Waves on a thin liquid layer falling down a solid wall, either vertical or inclined, are studied by means of a reduced equation. This equation is developed by the regularized long-wave expansion method, which is a combination of the Padé approximation and the long-wave expansion. Its numerical solutions are compared with the calculations of the full Navier–Stokes equation, simplified Navier–Stokes equation (the “boundary-layer” equation), and the traditional long-wave equations, as well as with experimental measurements. When the Reynolds number R is as small as unity, the present equation agrees with the Navier–Stokes equation and also with the traditional long-wave equations. For larger values of R, the traditional long-wave equations lose their validity and make a false prediction, while the present equation agrees with the Navier–Stokes equation, as long as the rescaled Reynolds number δ*=R/W1/3 does not exceed unity in the case of vertical films. Unlike the “boundary-layer” equation developed by previous researchers and expected to be valid at moderate and large Reynolds number, the present equation governs the surface evolution alone without postulating to resolve the velocity field. The structure of the present equation, however, has a correspondence to the depth-averaged equations, which facilitates discussing the physical mechanism of the wave dynamics. In particular, the physical origin of the instability mechanism and the wave suppression mechanism are discussed in terms of Whitham’s wave hierarchy theory. The balance of several physical effects such as drag, gravity, and inertia are also discussed in this connection. The analysis of the tail structure of permanent solitary waves predicts that the R dependence of its tail length λ exhibits two distinct regimes in the λ-R diagram. The second regime, which is not predicted by the traditional long-wave equations, arises when the inertia effect becomes dominant.

As an indicator of cooperative motion in a system of Brownian particles that models two-dimensional colloidal liquids, a displacement correlation tensor is calculated analytically and compared with numerical results. The key idea for the analytical calculation is to relate the displacement correlation tensor, which is a kind of four-point space-time correlation, to the Lagrangian two-time correlation of the deformation gradient tensor. Tensorial treatment of the statistical quantities, including the displacement correlation itself, allows capturing the vortical structure of the cooperative motion. The calculated displacement correlation also implies a negative long-time tail in the velocity autocorrelation, which is a manifestation of the cage effect. Both the longitudinal and transverse components of the displacement correlation are found to be expressible in terms of a similarity variable, suggesting that the cages are nested to form a self-similar structure in the space-time.

Pattern formation of desiccation cracks on a layer of a calcium carbonate paste is studied experimentally. This paste is known to exhibit a memory effect, which means that a short-time application of horizontal vibration to the fresh paste predetermines the direction of the cracks that are formed after the paste is dried. While the position of the cracks (as opposed to their direction) is still stochastic in the case of horizontal vibration, the present work reports that their positioning is also controllable, at least to some extent, by applying vertical vibration to the paste and imprinting the pattern of Faraday waves, thus breaking the translational symmetry of the system. The experiments show that the cracks tend to appear in the node zones of the Faraday waves: in the case of stripe-patterned Faraday waves, the cracks are formed twice more frequently in the node zones than in the anti-node zones, presumably due to the localized horizontal motion. As a result of this preference of the cracks to the node zones, the memory of the square lattice pattern of Faraday waves makes the cracks run in the oblique direction differing by 45 degrees from the intuitive lattice direction of the Faraday waves.

Diffusion in colloidal suspensions can be very slow due to the cage effect, which confines each particle within a short radius on one hand, and involves large-scale cooperative motions on the other. In search of insight into this cooperativity, here the authors develop a formalism to calculate the displacement correlation in colloidal systems, mainly in the two-dimensional case. To clarify the idea for it, studies are reviewed on cooperativity among the particles in the one-dimensional case, i.e. the single-file diffusion (SFD). As an improvement over the celebrated formula by Alexander and Pincus on the mean-square displacement (MSD) in SFD, it is shown that the displacement correlation in SFD can be calculated from Lagrangian correlation of the particle interval in the one-dimensional case, and also that the formula can be extended to higher dimensions. The improved formula becomes exact for large systems. By combining the formula with a nonlinear theory for correlation, a correction to the asymptotic law for the MSD in SFD is obtained. In the two-dimensional case, the linear theory gives description of vortical cooperative motion. Special Issue Comments:This article presents methodological insights into mathematical treatment of particle cooperativity in colloidal liquids. This article is related to the Special Issue articles [in Biophys. Rev. Lett.] about mathematical approaches to single-file diffusion 1 and Fourier-based analysis of quasi-onedimensional systems.

Plastic materials can carry memory of past mechanical treatment in the form of internal stress. We introduce a natural definition of the vorticity of internal stress in a simple two-dimensional model of elastoplastic fluids, which generates the internal stress. We demonstrate how the internal stress is induced under external loading, and how the presence of the internal stress modifies the plastic behavior.