Unsteady-state filtration of a liquid in a fractured-porous stratum to a well with a hemispherical end-face

A study is made of the motion of a liquid toward a well with a hemispherical end-face with unsteady-state spherical-radial filtration in a fractured-porous stratum consisting of hemispherical regions with different values of the permeability of the system of cracks, superposed one on another. A Laplace transform is used to find exact solutions to the problem of the lowering of the stratum pressure as a function of time and distance as well as of the output of a well working with a constant end-face pressure. The article discusses partial cases corresponding to the exploitation of closed and bounded open fractured-porous strata by a central well with a hemispherical end-face. On the basis of numerical calculations, the effect of the parameters of fractured-porous strata on the change in the indices of the process of their exploitation is established. It is established that, with the exploitation of fractured-porous strata, the process of the lowering of the end-face pressure of the well and its output become stabilized with sufficiently large values of the time.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 115–123, November–December, 1973.     

On the interaction between a dislocation and a crack in a material with a nonlinear stress-strain law

The problem of the interaction of a crack and a dislocation in a medium with a nonlinear stress-strain law is considered for the case of a semi-infinite crack in a displacement loaded strip under longitudinal shear deformation. A power law stress-strain relation is considered and the dislocation is assumed positioned so that the net effect of its interaction with the crack is to produce a zero stress intensity factor when combined with the effect of the applied displacements. Thus the Atkinson-Kay superdislocation model of a relaxed crack tip is extended to the situation where the material satisfies a power-law stress-strain relationship.     

Interaction of a rigid punch and a circular plate with a fixed side and a stress-free face

The axisymmetric contact problem of a rigid punch indentation into an elastic circular plate with a fixed side and a stress-free face is considered. The problem is solved by a method developed for finite bodies which is based on the properties of a biorthogonal system of vector functions. The problem is reduced to a Volterra integral equation (IE) of the first kind for the contract pressure function and to a system of two Volterra IE of the first kind for functions describing the derivative of the displacement of the plate upper surface outside the punch and the normal (or tangential) stress on the plate lower fixed surface. The last two functions are sought as the sum of a trigonometric series and a power-law function with a root singularity. The obtained ill-conditioned systems of linear algebraic equations are regularized by introducing small parameters and have a stable solution. A method for solving the Volterra IE is given. The contact pressure functions, the normal and tangential stresses on the plate fixed surface, and the dimensionless indentation force are found. Several examples of a plane punch computation are given.     

Pumping of a fluid through a microchannel by means of a bubble driven by a light beam

A new method of pumping a fluid through a microchannel device using a gas bubble-piston, set in motion by the thermocapillary force induced by a light beam, is proposed. To demonstrate the method, a model micropump has been assembled. The model consists of two reservoirs connected by two channels with a bubble-piston driven by a light beam. The pumping rate and the volume per piston stroke are evaluated experimentally. The method proposed is compared with known microfluid pumping methods. Some advantages of the new method are indicated.     

Simulation by a source of a thin jet from a body

An exact analytic solution is found to the following plane hydrodynamic problem. An unbounded flow of an ideal incompressible fluid flows around a plate BB' placed at right angles to the velocity vector of the flow at infinity. The pressure on the free boundary P is equal to the pressure in the flow. From an opening in the center of the plate, a jet with flow rate Q from a cavity with pressure P₀ encounters the flow head-on. As a result of the solution, it is found that for fixed width of the opening the values of Q allowed by the scheme are limited. In the limiting case Q = 0 Chaplygin's flow is obtained with stagnation region at the front [1], and in the limiting case Q = Q_max a jet out of a cavity with pressure P₀ into a cavity with pressure P. As Q varies in this interval, the total drag, regarded as the drag of the plate and the chamber from which the jet emerges, takes a minimal value at a certain point. If the width of the opening tends to the length of the slab, the problem of the collision of two jets is obtained; if the width of the opening tends to zero (Q o), the problem of jet flow past a slab with a source is obtained. It is shown that the replacement of the jet by the source gives a good approximation in both the sense of the force characteristics and in the sense of the behavior of the free streamlines.Translated from Izvestiya Akademii Nauk SSSR, Hekhanika Shidkosti i Gaza, No. 5, pp. 47–54, September–October, 1979.We thank L. I. Sedov for his interest in the work and G. Yu. Stepanov for proposing the method of solution and for a helpful discussion.     

