Equilibrium Shape of Droplets at the Wetted Surface in Strong Gravitational Fields

In this work, we develop a novel model describing the equilibrium shape of sessile droplets on a wetted horizontal surface in a gravitational field. The model takes into account the intermolecular Lenard-Jones forces between solid and the liquid molecules using the standard disjoining pressure approximation. These forces lead to the formation of a thin, non-removable fluid layer covering the solid substrate. Balancing the disjoining pressure against the surface tension and the gravitational force we calculate the smooth shape of the surface of the liquid. We obtain a criterion when the gravitational forces are so large that they level the droplets completely. We show that, in the case of weak gravitational forces, the maximum height of the droplets is described by the classical Quincke formula.Comment: 19 pages, Fig. 1

Development of a methodology to assess the hydrocyclone process with account of the rheological properties of the mineral slurry

The paper studies the possibility of assessing the separation of mineral raw materials, taking into account the rheology of the mineral slurry. The ores of the Mayskoye deposit were chosen as the object of the study, characterized by a thin impregnation of the valuable component – gold in the host minerals, which determines the use of fine and ultrafine milling. This fact is essential because the presence of a fine grade seriously affects the rheology of the mineral slurry used in subsequent mineral processing stages. This predetermines the necessity to take into account rheological parameters. The research performed provides the development of a methodology for assessing the separation of minerals in the hydrocyclone based on the interpretation of numerical and mathematical modeling data. using the object-oriented programming language Python, a program for calculating empirical coefficients of the rheological equation, theoretically describing the dynamics of internal transformations of the mineral slurry, was developed. Taking into account the process parameters of the laboratory unit with hydrocyclone and ore properties, three concentrations of solids in the mineral slurry were selected, conditionally corresponding to the minimum, average and maximum values. Rheological equations successively composed for three concentrations, i.e., 400, 500, and 700 g/l, made it possible to calculate the critical shear rates corresponding to the maximum dispersion of the mineral slurry in the hydrocyclone flow. Subsequent numerical simulation using Ansys Fluent software, as well as statistical evaluation of the shear rates at different levels of solids content showed that the shear rate profile in the cross-section of the hydrocyclone corresponding to the maximum dispersion of the mineral slurry is obtained at the content of 400 g/l

Transition in a numerical model of contact line dynamics and forced dewetting

We investigate the transition to a Landau-Levich-Derjaguin film in forced dewetting using a quadtree adaptive solution to the Navier-Stokes equations with surface tension. We use a discretization of the capillary forces near the receding contact line that yields an equilibrium for a specified contact angle $\theta_\Delta$ called the numerical contact angle. Despite the well-known contact line singularity, dynamic simulations can proceed without any explicit additional numerical procedure. We investigate angles from $15^\circ$ to $110^\circ$ and capillary numbers from $0.00085$ to $0.2$ where the mesh size $\Delta$ is varied in the range of $0.0035$ to $0.06$ of the capillary length $l_c$. To interpret the results, we use Cox's theory which involves a microscopic distance $r_m$ and a microscopic angle $\theta_e$. In the numerical case, the equivalent of $\theta_e$ is the angle $\theta_\Delta$ and we find that Cox's theory also applies. We introduce the scaling factor or gauge function $\phi$ so that $r_m = \Delta/\phi$ and estimate this gauge function by comparing our numerics to Cox's theory. The comparison provides a direct assessment of the agreement of the numerics with Cox's theory and reveals a critical feature of the numerical treatment of contact line dynamics: agreement is poor at small angles while it is better at large angles. This scaling factor is shown to depend only on $\theta_\Delta$ and the viscosity ratio $q$. In the case of small $\theta_e$, we use the prediction by Eggers [Phys. Rev. Lett., vol. 93, pp 094502, 2004] of the critical capillary number for the Landau-Levich-Derjaguin forced dewetting transition. We generalize this prediction to large $\theta_e$ and arbitrary $q$ and express the critical capillary number as a function of $\theta_e$ and $r_m$. An analogy can be drawn between $r_m$ and the numerical slip length.Comment: This version of the paper includes the corrections indicated in Ref. [1