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THE STABILITY OF GAUGE-UZAWA METHOD TO SOLVE NANOFLUID

  • JANG, DEOK-KYU (DEPARTMENT OF MATHEMATICS, KANGWON NATIONAL UNIVERSITY) ;
  • KIM, TAEK-CHEOL (DEPARTMENT OF MATHEMATICS, KANGWON NATIONAL UNIVERSITY) ;
  • PYO, JAE-HONG (DEPARTMENT OF MATHEMATICS, KANGWON NATIONAL UNIVERSITY)
  • Received : 2020.03.04
  • Accepted : 2020.05.04
  • Published : 2020.06.25

Abstract

Nanofluids is the fluids mixed with nanoscale particles and the mixed nano size materials affect heat transport. Researchers in this field has been focused on modeling and numerical computation by engineers In this paper, we analyze stability constraint of the dominant equations and check validate of the condition for most kinds of materials. So we mathematically analyze stability of the system. Also we apply Gauge-Uzawa algorithm to solve the system and prove stability of the method.

References

  1. S.U.S Choi and J.A. Eastman, Enhancing thermal conductivity of fluids with nanoparticles, ASME International Mechanical Engineering Congress & Exposition, 1995.
  2. B.C. Pak and Y.I. Cho, Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles, Exp. Heat Transfer 11(2) (1998) 151-170
  3. D. Wen and Y. Ding, Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions, Int. J. Heat Mass Transf. 47(24) (2004), 5181-5188
  4. Y. Xuan and Q. Li, Investigation on convective heat transfer and flow features on nanofluids, J. Heat Transfer 125(1) (2003), 151-155
  5. G. Roy, C.T. Nguyen and P.-R. Lajoie, Numerical investigation of laminar flow and heat transfer in a radial flow cooling system with the use of nanofluids, Superlattices and Microstructures 35(3-6) (2004), 497-511
  6. R.S. Vajjha, D.K. Das and P.K. Namburu, Numerical study of fluid dynamic and heat transfer performance of $Al_2O_3$ and CuO nanofluids in the flat tubes of a radiator, Int. J. Heat Fluid Flow 31(4) (2010) 613-621
  7. M.M. Rahman and I.A. Eltayeb, Radiative heat transfer in a hydromagnetic nanofluid past a non-linear stretching surface with convective boundary condition, Meccanica 48(3) (2013) 601-615
  8. M.M. Rahman, A.V. Rosca and I. Pop, Boundary layer flow of a nanofluid past a permeable exponentially shrinking/stretching surface with second order slip using Buongiorno's model, Int. J. Heat Mass Transf. 77 (2014) 1133-1143
  9. M.A. Sheremet and I. Pop, Mixed convection in a lid-driven square cavity filled by a nanofluid: Buongiorno's mathematical model, Appl. Math. Comput. 266 (2015) 792-808
  10. J. Buongiorno, Convective transport in nanofluids, J. Heat Transf. 128(3), (2006) 240-250
  11. M.J. Uddin, M.S. Alam and M.M. Rahman, Natural convective heat transfer flow of nanofluids inside a quarter-circular enclosure using nonhomogeneous dynamic model, Arab. J. Sci. Eng. 42(5) (2017) 1883-1901
  12. H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20(4), (1952) 571-571
  13. G.K. Batchelor, The effect of Brownian motion on the bulk stress in a suspension of spherical particles, J. Fluid Mech. 83(1) (1977) 97-117
  14. C.T. Nguyen, F. Desgranges, G. Roy, N. Galanis, T. Mare, S. Boucher and H. Angue Mintsa, Temperature and particle-size dependent viscosity data for water-based nanofluids-Hysteresis phenomenon, Int. J. Heat & Fluid Flow 28(6) (2007) 1492-1506
  15. C.T. Nguyen, F. Desgranges, N. Galanis, G. Roy, T. Mare, S. Boucher and H. Angue Mintsa, Viscosity data for $Al_2O_3$-water nanofluid-hysteresis: is heat transfer enhancement using nanofluids reliable?, Int. J. Therm. Sci. 47(2) (2008) 103-111
  16. J.C. Maxwell, A treatise on electricity and magnetism, Vol I, II, Oxford: Clarendon Press, 1873
  17. Y.Xuan, Q.Li and W.Hu, Aggregation structure and conductivity of nanofluids, AIChE J. 49(4) (2003) 1038-1043
  18. R.H. Nochetto and J.-H. Pyo, The gauge-Uzawa finite element method. part I: the Navier-Stokes equations, SIAM Journal on Numerical Analysis 43(3) (2005), 1043-1068
  19. R.H. Nochetto and J.-H. Pyo, A finite element gauge-Uzawa method part II : Boussinesq equations, Mathematical Models Methods in Applied Sciences 16, (2006), 1599-1626
  20. J.-H. Pyo and J. Shen, Gauge-Uzawa methods for incompressible flows with variable density, Journal of Computational Physics 221(1) (2007), 181-197
  21. R. Temam, Navier-Stokes equations, AMS Chelsea Publishing, 2001.
  22. A.J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp., 22, (1968) 745-762
  23. R. Temam, Sur l'approximation de la solution des equations de Navier-Stokes par la methode des pas fractionnaires (II), Arch. Rational Mech. Anal., 33(5) (1969), 377-385
  24. R.L. Hamilton and O.K. Crosser, Thermal conductivity of heterogeneous two-component systems, Ind. Eng. Chem. Fundamen. 1(3) (1962) 187-191