Evolving solitons in bubbly flows

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.