Evolving solitons in bubbly flows

L. van Wijngaarden

Research output: Contribution to journalArticleAcademic

8 Citations (Scopus)
51 Downloads (Pure)

Abstract

At the end of the sixties, it was shown that pressure waves in a bubbly liquid obey the KdV equation, the nonlinear term coming from convective acceleration and the dispersive term from volume oscillations of the bubbles. For a variableu, proportional to –p, wherep denotes pressure, the appropriate KdV equation can be casted in the formu t –6uu x +u xxx =0. The theory of this equation predicts that, under certain conditions, solitons evolve from an initial profileu(x,0). In particular, it can be shown that the numberN of those solitons can be found from solving the eigenvalue problem xx–u(x,0)=0, with(0)=1 and(0)=0.N is found from counting the zeros of the solution of this equation betweenx=0 andx=Q, say,Q being determined by the shape ofu(x,0). We took as an initial pressure profile a Shockwave, followed by an expansion wave. This can be realised in the laboratory and the problem, formulated above, can be solved exactly. In this contribution the solution is outlined and it is shown from the experimental results that from the said initial disturbance, indeed solitons evolve in the predicated quantity.
Original languageUndefined
Pages (from-to)507-516
JournalActa applicandae mathematicae
Volume39
Issue number1-3
DOIs
Publication statusPublished - 1995

Keywords

  • IR-50303

Cite this

van Wijngaarden, L. / Evolving solitons in bubbly flows. In: Acta applicandae mathematicae. 1995 ; Vol. 39, No. 1-3. pp. 507-516.
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Evolving solitons in bubbly flows. / van Wijngaarden, L.

In: Acta applicandae mathematicae, Vol. 39, No. 1-3, 1995, p. 507-516.

Research output: Contribution to journalArticleAcademic

TY - JOUR

T1 - Evolving solitons in bubbly flows

AU - van Wijngaarden, L.

PY - 1995

Y1 - 1995

N2 - At the end of the sixties, it was shown that pressure waves in a bubbly liquid obey the KdV equation, the nonlinear term coming from convective acceleration and the dispersive term from volume oscillations of the bubbles. For a variableu, proportional to –p, wherep denotes pressure, the appropriate KdV equation can be casted in the formu t –6uu x +u xxx =0. The theory of this equation predicts that, under certain conditions, solitons evolve from an initial profileu(x,0). In particular, it can be shown that the numberN of those solitons can be found from solving the eigenvalue problem xx–u(x,0)=0, with(0)=1 and(0)=0.N is found from counting the zeros of the solution of this equation betweenx=0 andx=Q, say,Q being determined by the shape ofu(x,0). We took as an initial pressure profile a Shockwave, followed by an expansion wave. This can be realised in the laboratory and the problem, formulated above, can be solved exactly. In this contribution the solution is outlined and it is shown from the experimental results that from the said initial disturbance, indeed solitons evolve in the predicated quantity.

AB - At the end of the sixties, it was shown that pressure waves in a bubbly liquid obey the KdV equation, the nonlinear term coming from convective acceleration and the dispersive term from volume oscillations of the bubbles. For a variableu, proportional to –p, wherep denotes pressure, the appropriate KdV equation can be casted in the formu t –6uu x +u xxx =0. The theory of this equation predicts that, under certain conditions, solitons evolve from an initial profileu(x,0). In particular, it can be shown that the numberN of those solitons can be found from solving the eigenvalue problem xx–u(x,0)=0, with(0)=1 and(0)=0.N is found from counting the zeros of the solution of this equation betweenx=0 andx=Q, say,Q being determined by the shape ofu(x,0). We took as an initial pressure profile a Shockwave, followed by an expansion wave. This can be realised in the laboratory and the problem, formulated above, can be solved exactly. In this contribution the solution is outlined and it is shown from the experimental results that from the said initial disturbance, indeed solitons evolve in the predicated quantity.

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