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Dynamics at low Reynolds numbers experiences recent revival in the fields of biophysics and active matter. While in bulk isotropic fluids it is exhaustively studied, this is less so in anisotropic fluids and in confined situations. Here, we combine the latter two by studying the rotation of a disk-like inclusion in a uniaxially anisotropic, globally oriented, incompressible two-dimensional fluid film. In terms of a perturbative expansion in parameters that quantify anisotropies in viscosity and in additional linear friction with a supporting substrate or other type of confinement, we derive analytical expressions for the resulting hydrodynamic flow and pressure fields as well as for the resistance and mobility coefficients of the rotating disk. It turns out that, in contrast to translational motion, the solutions remain well-behaved also in the absence of the additional linear friction. Comparison with results from finite-element simulations show very good agreement with those from our analytical calculations. Besides applications to describe technological systems, for instance, in the area of microfluidics and thin cells of aligned nematic liquid crystals, our solutions are important for quantitative theoretical approaches to fluid membranes and thin films in general featuring a preferred direction., Comment: 17 pages, 5 figures

We obtain the full equations of motion for a wide, pivoted, slider bearing. These are used to review the choice of the optimal position for the pivot point, to discuss its response to time-dependent sliding velocity, and to determine the stability of the motion. The case of an abrupt acceleration of the slider, which results in a large transient increase of the resistive force, is surprisingly complicated. We also discuss a general \emph{transversion} formula for changing the dependent variables in Stokes flow problems., Comment: 18 pages, 14 figures

We discuss a model for directed percolation in which the flux of material along each bond is a dynamical variable. The model includes a physically significant limiting case where the total flux of material is conserved. We show that the distribution of fluxes is asymptotic to a power law at small fluxes. We give an implicit equation for the exponent, in terms of probabilities characterising site occupations. In one dimension the site occupations are exactly independent, and the model is exactly solvable. In two dimensions, the independent-occupation assumption gives a good approximation. We explore the relationship between this model and traditional models for directed percolation., Comment: 30 pages, 16 figures

If a flat, horizontal, plate settles onto a flat surface, it is known that the gap $h$ decreases with time $t$ as a power-law: $h\sim t^{-1/2}$. We consider what happens if the plate is not initially horizontal, and/or the centre of mass is not symmetrically positioned: does one edge contact the surface in finite time, or does the plate approach the horizontal without making contact? The dynamics of this system is analysed and shown to be remarkably complex., Comment: 22 pages, 9 figures

We apply physics-informed neural networks (PINNs) to first-order two-scale periodic asymptotic homogenization of the property tensor in a generic elliptic equation. The problem of lack of differentiability of property tensors at the sharp phase interfaces is circumvented by making use of diffuse interface formulation. Periodic boundary conditions are incorporated strictly, through the introduction of an input-transfer layer (Fourier feature mapping), in which the sine and cosine of the inner product of position vectors and reciprocal lattice vectors are considered. This, together with the absence of Dirichlet boundary conditions, results in a lossless boundary condition application. The only loss terms are then due to the differential equation itself, which removes the necessity of scaling the loss entries. In demonstrating the formulation's versatility based on the reciprocal lattice vectors, crystalline arrangements defined with Bravais lattices are used. We also show that considering integer multiples of the reciprocal basis in the Fourier mapping leads to improved convergence of high-frequency functions. We consider applications in one, two, and three dimensions. Periodic composites, composed of embeddings of monodisperse inclusions in the form of disks/spheres in the two-/three-dimensional matrix, are considered. For demonstration purposes, stochastic monodisperse disk arrangements are also considered.

Experiments investigating particles floating on a randomly stirred fluid show regions of very low density, which are not well understood. We introduce a simplified model for understanding sparsely occupied regions of the phase space of non-autonomous, chaotic dynamical systems, based upon an extension of the skinny bakers' map. We show how the distribution of the sizes of voids in the phase space can be mapped to the statistics of the running maximum of a Wiener process. We find that the model exhibits a lacunarity transition, which is characterised by regions of the phase space remaining empty as the number of trajectories is increased., Comment: 16 pages, 8 figures

We investigate the evaporation of a two-dimensional droplet on a solid surface. The solid is flat but with smooth chemical variations that lead to a space-dependent local contact angle. We perform a detailed bifurcation analysis of the equilibrium properties of the droplet as its size is changed, observing the emergence of a hierarchy of bifurcations that strongly depends on the particular underlying chemical pattern. Symmetric and periodic patterns lead to a sequence of pitchfork and saddle-node bifurcations that make stable solutions to become saddle nodes. Under dynamic conditions, this change instability suggests that any perturbation in the system can make the droplet to shift laterally while relaxing to the nearest stable point, as is confirmed by numerical computations of the Cahn-Hilliard and Navier-Stokes system of equations. We also consider patterns with an amplitude gradient that creates a set of disconnected stable branches in the solution space, leading to a continuous change of the droplet's location upon evaporation., Comment: 13 pages, 8 figures

