For a pair of complex conjugate eigenvalues of a square matrix with 𝑛² real components, we associate √−1’s of geometric origin in ℝ^𝑛. We use these to provide a real, coordinate free decomposition of real diagonalizable matrices with a clear geometric interpretation.

Water scarcity has required constant water recycling, leading to a decline in water quality, further exacerbated by high concentrations of fine particles that reduce the efficiency of solid–liquid separation systems. Inclined settlers offer a viable secondary treatment option for high-turbidity water. Effective design requires understanding of operational conditions, geometry, and suspension properties. Using OpenFOAM, computational fluid dynamics simulations were performed for a continuous inclined countercurrent conduit to assess the influence of inlet particle concentration on efficiency, exploring various Surface Overflow Rates (SOR) and inclination angles. The results show that the steady state in which the flow settles is strongly dependent on the particle concentration. For very low particle concentrations, the flow is mostly stationary with little to no resuspension of particles. Increasingly unstable regimes are observed to emerge as the inlet concentration increases, leading to increased particle resuspension. Instabilities arise from overhanging zones at the tip of the suspension, generating recirculation zones that enlarge the resuspension region and induce entrainment within the bulk suspension. Shear instabilities become noticeable at large particle concentrations, further increasing resuspension. Different regimes were identified, influenced by the SOR and the inclination angles. Additionally, a Reynolds number characterizing these systems is proposed alongside a scale analysis. The findings highlight particle concentration as a critical parameter in inclined plate settler design.

We examine the buckling shape and critical compression of confined inhomogeneous composite sheets lying on a liquid foundation. The buckling modes are controlled by the bending stiffness of the sheet, the density of the substrate, and the size and the spatially dependent elastic coefficients of the sheet. We solve the beam equation describing the mechanical equilibrium of a sheet when its bending stiffness varies parallel to the direction of confinement. The case of a homogeneous bending stiffness exhibits a degeneracy of wrinkled states for certain lengths of the confined sheet; we explain this degeneracy using an asymptotic analysis valid for long sheets, and show that it corresponds to the switching of the sheet between symmetric and antisymmetric buckling modes. This degeneracy disappears for spatially dependent elastic coefficients. Medium length sheets buckle similarly to their homogeneous counterparts, whereas the wrinkled states in large length sheets concentrate the bending energy towards the soft regions of the sheet. We examine the buckling shape and critical compression of confined inhomogeneous composite sheets lying on a liquid foundation.

Steeply inclined settlers (SIS) are solid–liquid separators that feature close, inclined confining elements that allow significant settling enhancement when compared to vertical tanks, resulting in relatively low footprints at equal throughput. The present article reviews the working principle, flow configuration, capacity, and several technological advances related to this kind of equipment. The consistency of the Ponder, Nakamura and Kuroda theory, developed after the discovery of the Boycott effect in 1920, which is essential to the settling enhancement effect, and the engineering approach developed independently for the design of SIS settlers during the late 1960’s, based on particle trajectory analysis, is established. A discussion about potential developments for future improvement of the technology is made, with emphasis on three main topical areas: improvements of settling element array, optimization of inlet conditions, and potential improvements of the design of settling elements. The application of the technology to the mining industry is discussed in the context of increasing water scarcity and the progressive ore grade decrease.

We study the linear spatiotemporal stability of an infinite row of equal point vortices under symmetric confinement between parallel walls. These rows of vortices serve to model the secondary instability leading to the merging of consecutive (Kelvin-Helmholtz) vortices in free shear layers, allowing us to study how confinement limits the growth of shear layers through vortex pairings. Using a geometric construction akin to a Legendre transform on the dispersion relation, we compute the growth rate of the instability in different reference frames as a function of the frame velocity with respect to the vortices. This approach is verified and complemented with numerical computations of the linear impulse response, fully characterizing the absolute/convective nature of the instability. Similar to results by Healey on the primary instability of parallel tanh profiles [J. Fluid Mech. 623, 241 (2009)], we observe a range of confinement in which absolute instability is promoted. For a parallel shear layer with prescribed confinement and mixing length, the threshold for absolute/convective instability of the secondary pairing instability depends on the separation distance between consecutive vortices, which is physically determined by the wavelength selected by the previous (primary or pairing) instability. In the presence of counterflow and moderate to weak confinement, small (large) wavelength of the vortex row leads to absolute (convective) instability. While absolute secondary instabilities in spatially developing flows have been previously related to an abrupt transition to a complex behavior, this secondary pairing instability regenerates the flow with an increased wavelength, eventually leading to a convectively unstable row of vortices. We argue that since the primary instability remains active for large wavelengths, a spatially developing shear layer can directly saturate on the wavelength of such a convectively unstable row, by-passing the smaller wavelengths of absolute secondary instability. This provides a wavelength selection mechanism, according to which the distance between consecutive vortices should be sufficiently large in comparison with the channel width in order for the row of vortices to persist. We argue that the proposed wavelength selection criteria can serve as a guideline for experimentally obtaining plane shear layers with counterflow, which has remained an experimental challenge.