How smooth are particle trajectories in a Λ\LambdaCDM Universe?

It is shown here that in a flat, cold dark matter (CDM) dominated Universe with positive cosmological constant ($\Lambda$), modelled in terms of a Newtonian and collisionless fluid, particle trajectories are analytical in time (representable by a convergent Taylor series) until at least a finite time after decoupling. The time variable used for this statement is the cosmic scale factor, i.e., the "$a$-time", and not the cosmic time. For this, a Lagrangian-coordinates formulation of the Euler-Poisson equations is employed, originally used by Cauchy for 3-D incompressible flow. Temporal analyticity for $\Lambda$CDM is found to be a consequence of novel explicit all-order recursion relations for the $a$-time Taylor coefficients of the Lagrangian displacement field, from which we derive the convergence of the $a$-time Taylor series. A lower bound for the $a$-time where analyticity is guaranteed and shell-crossing is ruled out is obtained, whose value depends only on $\Lambda$ and on the initial spatial smoothness of the density field. The largest time interval is achieved when $\Lambda$ vanishes, i.e., for an Einstein-de Sitter universe. Analyticity holds also if, instead of the $a$-time, one uses the linear structure growth $D$-time, but no simple recursion relations are then obtained. The analyticity result also holds when a curvature term is included in the Friedmann equation for the background, but inclusion of a radiation term arising from the primordial era spoils analyticity.Comment: 16 pages, 4 figures, published in MNRAS, this paper introduces a convergent formulation of Lagrangian perturbation theory for LCD

Cauchy's almost forgotten Lagrangian formulation of the Euler equation for 3D incompressible flow

Two prized papers, one by Augustin Cauchy in 1815, presented to the French Academy and the other by Hermann Hankel in 1861, presented to G\"ottingen University, contain major discoveries on vorticity dynamics whose impact is now quickly increasing. Cauchy found a Lagrangian formulation of 3D ideal incompressible flow in terms of three invariants that generalize to three dimensions the now well-known law of conservation of vorticity along fluid particle trajectories for two-dimensional flow. This has very recently been used to prove analyticity in time of fluid particle trajectories for 3D incompressible Euler flow and can be extended to compressible flow, in particular to cosmological dark matter. Hankel showed that Cauchy's formulation gives a very simple Lagrangian derivation of the Helmholtz vorticity-flux invariants and, in the middle of the proof, derived an intermediate result which is the conservation of the circulation of the velocity around a closed contour moving with the fluid. This circulation theorem was to be rediscovered independently by William Thomson (Kelvin) in 1869. Cauchy's invariants were only occasionally cited in the 19th century --- besides Hankel, foremost by George Stokes and Maurice L\'evy --- and even less so in the 20th until they were rediscovered via Emmy Noether's theorem in the late 1960, but reattributed to Cauchy only at the end of the 20th century by Russian scientists.Comment: 23 pages, 6 figures, EPJ H (history), in pres

Hermann Hankel's "On the general theory of motion of fluids", an essay including an English translation of the complete Preisschrift from 1861

The present is a companion paper to "A contemporary look at Hermann Hankel's 1861 pioneering work on Lagrangian fluid dynamics" by Frisch, Grimberg and Villone (2017). Here we present the English translation of the 1861 prize manuscript from G\"ottingen University "Zur allgemeinen Theorie der Bewegung der Fl\"ussigkeiten" (On the general theory of the motion of the fluids) of Hermann Hankel (1839-1873), which was originally submitted in Latin and then translated into German by the Author for publication. We also provide the English translation of two important reports on the manuscript, one written by Bernhard Riemann and the other by Wilhelm Eduard Weber, during the assessment process for the prize. Finally we give a short biography of Hermann Hankel with his complete bibliography.Comment: 44 pages, see the companion paper by Frisch, Grimberg and Villone (2017), v2: minor revisions including change of title, accepted for publication in EPJ

L’intervento di medici ed infermieri nella Grande Guerra.

