High-dimensional partial-differential equations (PDEs) arise in a number of fields of science and engineering, where they are used to describe the evolution of joint probability functions. Their examples include the Boltzmann and Fokker-Planck equations. We develop new parallel algorithms to solve high-dimensional PDEs. The algorithms are based on canonical and hierarchical numerical tensor methods combined with alternating least squares and hierarchical singular value decomposition. Both implicit and explicit integration schemes are presented and discussed. We demonstrate the accuracy and efficiency of the proposed new algorithms in computing the numerical solution to both an advection equation in six variables plus time and a linearized version of the Boltzmann equation.

At atmospheric pressure, a drop of ethanol impacting on a solid surface produces a splash. Reducing the ambient pressure below its atmospheric value suppresses this splash. The origin of this so-called pressure effect is not well understood and this is the first study to present an in-depth comparison between various theoretical models that aim to predict splashing and simulations. In this work the pressure effect is explored numerically by resolving the Navier-Stokes equations at a 3-nm resolution. In addition to reproducing numerous experimental observations, it is found that different models all provide elements of what is observed in the simulations. The skating droplet model correctly predicts the existence and scaling of a gas film under the droplet, the lamella formation theory is able to correctly predict the scaling of the lamella ejection velocity as function of the impact velocity for liquids with different viscosity, and lastly, the dewetting theory's hypothesis of a lift force acting on the liquid sheet after ejection is consistent with our results.

A liquid drop impacting a dry solid surface with sufficient kinetic energy will splash, breaking apart into numerous secondary droplets. This phenomenon shows many similarities to forced wetting, including the entrainment of air at the contact line. Because of these similarities and the fact that forced wetting has been shown to depend on the wetting properties of the surface, existing theories predict splashing to depend on wetting properties as well. However, using high-speed interference imaging we observe that at high capillary numbers wetting properties have no effect on splashing for various liquid-surface combinations. Additionally, by fully resolving the Navier-Stokes equations at length and time scales inaccessible to experiments, we find that the shape and motion of the air-liquid interface at the contact line at the edge of the droplet are independent of wettability. We use these findings to evaluate existing theories and to compare splashing with forced wetting. arXiv:1607.08867v2 [physics.flu-dyn]

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In this work, an analytical Volume Of Fluid (VOF) implementation of the Generalized Navier Boundary Condition is presented based on the Brackbill surface tension model. The model is validated by simulations of droplets on a smooth surface in a planar geometry. Looking at the static behavior of the droplets, it is found that there is a good match between the droplet shape resolved in the simulations and the theoretically predicted shape for various values of the Young's angle. Evaluating the spreading of a droplet on a completely wetting surface, the Voinov-Tanner-Cox law (θ ∝ Ca 1/3 ) can be observed. At later times scaling follows r ∝ t 1/2 , suggesting spreading is limited by inertia. These observations are made without any fitting parameters except the slip