Nikolay Kuznetsov

Uniqueness and trapped modes for surface-piercing cylinders in oblique waves

Aspects of the solution to the linearized water-wave problem involving a pair of surface-piercing cylinders in oblique waves and infinite water depth are examined. In particular, the solution is proved to be unique for certain geometrical arrangements and wave parameters depending on whether the wave frequency is above or below the cut-off frequency. Outside these regions of uniqueness are constructed examples of non-uniqueness using ideas developed in McIver (1996) in the normal incidence case. Although non-uniqueness examples are obtained numerically, we are able to prove the existence of non-uniqueness under the assumption that the wave obliqueness is small.

Trapped modes of internal waves in a channel spanned by a submerged cylinder

A horizontal channel of infinite length and depth and of constant width contains inviscid, incompressible, two-layer fluid under gravity. The upper layer has constant finite depth and is occupied by a fluid of constant density ρ * . The lower layer has infinite depth and is occupied by a fluid of constant density ρ > ρ * . The parameter e = (ρ/ρ * )–1 is assumed to be small. The lower fluid is bounded internally by an immersed horizontal cylinder which extends right across the channel and has its generators normal to the sidewalls. The free, time-harmonic oscillations of fluid, which have finite kinetic and potential energy (such oscillations are called trapped modes), are investigated. Trapped modes in homogeneous fluid above submerged cylinders and other obstacles are well known. In the present paper it is shown that there are two sets of frequencies of trapped modes for the two-layer fluid. The frequencies of the first finite set are close to the frequencies of trapped modes in the homogeneous fluid (when ρ * = ρ). They correspond to the trapped modes of waves on the free surface of the upper fluid. The frequencies of the second finite set are proportional to e, and hence, are small. These latter frequencies correspond to the trapped modes of internal waves on the interface between two fluids. To obtain these results the perturbation method for a quadratic operator family was applied. The quadratic operator family with bounded, symmetric, linear, integral operators in the space L 2 (−∞, +∞) arises as a result of two reductions of the original problem. The first reduction allows to consider the potential in the lower fluid only. The second reduction is the same as used by Ursell (1987).

On the two-dimensional sloshing problem

We study an eigenvalue problem with a spectral parameter in a boundary condition. This problem for the two–dimensional Laplace equation is relevant to sloshing frequencies that describe free oscillations of an inviscid, incompressible, heavy fluid in a canal having uniform cross–section and bounded from above by a horizontal free surface. It is demonstrated that there exist domains such that at least one of the eigenfunctions has a nodal line or lines with both ends on the free surface (earlier, Kuttler tried to prove that there are no such nodal lines for all domains but his proof is erroneous). It is also shown that the fundamental eigenvalue is simple, and for the corresponding eigenfunction the behaviour of the nodal line is characterized. For this purpose, a new variational principle is proposed for an equivalent statement of the sloshing problem in terms of the conjugate stream function.

On uniqueness and solvability in the linearized two-dimensional problem of a supercritical stream about a surface-piercing body

Uniqueness and solvability theorems are proved for a well-posed formulation of the two-dimensional Neumann-Kelvin problem (the modified Neumann-Kelvin problem) in the case, when a body is partly immersed in a supercritical stream. Uniqueness is provided by two supplementary conditions which prescribe (i) additional flux at infinity downstream due to presence of body and (ii) a linear relation between the free-surface elevation at stern point and the velocity circulation along wetted contour. Two versions of source method are developed to find a solution. The first version is simpler, but it fails for some irregular values of the body’s velocity. In the second version complex sources’ strengths are used, avoiding irregular values and establishing the unique existence of solutions.

