Air cushioning in droplet impacts with liquid layers and other droplets

Abstract

Air cushioning of a high-speed liquid droplet impact with a finite-depth liquid layer sitting upon a rigid impermeable base is investigated. The evolution of the droplet and liquid-layer free-surfaces is studied alongside the pressure in the gas film dividing the two. The model predicts gas bubbles are trapped between the liquid free-surfaces as the droplet approaches impact. The key balance in the model occurs when the depth of the liquid layer equals the horizontal extent of interactions between the droplet and the gas film. For liquid layer depths significantly less than this a shallow liquid limit is investigated, which ultimately tends towards the air-cushioning behavior seen in droplet impact with a solid surface. Conversely, for liquid layer depths much deeper than this, the rigid base does not affect the air-cushioning of the droplet. The influence of compressibility is discussed and the relevant parameter regime for an incompressible model is identified. The size of the trapped gas bubble as a function of the liquid layer depth is investigated. The deep water model is extended to consider binary droplet collisions. Again, the model predicts gas bubbles will be trapped as the result of air cushioning in high-speed binary droplet impacts

Gas-cushioned droplet impacts with a thin layer of porous media

“…This formula, which has never been explicitly verified experimentally [30], includes shear stress instead of just velocity shear. However, because of the disparate length and velocity scales (15)(16) in the gas film, the second term on the left-hand side is O ε 2 smaller than the first term. Consequently if this extra term was included in the analysis, then it would be neglected at leading order leaving the Beavers-Joseph condition (8) as used.…”

“…For substrate heightsh εR, the leading-order gas cushioning is no-longer influenced by the impermeable base of the substrate. The porous substrate height O(εR) marking the limit of the regime in which the impermeable base effects the leading-order gas cushioning is the same height as previously determined for a liquid layer coating an impermeable base [16]. Finally, through extensions of the analysis of Hicks and Purvis [15], three-dimensional cushioned impacts with anisotropic substrates could be investigated.…”

“…An iteration is then performed between these two equations; convergence of the solution to within 10 −6 typically takes a limited number of iterations. This method offers a considerable saving in simulation time over previous work [15,16] that discretised (37), particularly when surface tension is included.…”

“…For surface tension dominated droplet deformation, the largest viscous term in the horizontal gas momentum equation (18b) is balanced by the horizontal derivative of the capillary pressure, as a result of the normal stress balance (6b), rather than the horizontal derivative of the liquid pressure. Using the gas non-dimensionalization (15)(16) this implies…”

“…where µ g is the dynamic viscosity of the gas, then the gas is also incompressible to leading order [28,16] and is governed by the incompressible Navier-Stokes equations ∇ ·ũ u u g = 0, ∂ũ u u g ∂t +ũ u u g ·∇ũ u u g = − 1…”