech. 76, 481–499 (1976)

    Article  MATH  Google Scholar 

  74. Tzou, D.Y.: Instability of nanofluids in natural convection. J. Heat Transf. 130, 1–9 (2008)

    Article  MATH  Google Scholar 

  75. Tzou, D.Y.: Thermal instability of nanofluids in natural convection. Int. J. Heat Mass Transf. 51, 2967–2979 (2008)

    Article  MATH  Google Scholar 

  76. Wang, X.Q., Mujumdar, A.S.: Heat transfer characteristics of nanofluids: A review. Int. J. Therm. Sci. 46, 1–19 (2007)

    Article  Google Scholar 

  77. Wen, D., Lin, G., Vafaei, S., Zhang, K.: Review of nanofluids for heat transfer applications. Particuology 7, 141–150 (2009)

    Article  Google Scholar 

  78. Xuan, Y., Roetzel, W.: Conceptions for heat transfer correlation of nanofluids. Int. J. Heat Mass Transf. 43, 3701–3707 (2000)

    Article  MATH  Google Scholar 

  79. Yadav, D., Agrawal, G.S., Bhargava, R.: Rayleigh–Bénard convection in nanofluid. Int. J. Appl. Math. Mech. 7, 61–76 (2011)

    MATH  Google Scholar 

  80. Yadav, D., Agrawal, G.S., Bhargava, R.: Thermal instability of rotating nanofluid layer. Int. J. Eng. Sci. 49, 1171–1184 (2011)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

One of us (MN) is grateful to the Department of Science and Technology, Government of India, for awarding a junior research fellowship to carry out her research under the “Promotion for University Research and Scientific Excellence (PURSE)” programme. She is also grateful to the Bangalore University for supporting her research. The authors are grateful to the four anonymous referees for their most useful comments that helped them refine the paper to the present form.

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  1. N. Meenakshi

    Present address: Department of Mathematics, Ramaiah University of Applied Sciences, Peenya Campus, Bangalore, 560 058, India

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  1. DST - PURSE Program, Department of Mathematics, Bangalore University, Jnanabharathi Campus, Bangalore, 560 056, India

    N. Meenakshi

  2. Department of Mathematics, Bangalore University, Jnanabharathi Campus, Bangalore, 560 056, India

    P. G. Siddheshwar

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Meenakshi, N., Siddheshwar, P.G. A theoretical study of enhanced heat transfer in nanoliquids with volumetric heat source. J. Appl. Math. Comput. 57, 703–728 (2018). https://doi.org/10.1007/s12190-017-1129-9

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