Helicity generation in uniform helical flows

Helicity generation conditions are derived for helical flows of Joukowski type with allowance for effects due to viscosity, buoyancy, temperature nonuniformity, and solid-body rotation. The upper and lower limits are determined for the rotation-frequency interval in which helicity can be generated by viscous forces. These conditions correspond to the regime of an isolated tornado-like vortex. An exact solution to the time-independent equations of motion for inviscid incompressible flow is obtained. The solution describes a generalized Kelvin-Helmholtz vortex having the form of a localized cylindrical vortex with nontrivial stable topological vortex-core structure determined by a finite value of helicity. For linear traveling inertia waves, which must have uniform helical structure, a general representation is found that characterizes helical structures of different origin.     

Energy-optimal time-dependent regimes of viscous incompressible fluid flow

The hydrodynamic equations of a viscous incompressible fluid are modified for axisymmetric flows in a pipe of time-varying radius. A new exact time-dependent solution of these equations which generalizes the well-known classic steady-state Hagen–Poiseuille solution for flow in a pipe of constant radius (independent of time) is obtained. It is shown that the law of time variation in the pipe radius can be determined from the condition of the minimum work done to pump a given fluid volume through such a pipe during the radius variation cycle period. A generalization of the optimal branching pipeline in which, instead of the Poiseuille law, its modification based on the use of the exact solution corresponding to the time-dependent M-shaped regime is employed is suggested. It is shown that the hydraulic resistance can be reduced over a certain range of the parameters of the time-dependent flow regime as compared with the steady-state pipe flow regime. The conclusion obtained can be used for the development of the hydrodynamic basis for simulating the optimal hydrodynamic blood flow regime.     

A new approach to the theory of Cherenkov radiation based on relativistic generalization of the Landau criterion

Relativistic generalization of the Landau criterion is obtained which, in contrast to the classical Tamm-Frank and Ginzburg theories, determines the primary energy mechanism of emission of nonbremsstrahlung Cherenkov radiation. It is shown that Cherenkov radiation may correspond to a threshold energetically favorable conversion of the condensate (ultimately long-wavelength) elementary Bose perturbations of a medium into transverse Cherenkov photons emitted by the medium proper during its interaction with a sufficiently fast charged particle. The threshold conditions of emission are determined for a medium with an arbitrary refractive index n, including the case of isotropic plasma with n<1 for which the classical theory of Cherenkov radiation prohibits such direct and effective nonbremsstrahlung emission of these particular transverse high-frequency electromagnetic waves. It is established that these conditions of emission agree with the data of well-known experiments on the threshold for observation of Cherenkov radiation, whereas the classical theory only corresponds to the conditions of observation of the interference maximum of this radiation. The possibility of direct effective emission of nonbremsstrahlung Cherenkov radiation, not taken into account in the classical theory, is considered for many observed astrophysical phenomena (type III solar radio bursts, particle acceleration by radiation, etc.).     

Utility of the tandem Pauson-Khand reaction in the construction of tetracycles

The scope of the tandem Pauson-Khand reaction has been explored for the regiospecific construction of [5.5.5.5]- and [5.6.6.5]tetracyclic systems via the photolytic method of Livinghouse. The rapid regiospecific entry into the two dicyclopentapentanoid systems 17 and 29 was accomplished from the key diene-diynes 11 and 19b. A photochemically mediated catalytic tandem Pauson-Khand cyclization was employed to prepare the parent ring systems of dicyclopenta[a,e]pentalene (from 19b) and dicyclopenta[a,f]pentalene (from 11) in regiospecific fashion in a one-pot process. Under these conditions, conversion of acyclic diene-diyne 16 into tetracyclic system 17 was achieved in 74% yield, while a similar process was employed to convert 28 into tetracycle 29 in 90% yield. This is much improved over the previous conditions that employed NMO. Six carbon-carbon bonds were generated in this process constituting up to 98% yield for each carbon-carbon bond so formed. Furthermore, tetracyclic [5.6.6.5] systems such as dicyclopenta[b,g]decalins 37, 38, and 40 were prepared from similar diene-diyne precursors via the tandem Pauson-Khand cyclization. Importantly, acetal 36 provided the desired cis-fused [5.6.6.5] system 38a (via 40a/b) in stereospecific fashion. This reaction is unique in that it provides a cis-decalin ring system; moreover, the yield of each of the six carbon-carbon bonds formed in this process was at least 89%. The structure of cis diol 38a was confirmed by X-ray crystallography.     

Solution to the paradox of the linear stability of the Hagen-Poiseuille flow and the viscous dissipative mechanism of the emergence of turbulence in a boundary layer

It has been shown that the conclusion of the linear instability of the Hagen-Poiseuille flow at finite Reynolds numbers requires the refusal of the use of the traditional “normal” form of the representation of disturbances, which implies the possibility of separation of variables describing disturbances as functions of the radial and longitudinal (along the axis of a tube) coordinates. In the absence of such separation of variables in the developed linear theory, it has been proposed to use a modification of the Bubnov-Galerkin theory that makes it possible to take into account the difference between the periods of the longitudinal variability for different radial modes preliminarily determined by the standard application of the Galerkin-Kantorovich method to the evolution equation of extremely small axisymmetric disturbances of the tangential component of the velocity field. It has been shown that the consideration of even two linearly interacting radial modes for the Hagen-Poiseuille flow can provide linear instability only in the presence of the mentioned conditionally periodic longitudinal variability of disturbances along the axis of the tube, when the threshold Reynolds number Re_th(p) is very sensitive to the ratio p of two longitudinal periods each describing longitudinal variability for its radial disturbance mode. In this case, the threshold Reynolds number can tend to infinity, Re_th(p) → ∞, only at p = p _k = k, p = p _1/k = 1/k, and \(p = p_{\sqrt k } = [k + 1 \pm \sqrt {(k + 1)^2 - 4} ]/2\) , where k = 1, 2, 3, …. The minimum Reynolds number Re_th(p) ≈ 448 (at which p ≈ 1.527) for the linear instability of the Hagen-Poiseuille flow quantitatively corresponds to the condition of the excitation of Tollmien-Schlichting waves in the boundary layer, where Re_th = 420. Similarity of the mechanisms of linear viscous dissipative instability for the Hagen-Poiseuille flow and Tollmien-Schlichting waves has been discussed. Good quantitative agreement has been obtained between the phase velocities of the vortex disturbances and the experimental data on the velocities of the leading and trailing edges of turbulent “puffs” propagating along the axis of the tube.     

Vavilov–Cherenkov radiation when cosmic rays pass through the relic photon gas and when fast charged particles traverse an optical laser beam

Using a new [9, 10] quantum theory of Vavilov–Cherenkov radiation (VCR) based on Abraham’s theory, we show that a threshold VCR effect can be excited by the relic photon gas when relativistic charged cosmic-ray particles with γ ≥ γ_th ≈ 1.9 × 10¹⁰ (where γ^–2 = 1–v²/c², v is the particle speed, and c is the speed of light in a vacuum) pass through it. This is compatible with the well-known GZK cutoff [7, 8] at γ ≈ 10¹¹. We have obtained the condition γ > γ_th ≈ 2.1 × 10² for the appearance of VCR when a sufficiently fast charged particle (an electron, a proton, or a nucleus) passes through intense laser radiation. This condition ensures that VCR can be observed experimentally (e.g., on the Large Hadron Collider) without invoking any additional conditions required from the currently existing estimate of γth > 8.8 × 10⁴ [13] based on the now universally accepted quantum theory of VCR, which follows from Minkowski’s theory (and which gives an estimate of γ > 10²¹ when excited by the relic photon gas).