We discuss the combined effects of overdamped motion in a quenched random potential and diffusion, in one dimension, in the limit where the diffusion coefficient is small. Our analysis considers the statistics of the mean first-passage time $T(x)$ to reach position $x$, arising from different realisations of the random potential: specifically, we contrast the median $\bar T(x)$, which is an informative description of the typical course of the dispersion, with the expectation value $\langle T(x)\rangle$, which is dominated by rare events where there is an exceptionally high barrier to diffusion. We show that at relatively short times the median $\bar T(x)$ is explained by a 'flooding' model, where $T(x)$ is predominantly determined by the highest barriers which is encountered before reaching position $x$. These highest barriers are quantified using methods of extreme value statistics., Comment: 17 pages, 5 figures

Many physical processes result in very uneven, apparently random, distributions of matter, characterized by fluctuations of the local density over orders of magnitude. The density of matter in the sparsest regions can have a power-law distribution, with an exponent that we term the lacunarity exponent. We discuss a mechanism which explains the wide occurrence of these power laws, and give analytical expressions for the exponent in some simple models., Comment: 5 pages, 4 figures

This paper analyses a model for the intensity of distribution for rays propagating without absorption in a random medium. The random medium is modelled as a dynamical map. After $N$ iterations, the intensity is modelled as a sum $S$ of ${\cal N}$ contributions from different trajectories, each of which is a product of $N$ independent identically distributed random variables $x_k$, representing successive focussing or de-focussing events. The number of ray trajectories reaching a given point is assumed to proliferate exponentially: ${\cal N}=\Lambda^N$, for some $\Lambda>1$. We investigate the probability distribution of $S$. We find a phase transition as parameters of the model are varied. There is a phase where the fluctuations of $S$ are suppressed as $N\to \infty$, and a phase where the $S$ has large fluctuations, for which we provide a large deviation analysis., Comment: 14 pages, 3 figures

Intrinsic instability of trajectories characterizes chaotic dynamical systems. We report here that trajectories can exhibit a surprisingly high degree of stability, over a very long time, in a chaotic dynamical system. We provide a detailed quantitative description of this effect for a one-dimensional model of inertial particles in a turbulent flow using large-deviation theory. Specifically, the determination of the entropy function for the distribution of finite-time Lyapunov exponents reduces to the analysis of a Schr\"odinger equation, which is tackled by semi-classical methods., Comment: 6 pages, 4 figures

We investigate statistical properties of trails formed by a random process incorporating aggregation, fragmentation, and diffusion. In this stochastic process, which takes place in one spatial dimension, two neighboring trails may combine to form a larger one and also, one trail may split into two. In addition, trails move diffusively. The model is defined by two parameters which quantify the fragmentation rate and the fragment size. In the long-time limit, the system reaches a steady state, and our focus is the limiting distribution of trail weights. We find that the density of trail weight has power-law tail $P(w) \sim w^{-\gamma}$ for small weight $w$. We obtain the exponent $\gamma$ analytically, and find that it varies continuously with the two model parameters. The exponent $\gamma$ can be positive or negative, so that in one range of parameters small-weight tails are abundant, and in the complementary range, they are rare., Comment: 8 pages, 8 figures

Chaos is widely understood as being a consequence of sensitive dependence upon initial conditions. This is the result of an instability in phase space, which separates trajectories exponentially. Here, we demonstrate that this criterion should be refined. Despite their overall intrinsic instability, trajectories may be very strongly convergent in phase space over extremely long periods, as revealed by our investigation of a simple chaotic system (a realistic model for small bodies in a turbulent flow). We establish that this strong convergence is a multi-facetted phenomenon, in which the clustering is intense, widespread and balanced by lacunarity of other regions. Power laws, indicative of scale-free features, characterise the distribution of particles in the system. We use large-deviation and extreme-value statistics to explain the effect. Our results show that the interpretation of the 'butterfly effect' needs to be carefully qualified. We argue that the combination of mixing and clustering processes makes our specific model relevant to understanding the evolution of simple organisms. Lastly, this notion of 'convergent chaos', which implies the existence of conditions for which uncertainties are unexpectedly small, may also be relevant to the valuation of insurance and futures contracts., Comment: 9 pages, 7 figures

We present a novel control methodology to control the roughening processes of semilinear parabolic stochastic partial differential equations in one dimension, which we exemplify with the stochastic Kuramoto-Sivashinsky equation. The original equation is split into a linear stochastic and a nonlinear deterministic equation so that we can apply linear feedback control methods. Our control strategy is then based on two steps: first, stabilize the zero solution of the deterministic part and, second, control the roughness of the stochastic linear equation. We consider both periodic controls and point actuated ones, observing in all cases that the second moment of the solution evolves in time according to a power-law until it saturates at the desired controlled value.

