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 problemψ xx−u(x,0)ψ=0, withψ(0)=1 andψ′(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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References
Hammack, J. L. and Segur, H.: The Korteweg-de Vries equation and water waves. Part 2, Comparison with experiments,J. Fluid Mech. 65(2), (1974), 289–314.
Ince, E. L.:Ordinary Differential Equations, Dover Publications, 1956.
Korteweg, D. J. and de Vries, G.: On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves,Phil. Mag. 39(240) (1895), 422.
Minnaert, M.: Musical air bubbles and the sound of running water,Phil. Mag. 16 (1933), 235.
Nakoryakov, V. E., Pokusaev, B. C., and Shreiber, I. R.:Wave Propagation in Gas-Liquid Media, Ch. 3, 1993.
Noordzij, L.: Shock waves in bubble-liquid mixtures, in L. I. Sedov (ed.),Non-Steady Flow of Water at High Speeds, Proc. IUTAM Symp., Nauka, Moscow, 1973.
Noordzij, L. and Wijngaarden, L. van: Relaxation effects, caused by relative motion, on shock waves in gas-bubble/liquid mixtures,J. Fluid Mech. 66(1) (1974), 115–143.
Watanabe, M. and Prosperetti, A.: Shock waves in dilute bubbly liquids,J. Fluid Mech. 274 (1994), 349–381.
Whitham, G. B.:Linear and Nonlinear Waves, Wiley, New York, 1974.
Whittaker, E. T. and Watson, G. N.:A Course of Modern Analysis, Cambridge University Press, Cambridge, 1952.
Wijngaarden, L. van: On the equations of motion for mixtures of liquid and gas bubbles,J. Fluid Mech. 33(3) (1968), 465–474.
Wijngaarden, L. van: One-dimensional flow of liquids containing small gas bubbles,Ann. Rev. Fluid Mech. 4 (1972), 369–396.