Numerical study of stagnation-point flows of incompressible fluid

  • Jinghui Zhu


The nonstationary stagnation-point flows of incompressible viscous fluid are studied by a finite difference method. By some ansatz, the two-dimensional equations are reduced to a one-dimensional equation and the three-dimensional ones are reduced to a coupled equations in one-dimension. We perform numerical experiments with various initial values to find the following results. In two dimensions, we find the formation and development of the boundary layers, while no blow-up is observed. In three dimensions, blow-ups are found if both of the initial values are sufficiently large. Some modified equations are also discussed and compared with the original equations. Different roles of the convection, stretching terms are also discussed.

Key words

Navier-Stokes equations finite difference method boundary layer blow-up 


  1. [1]
    Y.-G. Chen, Asymptotic behaviors of blowing up solution tou t = uxx +u1+α. J. Fac. Sci. Univ. Tokyo, sect. IV,33 (1986), 541–574.MATHGoogle Scholar
  2. [2]
    S. Childress, G.R. Ierley, E.A. Spiegel and W.R. Young, Blow up of unsteady two-dimensional Euler and Navier-Stokes solution having stagnation-point form. J. Fluid Mech.,203 (1989), 1–22.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    P. Constantin, P.D. Lax and A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation. Comm. Pure Appl. Math.,38 (1985), 715–724.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    T.Y. Hou and B.T.R. Wetton, Convergence of a difference scheme for the N-S equations using vorticity boundary condition. SIAM J. Numer. Anal.,29, No. 3 (1992), 615–639.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    T. Nakagawa, Blow up of a finite difference solution to ut = uxx +u2. Appl. Math. Optim.,2 (1976), 337–350.CrossRefMathSciNetGoogle Scholar
  6. [6]
    H. Okamoto, On a model equation of 2D incompressible fluid which shows the evolution of a boundary layer. Report on Annual Meeting of Applied Mathematics, 1995,90:1–90:4 (in Japanese).Google Scholar
  7. [7]
    H. Okamoto and M. Shōji, A spectral method for unsteady two-dimensional Navier-Stokes equations. Proc. 3-rd China-Japan Seminar on Numerical Mathematics (eds. Z.-C. Shi and M. Mori), Science Press, Beijing, 1988, 253–260.Google Scholar
  8. [8]
    H. Okamoto and M. Shōji, Boundary layer in unsteady two-dimensional Navier-Stokes equation. Recent Developments in Domain Decomposition Methods and Flow Problems, GAKUTO International Series, Mathematical Science and Applications, Vol.11, 1998, 171–180.Google Scholar
  9. [9]
    J.T. Stuart, Nonliear Euler partial differential equations: singularities in their solution. Symposium to honor C.C. Lin (eds. D.J. Benney, F.H. Shu and C. Yuan), World Scientific, 1987, 81–95.Google Scholar

Copyright information

© JJIAM Publishing Committee 2000

Authors and Affiliations

  • Jinghui Zhu
    • 1
  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

Personalised recommendations