Analysis for an N-stepped Rayleigh bar with sections of complex geometry

In this paper we analyse the vibrations of an N-stepped Rayleigh bar with sections of complex geometry, supported by end lumped masses and springs. Equations of motion and boundary conditions are derived from the Hamilton’s variational principal. The solutions for tapered and exponential sections are given. Two types of orthogonality for the eigenfunctions are obtained. The analytic solution to the N-stepped Rayleigh model is constructed in terms of Green function.     

Diffusional interaction of drops in a liquid

The method of combining asymptotic expansions (with respect to a large Peclet number) is used to investigate the three-dimensional problem of steady-state convective diffusion to the surface of drops, around which flows a laminar stream of a viscous incompressible liquid whose velocity field is assumed to be known from the solution of the corresponding hydrodynamic problem. It is shown that for large Peclet numbers the heat and mass transfer between drops is completely determined by the mutual arrangement of special (starting or ending at the surface of a drop) lines of flow; under these circumstances, in the flow there are chains of drops which have no mutual diffusional effect on one another, and the total diffusional flow to a drop is determined by diffusion to particles located upstream in the same chain. For the case where the distance between the drops in the chain is much leas than P^1/2 (P is the Peclet number), formulas for the distribution of the concentration and the total diffusional flow to the surface of each drop are obtained. It is shown that the total diffusional flow to the surface of a drop approaches zero in inverse proportion to its order number in a chain, which generalizes [1], in which the axisymmetric case is considered. A solution of the diffusional case is obtained for the case where there are critical lines at the surface of the drop. The problem is solved to the end if the singular flow lines are not closed and depart to infinity. With the presence of a region of closed circulation behind the drops, the problem is reduced to an integral equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika, Zhidkosti i Gaza, No. 2, pp. 44–56, March–April, 1978.The author thanks Yu. P. Gupalo and Yu. S. Ryazantsev for their interest in the work.     

Two problems of convective diffusion to the surfaces of poorly streamlined bodies

Problems of diffusion to particles of nonspherical shape at large Peclet numbers have been analyzed in many papers (see [1–7], for example). The solution of the problem of mass exchange of an ellipsoidal bubble at low Reynolds numbers was obtained in [1] while the solution at high Reynolds numbers was obtained in [2, 3]. In [4] an expression is given for the diffusional flux to the surface of a solid ellipsoidal particle over which a uniform Stokes stream flows. Generalization to the case of particles of arbitrary shape was done in [5, 6], while generalization to any number of critical lines on the surface of the body was done in [7, 8]. The two-dimensional problem of steady convective diffusion to the surface of a body of arbitrary shape is analyzed in the approximation of a diffusional boundary layer (ADBL). The simple analytical expressions obtained are more suitable for practical calculations than those in [5-8], since they allow one to determine at once, in the same coordinate system in which the field of flow over the particle was analyzed, the value of the diffusional flux to its surface (from the corresponding hydrodynamic characteristics). The plane problem of the diffusion to an elliptical cylinder in a uniform Stokes stream is solved. The problems of the diffusion to a plate of finite dimensions (in the plane case) and a disk (in the axisymmetric case) whose planes are normal to the direction of the incident stream are considered. It is shown that, in contrast to the results known earlier (see [4, 6-15], for example), where the total diffusional flux was proportional to the cube root of the Peclet number, here it is proportional to the one-fourth power.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 104–109, November–December, 1978.The authors thank Yu. P. Gupalo, Yu. S. Ryazantsev, and Yu. A. Sergeev for a useful discussion.     

Mass transfer of particles located on the flow axis at large peclet number

Diffusional influx of a substance dissolved in a medium into absorbent particles moving relative to one another in a viscous incompressible liquid is examined. An approximate analytical expression is obtained for the differential and integral flux of material into the surface of each particle, accounting for variations in the velocity and concentration fields due to the presence of the other particles. The results obtained can be applied to a lattice of spheres washed directly and uniformly in an infinite flow and located at distance 1 l P^1/3 relative to each other. It is shown that the diffusional flux of material into the first sphere is almost twice as large as into the other, and for a large number of spheres k the total diffusion flux tends to zero inversely as the 1/3 power of k.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 64–74, March–April, 1977.     

Limiting equilibrium in problems of nonlinear filtration

The article indicates a method for determining the field of the pressure of a non-Newtonian liquid, existing in a state of limiting equilibrium in a porous medium. This kind of question arises in the solution of problems in the theory and practice of the exploitation of petroleum deposits containing petroleum with an initial pressure gradient.