Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Global solutions describing the collapse of a spherical or cylindrical cavity

  • 37 Accesses

Abstract

The collapse of a spherical (cylindrical) cavity in air is studied analytically. The global solution for the entire domain between the sound front, separating the undisturbed and the disturbed gas, and the vacuum front is constructed in the form of infinite series in time with coefficients depending on an “appropriate” similarity variable. At timet=0+, the exact planar solution for a uniformly moving cavity is assumed to hold. The global analytic solution of this initial boundary value problem is found until the collapse time (=(γ−1)/2) for γ ≤ 1+(2/(1+v)), wherev=1 for cylindrical geometry, andv=2 for spherical geometry. For higher values of γ, the solution series diverge at timet — 2(β−1)/ (v(1+β)+(1−β)2) where β=2/(γ−1). A close agreement is found in the prediction of qualitative features of analytic solution and numerical results of Thomaset al. [1].

This is a preview of subscription content, log in to check access.

References

  1. [1]

    L. P. Thomas, V. Pais, R. Gratton and J. Diez. Phys. Fluids21, 676 (1986).

  2. [2]

    P. L. Sachdev,Nonlinear Diffusive Waves, Cambridge University Press, New York 1987.

  3. [3]

    K. P. Stanyokovich,Unsteady Motion of Continuous Media, Pergamon Press, New York 1960.

  4. [4]

    H. P. Greenspan and D. S. Butler, J. Fluid Mech.13, 101 (1962).

  5. [5]

    S. P. Bautin, J. Appl. Math. & Mech.46, 50 (1982).

  6. [6]

    E. Hille,Ordinary Differential Equations in the Complex Plane, John Wiley & Sons, New York 1972.

  7. [7]

    C. Hunter, J. Fluid Mech.15, 289 (1962).

  8. [8]

    J. Abiowitz, A. Ramani and H. Segur, J. Math. Phys.21, 715 (1980).

  9. [9]

    R. B. Lazarus, Phys. Fluids25, 1146 (1982).

  10. [10]

    S. P. Bautin, Diff. Eqs.12, 2052 (1976).

  11. [11]

    R. B. Lazarus, SIAM J. Num. Anal.18, 316 (1981).

  12. [12]

    A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier,Qualitative Theory of Second Order Dynamic Systems, John Wiley & Sons, New York 1973.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sachdev, P.L., Gupta, N. & Ahluwalia, D.S. Global solutions describing the collapse of a spherical or cylindrical cavity. Z. angew. Math. Phys. 43, 856–874 (1992). https://doi.org/10.1007/BF00913411

Download citation

Keywords

  • Mathematical Method
  • Close Agreement
  • Global Solution
  • Similarity Variable
  • Solution Series