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Notes: Abstract Ultrasonic wave propagation is studied in the framework of the continual model of a nonconducting perfect solid with frozen-in magnetization. The theoretical results are found to agree well with the experimental data.

Notes: Abstract It is shown that in flow past a system of spheres of radius a situated at the nodes of a cubic lattice with the period b in the direction of one of the principal translations of the lattice under the condition (a/b) · · P1/3≪1 (P is the Péclet number, P≫1), the concentration of dissolved material absorbed by the sphere surfaces diminishes logarithmically at distances large compared with b, but small compared with L=Pb2/4πa. At distances considerably larger than L, the decrease is described by an exponential law which coincides with the law of concentration decrease at distances much larger than b in the case of a spatially homogeneous distribution of the spheres. We consider the flow of an incompressible fluid with the velocity U past a system of spheres of radius a. We assume that the Reynolds number R=Ua/ν (where ν, the kinematic viscosity coefficient, is much larger than unity). Dissolved in the fluid is a material of concentration c which is absorbed by the sphere surfaces. The diffusion coefficient D is assumed to be sufficiently small for the Péclet number P=Ua/D to be much larger than unity. The spheres are situated at the nodes of a cubic lattice with the period b. As will be shown below, it is necessary that P(a/b)3≪1. Under these assumptions the concentration varies in a thin (of the order aP−1/3) diffusion layer near the surface of each sphere. A diffusion wake is formed behind each sphere. The transverse dimensions of this wake for a sufficiently widely spaced lattice (aP1/3≪ ≪b) exceed the effective thickness of the diffusion boundary layer, which enables us to reduce the problem of concentration absorption on the surface of the system of spheres to the problem considered by Levich [1] concerning the convective diffusion of a material of constant constant concentration flowing past a single sphere. Hasimoto [2] considers the solution of the Stokes equation describing the motion of a viscous fluid past an array of spheres situated at the nodes of a cubic lattice. However, he does not give an expression for the velocity field applicable near the surface of some single sphere which is necessary to the solution of the diffusion problem. In the method of Lamb [3] (§336) and Burgers [4], in dealing with the flow of a viscous stream past a single sphere, one considers the equation of motion in space, including the interior of the sphere, and not just the solution of the equation in the space outside the sphere with boundary conditions at the sphere surface. At the center of the sphere one places a concentrated force and a system of multipoles whose magnitude is chosen in such a way as to ensure fulfillment of the required boundary conditions. This idea of introducing an effective potential is used in [2] to find the velocity field of a fluid flowing past an array of spheres. We propose a treatment of the effective potential method somewhat different from that of [2].

Notes: Abstract This paper deals with the effect of the gas content and the shape of the space occupied by a system of bubbles on the rate at which the bubbles rise in an infinite medium and in a vertical cylindrical column. Deformations of the system favorable from the energy point of view ate considered for an assumed homogeneous and isotropic distribution of bubbles in the system. A theoretical description of the motion of gas bubble systems in liquids is necessary for the study of bubbling processes. This problem has been frequently considered for low Reynolds numbers (Re≪1) on the basis of the so-called cell model. In [1] an analogous model was used to describe the motion of a system ofmedium-size bubbles (Re≪300). It was assumed that each bubble is at any instant of time in the center of an imaginary spherical cell of liquid the radius of which equals the average distance between the centers of bubbles in the system. The normal component of the velocity of the liquid and the surface of the cell is zero. The former assumption is equivalent to the assumption of homogeneity and isotropy of the system, but the latter is by no means physically convincing. In the present paper we calculate the rising velocity of a system of medium-size bubbles in a liquid with low gas content; these results differ even qualitatively from those obtained on the basis of the cell model. This apparently suggests that the cell model is unsatisfactory, at least in the case of a liquid with low gas content.

Notes: Abstract A study is made of the charging of a low-conductivity liquid flowing through a cylindrical metal pipe with allowance for the finite rate of an electrochemical reaction that takes place on the walls of the pipe and is responsible for the charging. The solution of the problem uses the system of equations and boundary conditions derived by Gogosov, Tolmachev, and Nikiforovich [1]. The source of the space charge in the liquid is assumed to be a surface electrochemical reaction of ions of a contaminating impurity dissolved in the liquid [2–4]. The resulting current is found as a function of the flow velocity, the geometrical dimensions of the pipe, and the rate of the surface electrochemical reaction. It is shown that in the initial section of the pipe a kinetic regime is always realized in which the rate of the charging process is limited by the rate of the chemical reaction. Far from the initial section there is a diffusion regime, in which the rate of the charging process is determined by the rate of supply of ions to the surface of the pipe. With increasing velocity of the liquid the current tends to a limiting value, which depends only on the rate of the surface reaction, i.e., the current is saturated. The results given in the paper of experiments on the charging of hexane flowing through small-diameter pipes confirms the theory. The current was observed to be saturated in the experiments. A method is proposed for the experimental determination of the rate of a surface electrochemical reaction and the concentration of an ionic impurity.

Notes: Abstract A semiempirical method is proposed for taking into account the spin-orbit interaction in calculating line intensities for transitions in atoms and molecules. Use of the method is demonstrated for SX and CaXIV ions.

Notes: Abstract It is shown that the calculation of the matrix elements between degenerate states of arbitrary atomicmolecular systems can be reduced to a calculation of the matrix elements for some auxiliary operator. The perturbation theory for this calculation can be given a field form, and its matrix elements can be expressed in terms of the contributions from special Feynman diagrams. The exact rules are formulated for constructing the contributions from these diagrams.

Notes: Abstract A quantum-mechanical, three-dimensional, quasiclassical theory is developed for the effective cross section of the chemical reaction AB + CD → AC + BD. The quasiclassical approximation is used on the basis of the Born formula for the differential effective cross section. The steady-phase method is used to calculate the integrals appearing in the formula for the differential effective cross section. An analytic solution of the problem is found. The formulas obtained are convenient for use in performing numerical calculations.

Notes: Abstract Principles are presented for karstological municipal and project monitoring. Assumptions are formulated on the time prediction of sinkhole formation in karst-prone lands.

Notes: Abstract Rules are described for evaluating the contributions from the diagrams which arise in calculating the matrix elements of a one-particle tensor operator between two degenerate atomic levels.