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Register a new userMathematical modelling of fibre coating
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In this report we formulate and analyse a mathematical model describing the evolution of a thin liquid film coating a wire via an extrusion process. We consider the Navier-Stokes equations for a 2D incompressible Newtonian fluid coupled to the standard equation relating the fluid surface tension with the curvature. Taking the lubrication theory approximation and assuming steady state, the problem is reduced to a single third-order differential equation for the thin film height. An approximate analytical solution for the final film height is derived and compared with a numerical solution obtained by means of a shooting scheme. Good agreement between the two solutions is obtained, resulting in a relative error of around 5%. The approximate solution reveals that the key control parameters for the process are the initial film height, the fluid surface tension and viscosity, the wire velocity and the angle of exit at the extruder.
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The superheating that usually occurs when a solid is melted by volumetric heating can produce irregular solid-liquid interfaces. Such interfaces can be visualised in ice, where they are sometimes known as Tyndall stars. This paper describes some of the experimental observations of Tyndall stars and a mathematical model for the early stages of their evolution. The modelling is complicated by the strong crystalline anisotropy, which results in an anisotropic kinetic undercooling at the interface; it leads to an interesting class of free boundary problems that treat the melt region as infinitesimally thin.
Surface coatings and patterning technologies are essential for various physicochemical applications. In this Letter, we describe key parameters to achieve uniform particle coatings from binary solutions: First, multiple sequential Marangoni flows, set by solute and surfactant simultaneously, prevent non-uniform particle distributions and continuously mix suspended materials during droplet evaporation. Second, we show the importance of particle-surface interactions that can be established by surface-adsorbed macromolecules. To achieve a uniform deposit in a binary mixture, a small concentration of surfactant and surface-adsorbed polymer (0.05 wt% each) is sufficient, which offers a new physicochemical avenue for control of coatings.
Plants and insects use slender conical structures to transport and collect small droplets, which are propelled along the conical structures due to capillary action. These droplets can deposit a fluid film during their motion, but despite its importance to many biological systems and industrial applications the properties of the deposited film are unknown. We characterise the film deposition by developing an asymptotic analysis together with experimental measurements and numerical simulations based on the lubrication equation. We show that the deposited film thickness depends significantly on both the fiber radius and the droplet size, highlighting that the coating is affected by finite size effects relevant to film deposition on fibres of any slender geometry. We demonstrate that by changing the droplet size, while the mean fiber radius and the Capillary number are fixed, the thickness of the deposited film can change by an order of magnitude or more. We show that self-propelled droplets have significant potential to create passively coated structures.
It is known that an object translating parallel to a soft wall in a viscous fluid produces hydro- dynamic stresses that deform the wall, which, in turn, results in a lift force on the object. Recent experiments with cylinders sliding under gravity near a soft incline, which confirmed theoretical arguments for the lift force, also reported an unexplained steady-state rotation of the cylinders [Saintyves et al. PNAS 113(21), 2016]. Motivated by these observations, we show, in the lubrication limit, that an infinite cylinder that translates in a viscous fluid parallel to a soft wall at constant speed and separation distance must also rotate in order to remain free of torque. Using the Lorentz reciprocal theorem, we show analytically that for small deformations of the elastic layer, the angular velocity of the cylinder scales with the cube of the sliding speed. These predictions are confirmed numerically. We then apply the theory to the gravity-driven motion of a cylinder near a soft incline and find qualitative agreement with the experimental observations, namely that a softer elastic layer results in a greater angular speed of the cylinder.
We consider the deposition of a film of viscous liquid on a flat plate being withdrawn from a bath, experimentally and theoretically. For any plate speed $U$, there is a range of ``thick film solutions whose thickness scales like $U^{1/2}$ for small $U$. These solutions are realized for a partially wetting liquid, while for a perfectly wetting liquid the classical Landau-Levich-Derjaguin (LLD) film is observed, whose thickness scales like $U^{2/3}$. The thick film is distinguished from the LLD film by a dip in its spatial profile at the transition to the bath. We calculate the phase diagram for the existence of stationary film solutions as well as the film profiles, and find excellent agreement with experiment.
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