Interaction between a Decahydro-closo-Decaborate(2–) Anion and Aliphatic Carboxylic Acids

Interaction between a closo-decaborate anion B10H2–10and carboxylic acids RCOOH (R = H, CH3, C2H5, iso-C3H7, C4H9) is studied. The mono-, di- tri- and tetrasubstituted products B10H10 – n(OCOR)2–nare formed in sequence with the temperature growth. The reaction follows an essentially regioselective mechanism: only one of all possible isomers forms at every stage of the process. The respective hydroxy-closo-decaborates B10H10 – n(OH)2–nwere prepared by alkaline hydrolysis in aqueous and nonaqueous solutions. All the compounds were identified by chemical analysis and 11B NMR and IR spectroscopy. The crystal structure of [Pb(Bipy)(DMF)(B10H9OH)] · DMF was determined by X-ray diffraction.

Reversible single-crystal-to-single-crystal photoisomerization of a silver(I) macropolyhedral borane

We report a new family of photoactive molecular compounds for which reversible solid state transformations and linkage isomerism have been observed using in situ X-ray diffraction. The interaction of tris(triphenylphosphine)silver(I) with eicosaborate [B20H18]2− afforded a binuclear complex with a bridging boron cluster that underwent a series of isomerization transformations occurring in a single-crystal-to-single-crystal manner without crystal degradation and obeying the criteria for an effective molecular switch.

Compounds of Undecahydrodecaborate Anion B10H11

Conditions for the synthesis of salts of the B10H11–anion with different cations in the Cat2B10H10+ RCOOH (R = H, CF3; Cat = Me4N+, Et4N+, Bu4N+, Ph4P+, Ph4As+) systems were studied depending on the acid strength (pKa) and size of the cation. It was established that reactions with trifluoroacetic acid give compounds of this anion with any one of the quaternary ammonium, phosphonium, or arsonium cations, while formic acid can only give salts with the largest of these cations.

Cryogenian evolution of stigmasteroid biosynthesis

Biosynthesis of C29 sterols, enhancing eukaryotic temperature resistance, evolved during the global Neoproterozoic glaciations. Sedimentary hydrocarbon remnants of eukaryotic C26–C30 sterols can be used to reconstruct early algal evolution. Enhanced C29 sterol abundances provide algal cell membranes a density advantage in large temperature fluctuations. Here, we combined a literature review with new analyses to generate a comprehensive inventory of unambiguously syngenetic steranes in Neoproterozoic rocks. Our results show that the capacity for C29 24-ethyl-sterol biosynthesis emerged in the Cryogenian, that is, between 720 and 635 million years ago during the Neoproterozoic Snowball Earth glaciations, which were an evolutionary stimulant, not a bottleneck. This biochemical innovation heralded the rise of green algae to global dominance of marine ecosystems and highlights the environmental drivers for the evolution of sterol biosynthesis. The Cryogenian emergence of C29 sterol biosynthesis places a benchmark for verifying older sterane signatures and sets a new framework for our understanding of early algal evolution.

closo-Dodecaborate Intercalated Yttrium Hydroxide as a First Example of Boron Cluster Anion-Containing Layered Inorganic Substances

The first member, Y2(OH)5.46(B12H12)0.23Cl0.08·4.98H2O, of a new family of boron-containing substances, closo-dodecaborate intercalated layered rare-earth hydroxides, was synthesized using a microwave-assisted hydrothermal method. The structure and composition of this compound were studied by X-ray diffraction, transmission and scanning electron microscopy, thermal analysis, inductively coupled plasma mass spectrometry, IR spectroscopy, and X-ray photoelectron spectroscopy. The title compound had the composition Y2(OH)5.46(B12H12)0.23Cl0.08·4.98H2O and crystallized in a form of plate-like, aggregated particles less than 10 nm thick. The coordination of closo-dodecaborate anions with yttrium hydroxide host layers was demonstrated.

On bounds and non-existence in the problem of steady waves with vorticity

For the problem describing steady, gravity waves with vorticity on a two-dimensional, unidirectional flow of finite depth the following results are obtained. (i) Bounds for the free-surface profile and for Bernoullis constant. (ii) If only one parallel shear flow exists for a given value of Bernoullis constant, then there are no wave solutions provided the vorticity distribution is subject to a certain condition.