Cerebrospinal fluid (CSF) flow is modelled in human perivascular spaces (PVSs) that surround brain penetrating arteries. It is proposed that the outer PVS boundary oscillates due to brain pulsations, and the arterial wall motion is driven by a blood pressure wave. Lubrication theory is employed to derive a mathematical model for the CSF flow, which is then solved numerically. A parametric analysis is undertaken to investigate the effect of the brain pulsations, which shows that pulsations magnify the net axial CSF flows created by the arterial wall motion. The findings suggest that net axial CSF flows are almost entirely positive (deeper into the brain), with arterial wall motion highly dependent on PVS-penetrating artery configurations. Given the glymphatic hypothesis, the findings support the clinical practice of treating dilated PVSs as indicators of an increased likelihood of neurodegenerative conditions, such as dementia.

We study the emergence of strong interactions between dissipative coherent structures (pulses) in spatially extended systems. Focusing first on a prototypical model problem from fluid dynamics, that of liquid film flowing down a vertical plane, we show that under certain conditions, a two-pulse system undergoes a transition from a regime of decaying oscillatory dynamics to one with self-sustained oscillations. Intriguingly, such a transition is not governed by the standard Hopf bifurcation. Instead, a novel governing mechanism for transition to oscillatory dynamics is unraveled via a peculiar and atypical Hopf bifurcation in which a complex-conjugate resonance pair crosses the imaginary axis in the complex plane. Prior to crossing the essential spectrum (including at the bifurcation point), this resonance pair does not appear in the standard L_{C}^{2}-based spectral analysis but reveals itself when appropriate weighted functional spaces are used. We show that such a resonance pair originates from the splitting of a resonance pole of the single-pulse system. While this object is not part of the classical spectrum, it plays a vital role in shaping the system’s dynamics. We further demonstrate that this resonance-pole mechanism extends to a broad range of systems. Specifically, in the generalised Kuramoto-Sivashinsky equation—a model prototype applicable across a wide range of fields from fluid dynamics to geophysics and plasma physics—we observe the same bifurcation and resulting oscillatory pulse interactions. By contrast, in the FitzHugh-Nagumo model—a central model prototype in reaction-diffusion systems—the resonance pole splits into real eigenvalues, and monotonic pulse interactions occur. In addition, we illustrate that the resonance pole may induce oscillatory interactions in three-pulse systems and eventually lead to chaotic dynamics in strongly interacting multipulse systems, which can be quantified in terms of a positive Lyapunov exponent.

For homogeneous and isotropic linearly elastic solids and for incompressible fluids under low-Reynolds-number conditions, the fundamental solutions of the associated continuum equations were derived a long time ago for bulk systems. That is, the corresponding Green's functions are available in infinitely extended systems, where boundaries do not play any role. However, introducing boundaries renders the situation significantly more complex. Here, we derive the corresponding Green's functions for a linearly elastic homogeneous and isotropic material in a wedge-shaped geometry. Two flat boundaries confine the material and meet at a straight edge. No-slip (NOS) and free-slip (FRS) conditions are considered. The force is oriented in a direction parallel to the straight edge of the wedge. Assuming incompressibility, our expressions also apply to the situation of low-Reynolds-number hydrodynamic viscous fluid flows. Thus, they may be used, for instance, to describe the motion of self-propelled objects guided by an edge or the distortion of soft elastic actuators in wedge-shaped environments of operation.

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 a diffuse interface approach. Periodic boundary conditions are incorporated strictly through the introduction of an input -transfer layer (Fourier feature mapping), which takes the sine and cosine of the inner product of position and reciprocal lattice vectors. This, together with the absence of Dirichlet boundary conditions, results in a lossless boundary condition application. Consequently, the sole contributors to the loss are the locally -scaled differential equation residuals. We use crystalline arrangements that are defined via Bravais lattices to demonstrate the formulation's versatility based on the reciprocal lattice vectors. 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, including periodic composites, composed of embeddings of monodisperse inclusions in the form of disks/spheres, and stochastic monodisperse disk arrangements.

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 in the resistive force, is surprisingly complicated. We also discuss a general transversion formula for changing the dependent variables in Stokes flow problems.

