Navier–Stokes equations: an analysis of a possible gap to achieve the energy equality

The paper is concerned with the IBVP of the Navier–Stokes equations. The goal is to evaluate the possible gap between the energy equality and the energy inequality deduced for a weak solution. This kind of analysis is new and the result is a natural continuation and improvement of a result obtained by the same authors in Crispo et al. (Some new properties of a suitable weak solution to the Navier–Stokes equations. arXiv:1904.07641).

     

Evaluation of different light-curing lamps

The aim of this study has been to evaluate light-curing composites polymerization quality carried out by halogen and new-diode lamps through the thermal analysis (TG?CDTA). Samples have been polymerized at 3?C20?C40?C60?s by halogen lamp and 1?C3?C6?C9?s by new-diode lamp. The TG/DTA analysis shows that different light-curing times affect the degree of conversion of the composite, since by increasing the curing time the quantity of the monomer that has not reacted (residual) decreases. The new-diode lamp, according to the manufacturer, can cure composite restorations in few seconds; but at the conditions used in this study, the samples cured by the halogen lamp at the standard times of exposure, compared to the samples cured in few seconds by the new-diode lamp, show a lower mass loss.     

Speedy Motions of a Body Immersed in an Infinitely Extended Medium

We study the motion of a classical point body of mass M, moving under the action of a constant force of intensity E and immersed in a Vlasov fluid of free particles, interacting with the body via a bounded short range potential Ψ. We prove that if its initial velocity is large enough then the body escapes to infinity increasing its speed without any bound (runaway effect). Moreover, the body asymptotically reaches a uniformly accelerated motion with acceleration E/M. We then discuss at a heuristic level the case in which Ψ(r) diverges at short distances like gr ^−α , g,α>0, by showing that the runaway effect still occurs if α<2.     

Step bunching during the epitaxial growth of a generic binary-compound thin film

A discrete-continuum theory for the step-flow growth by chemical beam or metalorganic vapor-phase epitaxy of a generic binary-compound thin film is developed from basic considerations of continuum physics in accordance with the second law of thermodynamics. Our theory accounts for dissipation, chemical and otherwise; allows for departures from equilibrium; and generalizes the classical, variationally derived Gibbs-Thomson relation along the steps. In contrast to existing models, the diffusing species are coupled through a chemical reaction whereby bulk molecules are crystallized from adatoms attaching to the step edges. The linear stability analysis of the resulting free-boundary problem for a periodic train of rectilinear steps yields pairing in the presence of the normal Ehrlich-Schwoebel barrier for both species, counter to the predictions of standard Burton-Cabrera-Frank models for single-species growth. In particular, we show that the onset of step bunching occurs as long as the adatom equilibrium coverage of either species is sufficiently high, a condition met, e.g., during the epitaxy of gallium arsenide. The physical origin of this instability is to be found in the dependence of the step chemical potential on the jump in the adatom grand canonical potential, a term that couples adjacent terraces and—counter to elastic, entropic, or electrostatic interactions between steps—is attractive.     

Hypertractions and hyperstresses convey the same mechanical information

A strengthened and generalized version of the standard Virtual Work Principle is shown to imply, in addition to bulk and boundary balances, a one-to-one correspondence between surface and edge hypertractions and hyperstress fields in second-gradient continua. When edge hypertractions are constitutively taken null, the hyperstress is shown to take the form it has for a relevant example of second-gradient fluid-like material, referred to as a Navier–Stokes—α fluid.     

The adjoint representation inside the exterior algebra of a simple Lie algebra

For a simple complex Lie algebra g$\mathfrak{g}$ we study the space of invariants A=(?g^?⊗g^?)^g$A={(?{\mathfrak{g}}^{?}\otimes {\mathfrak{g}}^{?})}^{\mathfrak{g}}$, which describes the isotypic component of type g$\mathfrak{g}$ in ?g^?$?{\mathfrak{g}}^{?}$, as a module over the algebra of invariants (?g^?)^g${(?{\mathfrak{g}}^{?})}^{\mathfrak{g}}$. As main result we prove that A   is a free module, of rank twice the rank of g$\mathfrak{g}$, over the exterior algebra generated by all primitive invariants in (?g^?)^g${(?{\mathfrak{g}}^{?})}^{\mathfrak{g}}$, with the exception of the one of highest degree.