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Global solutions describing the collapse of a spherical or cylindrical cavity

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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].

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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

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