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1. Clustering in Euler–Euler and Euler–Lagrange simulations of unbounded homogeneous particle-laden shear

   https://doi.org/10.1017/jfm.2018.796  

   Kasbaoui, M. Houssem; Koch, Donald L.; Desjardins, Olivier (January 2019, Journal of Fluid Mechanics)

   Particle-laden flows of sedimenting solid particles or droplets in a carrier gas have strong inter-phase coupling. Even at low particle volume fractions, the two-way coupling can be significant due to the large particle to gas density ratio. In this semi-dilute regime, the slip velocity between phases leads to sustained clustering that strongly modulates the overall flow. The analysis of perturbations in homogeneous shear reveals the process by which clusters form: (i) the preferential concentration of inertial particles in the stretching regions of the flow leads to the formation of highly concentrated particle sheets, (ii) the thickness of the latter is controlled by particle-trajectory crossing, which causes a local dispersion of particles, (iii) a transverse Rayleigh–Taylor instability, aided by the shear-induced rotation of the particle sheets towards the gravity normal direction, breaks the planar structure into smaller clusters. Simulations in the Euler–Lagrange formalism are compared to Euler–Euler simulations with the two-fluid and anisotropic-Gaussian methods. It is found that the two-fluid method is unable to capture the particle dispersion due to particle-trajectory crossing and leads instead to the formation of discontinuities. These are removed with the anisotropic-Gaussian method which derives from a kinetic approach with particle-trajectory crossing in mind. 

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2. Euler characteristic of stable envelopes

   https://doi.org/10.1007/s00029-022-00788-w  

   Dinkins, Hunter; Smirnov, Andrey (September 2022, Selecta Mathematica)
3. Euler systems for GSp(4)

   https://doi.org/10.4171/JEMS/1124  

   Loeffler, David; Skinner, Christopher; Zerbes, Sarah Livia (January 2022, Journal of the European Mathematical Society)
4. Euler Transformation of Polyhedral Complexes

   https://doi.org/10.1142/S0218195920500090  

   Gupta, Prashant; Krishnamoorthy, Bala (January 2021, International Journal of Computational Geometry & Applications)

   null (Ed.)

   We propose an Euler transformation that transforms a given [Formula: see text]-dimensional cell complex [Formula: see text] for [Formula: see text] into a new [Formula: see text]-complex [Formula: see text] in which every vertex is part of the same even number of edges. Hence every vertex in the graph [Formula: see text] that is the [Formula: see text]-skeleton of [Formula: see text] has an even degree, which makes [Formula: see text] Eulerian, i.e., it is guaranteed to contain an Eulerian tour. Meshes whose edges admit Eulerian tours are crucial in coverage problems arising in several applications including 3D printing and robotics. For [Formula: see text]-complexes in [Formula: see text] ([Formula: see text]) under mild assumptions (that no two adjacent edges of a [Formula: see text]-cell in [Formula: see text] are boundary edges), we show that the Euler transformed [Formula: see text]-complex [Formula: see text] has a geometric realization in [Formula: see text], and that each vertex in its [Formula: see text]-skeleton has degree [Formula: see text]. We bound the numbers of vertices, edges, and [Formula: see text]-cells in [Formula: see text] as small scalar multiples of the corresponding numbers in [Formula: see text]. We prove corresponding results for [Formula: see text]-complexes in [Formula: see text] under an additional assumption that the degree of a vertex in each [Formula: see text]-cell containing it is [Formula: see text]. In this setting, every vertex in [Formula: see text] is shown to have a degree of [Formula: see text]. We also present bounds on parameters measuring geometric quality (aspect ratios, minimum edge length, and maximum angle of cells) of [Formula: see text] in terms of the corresponding parameters of [Formula: see text] for [Formula: see text]. Finally, we illustrate a direct application of the proposed Euler transformation in additive manufacturing. 

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5. Axial-Current Anomaly in Euler Fluids

   https://doi.org/10.1103/PhysRevLett.128.054501  

   Abanov, A. G.; Wiegmann, P. B. (February 2022, Physical Review Letters)