[email protected] al Convegno: “Pax in Bello … sempre dalla parte di chi soffre. Croce Rossa e Sanità Militare nella Grande Guerra”, organizzato dalla C.R.I. Comitato Provinciale di Campobasso, in collaborazione con l'Università degli Studi del Molise e la Regione Molise. Campobasso, 29 maggio 2015, Università degli Studi del Molise, Aula "Colozza"

Quantitative imaging of the complexity in liquid bubbles' evolution reveals the dynamics of film retraction

The dynamics and stability of thin liquid films have fascinated scientists over many decades. Thin film flows are central to numerous areas of engineering, geophysics, and biophysics and occur over a wide range of length, velocity, and liquid properties scales. In spite of many significant developments in this area, we still lack appropriate quantitative experimental tools with the spatial and temporal resolution necessary for a comprehensive study of film evolution. We propose tackling this problem with a holographic technique that combines quantitative phase imaging with a custom setup designed to form and manipulate bubbles. The results, gathered on a model aqueous polymeric solution, provide an unparalleled insight into bubble dynamics through the combination of full-field thickness estimation, three-dimensional imaging, and fast acquisition time. The unprecedented level of detail offered by the proposed methodology will promote a deeper understanding of the underlying physics of thin film dynamics

Singularities and the distribution of density in the Burgers/adhesion model

We are interested in the tail behavior of the pdf of mass density within the one and $d$-dimensional Burgers/adhesion model used, e.g., to model the formation of large-scale structures in the Universe after baryon-photon decoupling. We show that large densities are localized near ``kurtoparabolic'' singularities residing on space-time manifolds of codimension two ($d \le 2$) or higher ($d \ge 3$). For smooth initial conditions, such singularities are obtained from the convex hull of the Lagrangian potential (the initial velocity potential minus a parabolic term). The singularities contribute {\em \hbox{universal} power-law tails} to the density pdf when the initial conditions are random. In one dimension the singularities are preshocks (nascent shocks), whereas in two and three dimensions they persist in time and correspond to boundaries of shocks; in all cases the corresponding density pdf has the exponent -7/2, originally proposed by E, Khanin, Mazel and Sinai (1997 Phys. Rev. Lett. 78, 1904) for the pdf of velocity gradients in one-dimensional forced Burgers turbulence. We also briefly consider models permitting particle crossings and thus multi-stream solutions, such as the Zel'dovich approximation and the (Jeans)--Vlasov--Poisson equation with single-stream initial data: they have singularities of codimension one, yielding power-law tails with exponent -3.Comment: LATEX 11 pages, 6 figures, revised; Physica D, in pres

Dispersive stabilization of the inverse cascade for the Kolmogorov flow

It is shown by perturbation techniques and numerical simulations that the inverse cascade of kink-antikink annihilations, characteristic of the Kolmogorov flow in the slightly supercritical Reynolds number regime, is halted by the dispersive action of Rossby waves in the beta-plane approximation. For beta tending to zero, the largest excited scale is proportional to the logarithm of one over beta and differs strongly from what is predicted by standard dimensional phenomenology which ignores depletion of nonlinearity.Comment: 4 pages, LATEX, 3 figures. v3: revised version with minor correction

Quantitative imaging of the complexity in liquid bubbles’ evolution reveals the dynamics of film retraction

Thin liquid films: Seeing bubbles in a better light A procedure for imaging the complex fluid dynamics in bubbles could greatly assist efforts to understand and exploit thin liquid films in applications ranging through medicine, industrial chemistry and engineering. Thin liquid films are ubiquitous in nature, found in such varied systems as soap bubbles, biological membranes, detergents, oils, insulation, foods and geological magma. Researchers in Italy led by Biagio Mandracchia at the Institute of Applied Science and Intelligent Systems in Naples, devised a novel holographic phase imaging technique to watch bubbles as they form, develop, burst and retract. The researchers built customized apparatus to create and manipulate the bubbles. The unprecedented level of detail being revealed offers deeper understanding of the physics underlying thin film behavior. Insights into the complex fluid dynamics within bubbles could advance thin film technology for many applications