Closed equation model for cavity evolution in granular media

Cavity evolution in granular media is crucial in explosion-driven gas–particle flows, particularly in many engineering applications. In this study, a theoretical model was first proposed to describe the cavity evolution in granular media by extending the classical Rayleigh–Plesset model. A closed equation set comprising the radius, pressure and gas leak-off velocity equations was built by considering the gas expansion and non-Darcy gas-penetration effects. Both centrally symmetric and non-centrally symmetric cases of gas injection into granular media were investigated. Especially for modelling the non-symmetric scenario, the radius and gas leak-off velocity equations were proposed in each radial direction with angle θ , and then the pressure equation was built up based on the integral gas leak-off along the cavity outline. Through non-dimensionalizing the theoretical equations, four key dimensionless numbers ₁,\ ₄ were obtained to characterize the response time of cavity expansion and the intensity of non-Darcy effects for both cases. This allowed us to determine a scaling law of effective cavity radius Reff^*=√ 2 ₂/(7 π )t*1/2 and the critical time tcr^*=√12.5/ ₁ for two-dimensional cavity evolution. Additionally, the necessity of incorporating non-Darcy effects was ascertained under conditions of ₄>400 . The findings demonstrate that the proposed theoretical equations effectively predict the cavity evolution results under various operational conditions ( 0.7< ₁<7× 10², 3< ₄<1.1× 10³ ), as validated by refined Euler–Lagrange numerical simulations.