The shape and stability of a droplet in contact with a solid surface is affected by the chemical composition and topography of the solid, and crucially, by the droplet's size. During a variation in size, most often observed during evaporation, droplets on smooth patterned surfaces can undergo sudden shape and position changes. Such changes, called snaps, are prompted by the surface pattern and arise from fold and pitchfork bifurcations which respectively cause symmetric and asymmetric motions. Yet, which type of snap is likely to be observed is an open fundamental question that has relevance in the rational design of surfaces for managing droplets. Here we show that the likelihood of observing symmetric or asymmetric snaps depends on the distance between fold and pitchfork bifurcation points and on how this distance varies for droplets that grow or shrink in size on surfaces patterned with a smooth topography. Our results can help develop strategies to control droplets by exploiting smooth surface patterns but also have broader relevance in situations where different types of bifurcations compete in determining the stability of a system, for instance in snap-through instabilities observed in elastic media.

To make progress towards the development of a theory on the motion of inclusions in thin structured films and membranes, we here consider as an initial step a circular disk in a two-dimensional, uniaxially anisotropic fluid layer. We assume overdamped dynamics, incompressibility of the fluid, and global alignment of the axis of anisotropy. Motion within this layer is affected by additional linear friction with the environment, for instance, a supporting substrate. We investigate the induced flows in the fluid when the disk is translated parallel or perpendicular to the direction of anisotropy. Moreover, expressions for corresponding mobilities and resistance coefficients of the disk are derived. Our results are obtained within the framework of a perturbative expansion in the parameters that quantify the anisotropy of the fluid. Good agreement is found for moderate anisotropy when compared to associated results from finite-element simulations. At pronounced anisotropy, the induced flow fields are still predicted qualitatively correctly by the perturbative theory, although quantitative deviations arise. We hope to stimulate with our investigations corresponding experimental analyses, for example, concerning fluid flows in anisotropic thin films on uniaxially rubbed supporting substrates.

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.

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. We find that, depending upon the initial position of the plate and the position of the centre of force, the plate might either make contact in finite time or settle progressively without ever making contact. Our results show an excellent agreement between analytical exact solutions, asymptotic solutions and numerical studies of the lubrication equations.

We study the evolution and lifetime of droplets evaporating on a smooth chemical pattern, which is characterised by a spatially varying contact angle. We formulate a model that combines the evaporation rate of the droplet for a given volume with the static stability of the droplet as the volume changes in time quasi-statically. We derive an exact equation for the evaporation rate that is studied analytically under the limiting cases of nearly neutral wetting and highly hydrophilic conditions. We find that the evaporation rate of the droplet is highly dependent on the size and shape of the fictitious infinity where a far-field boundary conditions needs to be applied. We also study how the droplet’s lifetime depends on the averaged contact angle and strength $$\varepsilon $$ of the chemical pattern, observing that the lifetime of the droplet is maximised for a droplet with average contact angle $$\pi /2$$ and with $$\varepsilon \lesssim 0.1$$ .

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.

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.

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.

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.

We propose a new numerical method to solve the Cahn-Hilliard equation coupled with non-linear wetting boundary conditions. We show that the method is mass-conservative and that the discrete solution satisfies a discrete energy law similar to the one satisfied by the exact solution. We perform several tests inspired by realistic situations to verify the accuracy and performance of the method: wetting of a chemically heterogeneous substrate in three dimensions, wetting-driven nucleation in a complex two-dimensional domain and three-dimensional diffusion through a porous medium.

We study contact angle hysteresis in a chemically heterogeneous microchannel by tracking static meniscus configurations in the microchannel upon varying the volume of liquid. We first construct a graphical force balance similar to a previous approach by Joanny and de Gennes for this system, though here with a straight contact line. It is shown that hysteresis is induced by wettability gradients above a finite threshold value. This is also visualized in a phase-plane plot enabling to easily predict stick-slip events of the contact line and the occurrence of hysteresis. Above the threshold and for nonoverlapping Gaussian defects, we find good agreement with the expressions by Joanny and de Gennes for the hysteresis amplitude induced by a dilute system of defects. In particular, the hysteresis amplitude is found to be proportional to the square of the defect force and to the defect concentration. For a model sinusoidal heterogeneity, decreasing the ratio between the heterogeneity wavelength and the microchannel gap size brings the system from a subthreshold regime, to a stick-slip dominated regime, and finally to a regime with a quasiconstant advancing and receding angle. In the latter case, the hysteresis amplitude is found to be proportional to the defect force. We also consider an unusual heterogeneity for which the gradients of increasing and decreasing wettability are different. In such a situation breaking the left/right symmetry, whether or not hysteresis is observed will depend on the side the liquid enters the microchannel.