Falling of a body with a flat elastic bottom on a thin liquid layer at a small angle

Abstract

Nonlinear shallow water equations and the method of matched asymptotic expansions are used to solve the problem of the impact of a box-type body with a flat bottom on a thin elastic liquid layer at a small angle in the plane formulation. It is established that, at certain values of the input parameters of the problem, the liquid pressure near the body edges becomes less than atmospheric pressure, and the liquid separates from the bottom of the box. Calculations demonstrating the influence of elastic bottom and liquid separation on the body motion are performed. It is shown that the presence of an elastic bottom significantly changes the hydrodynamic pressure distribution and can cause loads higher than in the case of a rigid body.Keywords: impact, jet formation, nonlinear shallow water equations, separation of liquid from body, elastic plate, method of matched asymptotic expansions.The problem of the collision of bodies with liquid is of interest to researchers when it comes to the simulation of natural phenomena, such as asteroid falling or technical processes, e.g., emergency landing of an aircraft. Theoretical and experimental studies show that the greatest hydrodynamic pressures occur at the impact of bodies on a flat bottom at a small angle or smooth blunt bodies. Impact on a thin liquid layer is much more dangerous than impact on deep water. Thus, the case considered is the most dangerous. Theoretical and experimental studies [1-3] also show that the impact of a blunt body on a thin water layer does not produce a soliton. The liquid displaced by the body rises as a jet, whose thickness is comparable with that of the liquid layer.In addition to numerical algorithms, there are analytic approaches to solving the problem of impact. The first of them is the Wagner approximation for the case of a smooth blunt body and an infinitely deep liquid [4]. The second approach was proposed by Korobkin for the case of impact on a thin liquid layer [1,5]: the problem was solved in a two-dimensional formulation by the method of matched asymptotic expansions. The theory of impact on the surface of a liquid of finite depth was developed in [6][7][8]. Recent studies have been devoted to the asymmetric and oblique impact of bodies [8][9][10][11][12][13][14], the influence of structure deformability [13,15,16], and separation of liquid from a body during the impact [12][13][14]18]. A sloping smooth impact of a blunt body on a thin liquid layer was studied in [12,14], where the separation point was determined using the following conditions: p = p x = 0. In [13,17], the case of oblique impact of an elastic plate on a liquid of infinite depth was studied, different conditions at the separation point were investigated, the results were compared, and it was shown that the Brillouin-Will conditions are satisfied at p = p x = 0.Tkacheva [15] studied the problem of symmetric impact of a box-type body with a flat bottom at a zero angle taking into account bottom deformations. In [18], asymmetrical falling of a rigid box on a thin liquid l...