Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 33
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Cao, Chongsheng Ibrahim, Slim Nakanishi, Kenji and Titi, Edriss S. 2015. Finite-Time Blowup for the Inviscid Primitive Equations of Oceanic and Atmospheric Dynamics. Communications in Mathematical Physics, Vol. 337, Issue. 2, p. 473.

    Ebin, David G. and Preston, Stephen C. 2015. Riemannian Geometry of the Contactomorphism Group. Arnold Mathematical Journal, Vol. 1, Issue. 1, p. 5.

    Sarria, Alejandro and Wu, Jiahong 2015. Blowup in stagnation-point form solutions of the inviscid 2d Boussinesq equations. Journal of Differential Equations, Vol. 259, Issue. 8, p. 3559.

    Sarria, Alejandro 2015. Global estimates and blow-up criteria for the generalized Hunter-Saxton system. Discrete and Continuous Dynamical Systems - Series B, Vol. 20, Issue. 2, p. 641.

    Preston, Stephen C. and Sarria, Alejandro 2014. One-parameter solutions of the Euler-Arnold equation on the contactomorphism group. Discrete and Continuous Dynamical Systems, Vol. 35, Issue. 5, p. 2123.

    Rabinowitch, Alexander S. 2014. On some classes of nonstationary axially symmetric solutions to the Navier-Stokes equations. Journal of Mathematical Physics, Vol. 55, Issue. 9, p. 093102.

    Sarria, Alejandro and Saxton, Ralph 2013. Blow-up of Solutions to the Generalized Inviscid Proudman–Johnson Equation. Journal of Mathematical Fluid Mechanics, Vol. 15, Issue. 3, p. 493.

    Cho, Chien-Hong and Wunsch, Marcus 2012. Global weak solutions to the generalized Proudman-Johnson equation. Communications on Pure and Applied Analysis, Vol. 11, Issue. 4, p. 1387.

    Gilbert, Andrew Klapper, Isaac Thiffeault, Jean-Luc and Wang, Jane 2011. Preface. Physica D: Nonlinear Phenomena, Vol. 240, Issue. 20, p. 1565.

    Wunsch, Marcus 2011. The Generalized Proudman–Johnson Equation Revisited. Journal of Mathematical Fluid Mechanics, Vol. 13, Issue. 1, p. 147.

    Castro, A. and Córdoba, D. 2010. Infinite energy solutions of the surface quasi-geostrophic equation. Advances in Mathematics, Vol. 225, Issue. 4, p. 1820.

    Cho, Chien-Hong and Wunsch, Marcus 2010. Global and singular solutions to the generalized Proudman–Johnson equation. Journal of Differential Equations, Vol. 249, Issue. 2, p. 392.

    Kim, Sun-Chul and Okamoto, Hisashi 2010. Vortices of large scale appearing in the 2D stationary Navier–Stokes equations at large Reynolds numbers. Japan Journal of Industrial and Applied Mathematics, Vol. 27, Issue. 1, p. 47.

    Saxton, Ralph and Tiğlay, Feride 2008. Global Existence of Some Infinite Energy Solutions for a Perfect Incompressible Fluid. SIAM Journal on Mathematical Analysis, Vol. 40, Issue. 4, p. 1499.

    Okamoto, Hisashi and Ohkitani, Koji 2005. On the Role of the Convection Term in the Equations of Motion of Incompressible Fluid. Journal of the Physical Society of Japan, Vol. 74, Issue. 10, p. 2737.

    Pelz, R B and Ohkitani, K 2005. Linearly strained flows with and without boundaries—the regularizing effect of the pressure term. Fluid Dynamics Research, Vol. 36, Issue. 4-6, p. 193.

    Constantin, Peter 2003.

    Gibbon, J D Moore, D R and Stuart, J T 2003. Exact, infinite energy, blow-up solutions of the three-dimensional Euler equations. Nonlinearity, Vol. 16, Issue. 5, p. 1823.

    Hewitt, R E Duck, P W and Al-Azhari, M 2003. Extensions to three-dimensional flow in a porous channel. Fluid Dynamics Research, Vol. 33, Issue. 1-2, p. 17.

    Kumari, M. and Nath, G. 2002. Unsteady flow and heat transfer of a viscous fluid in the stagnation region of a three-dimensional body with a magnetic field. International Journal of Engineering Science, Vol. 40, Issue. 4, p. 411.

  • Journal of Fluid Mechanics, Volume 203
  • June 1989, pp. 1-22

Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form

  • S. Childress (a1), G. R. Ierley (a2), E. A. Spiegel (a3) and W. R. Young (a4)
  • DOI:
  • Published online: 01 April 2006

The time-dependent form of the classic, two-dimensional stagnation-point solution of the Navier-Stokes equations is considered. If the viscosity is zero, a class of solutions of the initial-value problem can be found in closed form using Lagrangian coordinates. These solutions exhibit singular behaviour in finite time, because of the infinite domain and unbounded initial vorticity. Thus, the blow-up found by Stuart in three dimensions using the stagnation-point form, also occurs in two. The singularity vanishes under a discrete, finite-dimensional ‘point vortex’ approximation, but is recovered as the number of vortices tends to infinity. We find that a small positive viscosity does not arrest the breakdown, but does strongly alter its form. Similar results are summarized for certain Boussinesq stratified flows.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *