Transient dynamics of an elastic Hele-Shaw cell due to external forces with application to impact mitigation

We study the transient dynamics of a viscous liquid contained in a narrow gap between a rigid surface and a parallel elastic plate. The elastic plate is deformed due to an externally applied time-varying pressure field. We model the flow field via the lubrication approximation and the plate deformation by the Kirchhoff-Love plate theory. We obtain a self-similarity solution for the case of an external point force acting on the elastic plate. The pressure and deformation field during and after the application of the external force are derived and presented by closed-form expressions. We examine a distributed external pressure, spatially uniform and linearly increasing with time, acting on the elastic plate over a finite region and during a finite time period, similar to the viscous-elastic interaction time-scale. The interaction between elasticity and viscosity is shown to reduce by an order of magnitude the pressure within the Hele-Shaw cell compared with the externally applied pressure. The results thus suggest that elastic Hele-Shaw configurations may be used to achieve significant impact mitigation.