STRUCTURE AND STABILITY OF A SPHERICAL SHOCK WAVE IN A VAN DER WAALS GAS

Strong spherical shock waves are studied for an imperfect gas, here modelled by a van der Waals equation of state. Similarity solutions of Guderley type are shown to exist, in which the radius of the shock is proportional to (–t)^α, where t is time measured from the moment at which the shock focuses. The exponent α depends on both the ratio of specific heats, γ, and on the van der Waals excluded volume, b. For small b, the solution resembles the Guderley solution and is well described by the Chester–Chisnell–Whitham (CCW) approximation. A new branch of solutions, which the CCW approximation fails to locate, is shown to exist for larger b. The linear stability of the similarity solutions is examined directly, without the use of the CCW approximation. Normal modes grow (Re (β) < 0) or decay (Re (β) > 0) as (−t)^αβ. where the ‘growth rate’, β, is a function of γ, b and n, the spherical harmonic wavenumber. No physically meaningful, discrete, spherically symmetric(n = 0) modes were found. This case was examined numerically and nonlinearly for shocks launched by a spherical ‘piston’; no evidence of instability was discovered. For n > 0, the existence of an infinite discrete spectrum of normal modes is indicated for all γ and b. In every case examined, the shock is unstable if b = 0 (the ideal gas), Re (β) for the most unstable mode being negative for all n ≥ 1 but tending to zero as n→∞; it is shown that β∼ in√[(γ − 1)/(γ + 1)] as n → ∞. It is found that, in general terms, stability is enhanced if either n or b is increased. Special attention is given to the structure and stability of solutions in the limit γ→ 1. The limit of the nearly incompressible fluid is also briefly considered. A way of increasing the light emission from a sonoluminescing bubble is suggested by the present analysis and is described.