When a plate is withdrawn from a liquid bath a coating layer is deposited whose thickness and homogeneity depend on the velocity and the wetting properties of the plate. Using a long-wave mesoscopic hydrodynamic description that incorporates wettability via a Derjaguin (disjoining) pressure we identify four qualitatively different dynamic transitions between microscopic and macroscopic coatings that are out-of-equilibrium equivalents of well known equilibrium unbinding transitions. Namely, these are continuous and discontinuous dynamic emptying transitions and discontinuous and continuous dynamic wetting transitions. We uncover several features that have no equivalent at equilibrium.The equilibrium and non-equilibrium behaviour of mesoscopic and macroscopic drops, meniscii and films of liquid in contact with static or moving solid substrates is not only of fundamental interest but also crucial for a large number of modern technologies. On the one hand, the equilibrium behaviour of films, drops and meniscii is studied by means of statistical physics. A rich substrate-induced phase transition behaviour is described even for simple liquids, e.g., related to wetting and emptying transitions that both represent unbinding transitions. In the former case the thickness of an adsorption layer of liquid diverges continuously or discontinuously at a critical temperature or strength of substrate-liquid interaction, i.e., the liquid-gas interface of the film unbinds from the liquid-solid interface [1]. In the case of the emptying transition a macroscopic meniscus in a tilted slit capillary develops a tongue (or foot) along the lower wall of a length that diverges logarithmically at a critical slit width, i.e., the tip of the foot unbinds from the meniscus [2].
Abstract. A liquid film is studied that is deposited onto a flat plate that is inclined at a constant angle to the horizontal and is extracted from a liquid bath at a constant speed. We analyse steady-state solutions of a long-wave evolution equation for the film thickness. Using centre manifold theory, we first obtain an asymptotic expansion of solutions in the bath region. The presence of an additional temperature gradient along the plate that induces a Marangoni shear stress significantly changes these expansions and leads to the presence of logarithmic terms that are absent otherwise. Next, we numerically obtain steady solutions and analyse their behaviour as the plate velocity is changed. We observe that the bifurcation curve exhibits collapsed (or exponential) heteroclinic snaking when the plate inclination angle is above a certain critical value. Otherwise, the bifurcation curve is monotonic. The steady profiles along these curves are characterised by a foot-like structure that is formed close to the meniscus and is preceded by a thin precursor film further up the plate. The length of the foot increases along the bifurcation curve. Finally, we prove with a Shilnikov-type method that the snaking behaviour of the bifurcation curves is caused by the existence of an infinite number of heteroclinic orbits close to a heteroclinic chain that connects in an appropriate three-dimensional phase space the fixed point corresponding to the precursor film with the fixed point corresponding to the foot and then with the fixed point corresponding to the bath.
During the spreading of a liquid over a solid substrate, the contact line can stay pinned at sharp edges until the contact angle exceeds a critical value. At (or sufficiently near) equilibrium, this is known as Gibbs' criterion. Here, we show both experimentally and theoretically that for completely wetting volatile liquids there also exists a dynamically-produced critical angle for depinning, which increases with the evaporation rate. This suggests that one may introduce a simple modification of the Gibbs' criterion for (de)pinning, that accounts for the non-equilibrium effect of evaporation.
Membranes have been shown to be exceptionally successfully in the challenging separation of stable oil-water emulsions, but suffer from severe fouling that limits their performance. Understanding the mechanisms leading to oil deposition on the membrane surface, as influenced by hydrodynamics and colloidal surface interactions is imperative for informing better engineered membrane surfaces and process conditions.Here, we study the the interactions between an oil droplet and a membrane surface.Hydrodynamics within the water film, confined between the droplet and the membrane, are captured within the framework of the lubrication approximation, coupled with the van der Waals (vdW) and electrostatic interactions through the droplet shape, which is governed by an augmented Young-Laplace equation. The model is used to calculate possible equilibrium positions, where the droplet is held at a finite distance from the membrane by a balance of the forces present. An equilibrium phase diagram is constructed as a function of various process parameters, and is shown in terms of the scaled permeation rate through the membrane. The phase diagram identifies the range of conditions leading to deposition, characterized by a 'critical' permeation rate, beyond which no equilibrium exists. When equilibrium positions are permitted, we find that these may be classified as stable/unstable, in the kinetic sense. Further, our arXiv:2003.10349v1 [physics.flu-dyn] 23 Mar 2020 results demonstrate the link between the deformation of the droplet and the stability of equilibria. An upward deflection of the droplet surface, owing to a dominant, long-range repulsion, has a stabilizing effect as it maintains the separation between droplet and membrane. Conversely, a downward deflection is de-stabilizing, due to the self-amplifying effect of strongly increasing attractive forces with separation distanceas the surfaces are pulled together due to deformation, the attractive force increases, causing further deformation. This is also manifested by a dependence of the bi-stable region on the deformability of the droplet, which is represented by a capillary number, modified so as to account for the effect of the permeable boundary. As the droplet becomes more easy to deform, the transition from an unconditionally stable region of the phase diagram, to a point beyond which there is no equilibrium (interpreted as deposition) becomes abrupt. These results provide valuable physical insight into the mechanisms that govern oil fouling of membrane surfaces.
This note is based on a relatively unknown paper of Albert Einstein published in 1941 in the Revista de la Universidad Nacional de Tucumán. That work can be regarded as the prequel of the one written in 1943 in collaboration with Wolfgang Pauli, where, along the same lines, a proof of the non-existence of certain non-singular solutions in Kaluza-Klein theory was given. More generally, disproving the existence of regular solutions in classical unified field theories became, specially after 1930, an important criterion leading Einstein's investigations on unified field theory. This is the context in which Einstein's paper of 1941 and its generalizations of 1943 and 1948 become important.
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