Obreschkow D, Bruderer M, Farhat M
The University of Western Australia, ICRAR, 35 Stirling Highway, Crawley, WA 6009, Australia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Jun;85(6 Pt 2):066303. doi: 10.1103/PhysRevE.85.066303. Epub 2012 Jun 5.
The Rayleigh equation 3/2R+RR+pρ(-1)=0 with initial conditions R(0)=R(0), R(0)=0 models the collapse of an empty spherical bubble of radius R(T) in an ideal, infinite liquid with far-field pressure p and density ρ. The solution for r≡R/R(0) as a function of time t≡T/T(c), where R(T(c))≡0, is independent of R(0), p, and ρ. While no closed-form expression for r(t) is known, we find that r(0)(t)=(1-t(2))(2/5) approximates r(t) with an error below 1%. A systematic development in orders of t(2) further yields the 0.001% approximation r(*)(t)=r(0)(t)[1-a(1)Li(2.21)(t(2))], where a(1)≈-0.01832099 is a constant and Li is the polylogarithm. The usefulness of these approximations is demonstrated by comparison to high-precision cavitation data obtained in microgravity.
瑞利方程(\frac{3}{2}R + RR + p\rho(-1) = 0)，初始条件为(R(0) = R(0))，(R(0) = 0)，用于模拟半径为(R(T))的空球形气泡在理想无限液体中的坍塌，该液体具有远场压力(p)和密度(\rho)。作为时间(t\equiv T/T(c))的函数，(r\equiv R/R(0))的解与(R(0))、(p)和(\rho)无关，其中(R(T(c))\equiv 0)。虽然(r(t))没有已知的封闭形式表达式，但我们发现(r(0)(t) = (1 - t(2))(2/5))以低于(1%)的误差近似(r(t))。按(t(2))的阶数进行系统展开进一步得到了精度为(0.001%) 的近似值(r(*)(t) = r(0)(t)[1 - a(1)Li(2.21)(t(2))])，其中(a(1)\approx -0.01832099)是一个常数，(Li)是多重对数函数。通过与在微重力条件下获得的高精度空化数据进行比较，证明了这些近似值的实用性
