Shape of a plane hydraulic fracture crack in an elastic impermeable medium at various injection rates

Using a representation of the solutions for the shape of the hydraulic fracture crack opening and the pressure distribution along the crack in the form of a series expansion in Tchebychev polynomials of the second kind, a technique of finding the expansion coefficients is proposed and the curves for crack opening shape and pressure distribution are constructed at various injection rates. This enables one to clarify the dependence of the crack shape on the injection rate.     

The growth of a single crystal of Sr3CuIrO6 and its magnetic behavior compared to polycrystals

We have grown single crystals of the psuedo-one-dimensional compound Sr₃CuIrO₆, a K₄CdCl₆-derived monoclinic structure with Cu-Ir chains along the [101] direction. We present the ac and dc magnetization behavior of the single crystals in comparison with that of the polycrystalline form reported earlier. There is a distinct evidence for at least two magnetic transitions, at 5 K (T ₁) and 19 K (T ₂), with different relative magnitudes in the single and polycrystals. The low temperature magnetic relaxation behavior of both the forms is found to be widely different, exhibiting unexpected time dependence.     

Study of internal waves in the case of rapid horizontal motion of cylinders and spheres

A sphere and a cylinder are simulated by mass sources distributed over their surfaces, and expressions are obtained for the wave resistance of such sources moving horizontally in an exponentially stratified fluid. It is shown that then, in contrast to the case of simulation of such bodies by point dipoles, the paradox of an infinite wave resistance does not arise. Different variants of the method of equivalent sources are discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 94–100, November–December, 1982.     

The effective conductivity of a randomly inhomogeneous medium

Using a representation of the solution to the diffusion equation in a randomly inhomogeneous medium in the form of a Feynman path integral an explicit expression is obtained for the effective conductivity in a space of arbitrary dimension. A calculation of the path integral only turns out to be possible in the case of a large-scale limit. In particular, it is shown that in the three-dimensional case the expression for the effective conductivity does not admit of an expansion in terms of the conductivity variance. This indicates that the use of standard perturbation theory in the form of an expansion in terms of the conductivity fluctuations is incorrect.     

Contribution of macroscopic van der Waals interactions to friction forces

Using an approximation linear in velocities, an expression is obtained for friction forces due to nonretarded macroscopic van der Waals interactions. These forces are generated when field drag effects by a moving medium are taken into account.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 102–107, November, 1976.The author is grateful to Yu. S. Barash for critical remarks and discussions.