We consider a dynamical system which has a stable attractor and which is perturbed by an additive noise. Under some quite typical conditions, the fluctuations from the attractor are intermittent and have a probability distribution with power-law tails. We show that this results from a stochastic cascade of amplification of fluctuations due to transient periods of instability. The exponent of the power-law is interpreted as a negative fractal dimension., Comment: 5 pages, 4 figures

Using thermodynamic and variational principles we examine a basic phase field model for a mixture of two incompressible fluids in strongly perforated domains. With the help of the multiple scale method with drift and our recently introduced splitting strategy for Ginzburg-Landau/Cahn-Hilliard-type equations [Schmuck et al., Proc. R. Soc. A 468:3705-3724, 2012.], we rigorously derive an effective macroscopic phase field formulation under the assumption of periodic flow and a sufficiently large P\'eclet number. As for classical convection-diffusion problems, we obtain systematically diffusion-dispersion relations (including Taylor-Aris-dispersion). Our results also provide a convenient analytical and computational framework to macroscopically track interfaces in porous media. In view of the well-known versatility of phase field models, our study proposes a promising model for many engineering and scientific applications such as multiphase flows in porous media, microfluidics, and fuel cells., Comment: 25 pages, 5 figures

We derive a new effective macroscopic Cahn-Hilliard equation whose homogeneous free energy is represented by 4-th order polynomials, which form the frequently applied double-well potential. This upscaling is done for perforated/strongly het- erogeneous domains. To the best knowledge of the authors, this seems to be the first attempt of upscaling the Cahn-Hilliard equation in such domains. The new homog- enized equation should have a broad range of applicability due to the well-known versatility of phase-field models. The additionally introduced feature of systemati- cally and reliably accounting for confined geometries by homogenization allows for new modeling and numerical perspectives in both, science and engineering. Our results are applied to wetting dynamics in porous media and to a single channel with strongly heterogeneous walls.

We examine the interaction of two-dimensional solitary pulses on falling liquid films. We make use of the second-order model derived by Ruyer-Quil and Manneville [Eur. Phys. J. B 6, 277 (1998); Eur. Phys. J. B 15, 357 (2000); Phys. Fluids 14, 170 (2002)] by combining the long-wave approximation with a weighted residuals technique. The model includes (second-order) viscous dispersion effects which originate from the streamwise momentum equation and tangential stress balance. These effects play a dispersive role that primarily influences the shape of the capillary ripples in front of the solitary pulses. We show that different physical parameters, such as surface tension and viscosity, play a crucial role in the interaction between solitary pulses giving rise eventually to the formation of bound states consisting of two or more pulses separated by well-defined distances and travelling at the same velocity. By developing a rigorous coherent-structure theory, we are able to theoretically predict the pulse-separation distances for which bound states are formed. Viscous dispersion affects the distances at which bound states are observed. We show that the theory is in very good agreement with computations of the second-order model. We also demonstrate that the presence of bound states allows the film free surface to reach a self-organized state that can be statistically described in terms of a gas of solitary waves separated by a typical mean distance and characterized by a typical density.

We investigate the complex spatio-temporal dynamics in avalanche driven surface growth by means of scaling theory. We study local activity statistics, avalanche kinetics, and temporal correlations in the global interface velocity, obtaining different scaling relationships among the involved critical exponents depending on how far from or close to a critical point the system is. Our scaling arguments are very general and connect local and global magnitudes through several scaling relationships. We expect our results to be applicable in a wide range of systems exhibiting interface kinetic roughening driven by avalanches of local activity, either critical or not. As an example we apply the scaling theory to analyze avalanches and roughening of forced-flow imbibition fronts in excellent agreement with phase-field numerical simulations., Comment: 6 pages, 3 eps figures

We study avalanche dynamics and local activity of forced-flow imbibition fronts in disordered media. We focus on the front dynamics as the mean velocity $\bar{v}$ of the interface is decreased and the pinning state is approached. Scaling arguments allow us to obtain the statistics of avalanche sizes and durations, which become power-law distributed due to the existence of a critical point at $\bar{v}= 0$. Results are compared with phase-field numerical simulations.

We show that time dependent couplings may lead to nontrivial scaling properties of the surface fluctuations of the asymptotic regime in non-equilibrium kinetic roughening models . Three typical situations are studied. In the case of a crossover between two different rough regimes, the time-dependent coupling may result in anomalous scaling for scales above the crossover length. In a different setting, for a crossover from a rough to either a flat or damping regime, the time dependent crossover length may conspire to produce a rough surface, despite the most relevant term tends to flatten the surface. In addition, our analysis sheds light into an existing debate in the problem of spontaneous imbibition, where time dependent couplings naturally arise in theoretical models and experiments., Comment: Accepted for publication in Physical Review E (Rapid Comm.)