We address numerically the deformation and fragmentation dynamics of a single liquid drop subject to impulsive acceleration by a unidirectional gas stream. The density ratio between the liquid and gaseous phases is varied in the 10 − 2000 range and comparisons are made with recent drop breakup experiments. We show for low Ohnesorge numbers that the liquid-gas density contrast significantly modifies the critical Weber number for the transition between bursting and stripping fragmentation regimes on the one hand, for drop fragmentation on the other hand. We suggest a simple theoretical argument to predict the transitional Weber number as a function of the density contrast and show that the stabilising influence of small contrasts can be explained by the effect of inertia in the nonlinear coupling between drop stretching and centroid acceleration.
We consider the nonlinear optimisation of irreversible mixing induced by an initial finite amplitude perturbation of a statically stable density-stratified fluid with kinematic viscosity ν and density diffusivity κ. The initial diffusive error function density distribution varies continuously so that ρ ∈ [ρ− 1 2 ρ 0 ,ρ+ 1 2 ρ 0 ]. A constant pressure gradient is imposed in a plane two-dimensional channel of depth 2h. We consider flows with a finite Péclet number Pe = U m h/κ = 500 and Prandtl number Pr = ν/κ = 1, and a range of bulk Richardson numbers Ri b = gρ 0 h/(ρU 2 ) ∈ [0, 1] where U m is the maximum flow speed of the laminar parallel flow, and g is the gravitational acceleration. We use the constrained variational direct-adjoint-looping (DAL) method to solve two optimization problems, extending the optimal mixing results of Foures et al. (2014) to stratified flows, where the irreversible mixing of the active scalar density leads to a conversion of kinetic energy into potential energy. We identify initial perturbations of fixed finite kinetic energy which maximize the time-averaged perturbation kinetic energy developed by the perturbations over a finite time interval, and initial perturbations that minimise the value (at a target time, chosen to be T = 10) of a 'mix-norm' as first introduced by Mathew, Mezic & Petzold (2005), further discussed by Thiffeault (2012) and shown by Foures et al. (2014) to be a computationally efficient and robust proxy for identifying perturbations that minimise the long-time variance of a scalar distribution. We demonstrate, for all bulk Richardson numbers considered, that the time-averaged-kineticenergy-maximising perturbations are significantly suboptimal at mixing compared to the mix-norm-minimising perturbations, and also that minimising the mix-norm remains (for density-stratified flows) a good proxy for identifying perturbations which minimise the variance at long times. Although increasing stratification reduces the mixing in general, mix-norm-minimising optimal perturbations can still trigger substantial mixing for Ri b 0.3. By considering the time evolution of the kinetic energy and potential energy reservoirs, we find that such perturbations lead to a flow which, through Taylor dispersion, very effectively converts perturbation kinetic energy into 'available potential energy', which in turn leads rapidly and irreversibly to thorough and efficient mixing, with little energy returned to the kinetic energy reservoirs.
We investigate both experimentally and numerically the impact of liquid drops on deep pools of aqueous glycerol solutions with variable pool viscosity and air pressure. With this approach we are able to address drop impacts on substrates that continuously transition from low-viscosity liquids to almost solids. We show that the generic corolla spreading out from the impact point consists of two distinct sheets, namely an ejecta sheet fed by the drop liquid and a second sheet fed by the substrate liquid, which evolve on separated timescales. These two sheets contribute to a varying extent to the corolla overall dynamics and splashing, depending in particular on the viscosity ratio between the two liquids.Throwing a stone in a stagnant pond or letting a waterdrop fall onto a dry plate equally contribute to the active pleasures of water splashing [1], as does the rewarding observation of the short-lived liquid corolla which, in both cases, blooms on the impact point [2]. From a comprehensive point of view however, the dynamics of the two events remarkably differif only for the matter-of-fact reason that the splashed liquid belongs to the projectile in the latter, and to the impacted substrate in the former. The case of a liquid drop hitting a liquid surface therefore raises a natural question: which of the two liquids feeds the corolla as it spreads out, develops and eventually disintegrates? How does the splashing dynamics relate to that of the two first problems? Splashes are formally defined as the ejection of small droplets due to the large deformation of a liquid interface following an impact, and occur in a large diversity of problems related to challenging environmental and industrial applications [3][4][5][6][7][8][9]. In particular, two manifestations of splashing are discussed at length in the literature, referred to as 'prompt splash' [10][11][12] and 'crown splash' [12,13], and mostly discriminated by the dynamics, shape and behaviour of the liquid sheet (which we will generically refer to as corolla) whose desintegration results in the ejection of droplets. Prompt splash is associated with the early destabilisation of a thin ejecta sheet shooting out almost horizontally from the impact point: this axisymmetric liquid jet expands radially, bends upwards and disintegrates into small and fast droplets [10,12,14,15]. On the other hand, crown splash originates in the destabilisation of an almost vertically expanding liquid sheet (sometimes referred to as Peregrine sheet [12]) rising out of the impact region [1,12,16]. Here the 'crown' emerges through the fingering of the liquid rim at the leading edge of the Peregrine sheet, owing to coupled Rayleigh-Taylor (RT) and Rayleigh-Plateau (RP) instabilities, and produces somewhat larger droplets [12,[16][17][18]. However, some considerable confusion remains regarding the precise characterisation of these two splashing regimes and associated corolla dynamics. Indeed, the complicated splashing phenomenology rarely allows for such a clear separation between the coroll...
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