Torsion problems for a cylinder with a rectangular hole and a rectangular cylinder with a crack

Using the method of singular integral equation and the crack-cutting technique, the rigorous solutions are obtained for a cylinder with a rectangular hole and a rectangular cylinder with a crack, which exactly satisfy the boundary conditions and the conditions at the corner points. After that the torsional rigidities and the stress intensity factors at the crack tip are determined. Next, for the doubly connected circular cylinder with a rectangular hole the expressions for the singular stresses around the concave corner points are derived and the generalized stress intensity factors are then defined. Since the crack-cutting technique is used in this paper, the solution of the matching rectangular cylinder is also obtained and its numerical results coincide with those in references. Thus the method proposed here is verified. The project supported by National Natural Science Foundation of China     

Motion of a slender body in a fluid under a floating plate

This paper studies the three-dimensional unsteady problem of the hydroelastic behavior of a floating infinite plate under the impact of waves generated by horizontal rectilinear motion of a slender solid in a fluid of infinite depth. An analytic solution of the problem is found based on the known solutions for the unsteady motion of a point source of mass in a fluid of infinite depth under a floating plate. Asymptotic formulas are obtained which model the motion of a solid slender body in a fluid by replacing the body with a source-sink system. These formulas are used to numerically analyze the effect of plate thickness, depth of the body, its dimensions and the velocity of rectilinear motion on the amplitude of deflection of the floating plate. The motion of a submarine under a nonbreakable plate was modeled experimentally. Theoretical and experimental data are in good agreement.     

Interaction of a cylinder with a free surface and a jet

The problem of the asymmetric flow around a cylinder of a nonviscous jet is considered and, in particular, of flow with one free boundary. The problem of the choice of circulation is posed, based on a generalization of the Zhukovskii-Chaplygin postulate. Several possibilities are considered and the principle of the minimum of the maximum velocity on a contour is proposed, which qualitatively and quantitatively truly reflects certain aspects of the interaction of the cylinder with the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 50–58, September–October, 1977.     

Divergence of a helicoidal shell in a pipe with a flowing fluid

This paper considers a solution of the problem of coupled hydroelasticity for a helicoidal shell in a rigid tube with a flowing ideal incompressible fluid, which is of interest for the design of heat exchange systems. The flow is considered potential, and boundary conditions are imposed on the deformed surface. The version of the classical theory of elastic shells as the Lagrangian mechanics of deformable surfaces is used. The longitudinal-torsional vibrations of a long shell and a naturally twisted rod are studied. It is established that the obtained hydrodynamic loads are conservative, so that a divergence type instability is possible. A critical combination of parameters is determined.     

Impact of a body with a plane bottom on a thin liquid layer at a small angle

The problem of the impact of a body with a plane bottom (of the type of a box) on a thin liquid layer at a small angle is solved in the two-dimensional formulation. The nonlinear shallow water equations are used, together with the method of matched asymptotic expansions. It is found that at certain values of the input parameters of the problem the liquid pressure diminishes near the lower end of the body and becomes smaller than the atmospheric pressure, which results in liquid separation from the box bottom. The numerical results show that all input parameters of the problem have a considerable effect on the nature of body motion. The liquid separation effect on body motion is analyzed.     

Motion of a cylindrical cavity near a free boundary of a liquid

The dynamics of a cylindrical cavity in a liquid with no allowance for vertical displacement are considered in [1–3]. The present study investigates the pulsation motion of a cylindrical cavity near a free boundary of a liquid, allowing for displacement of the axis of the cavity in the vertical direction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 173–176, November–December, 1988.     

Force Action of a Body with a Permeable Boundary on a Wall in a Fluid

The ideal fluid flow due to fluid penetration through the boundary of an infinitely long solid cylinder in contact with a solid wall is determined. A formula is derived according to which the force exerted by a finite-length part of the cylinder on the wall is directed into the wall and can thus have an arbitrarily large absolute value. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 1, pp. 82–84, January–February, 2006.     

Interaction of a supersonic flow with a transverse jet injected through a circular aperture in a plate

A flow pattern created by the interaction of a supersonic flow with a transverse sonic or supersonic jet injected normally to the direction of the main flow through a circular aperture in a plate is considered. The pressure rises in front of the jet owing to the retarding action of the incident flow. The boundary layer building up on the wall in front of the injection nozzle is accordingly detached. The flow pattern in the region of interaction between the jet and the external flow is illustrated in Fig. 1. The three-dimensional zone of detachment thus formed deflects the incident flow from the wall, and in front of the jet a complicated system of sharp jumps in contraction develops. A three-dimensional system of jumps also develops in the jet itself.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 5, pp. 193–197, September–October, 1970.     

Mixing of a supersonic stream and a transverse jet injected through a circular hole in a plate

An experimental investigation is conducted of the propagation and mixing of a transverse sonic jet injected into a supersonic gas stream through a circular hole in a plate. The picture of the interaction at the initial section of the injected jet is examined in the majority of papers devoted to this question [1–3]. The aim of this paper is to investigate the mixing of the injected jet with the free stream, and to establish criteria governing the jet insertion and mixing processes.     

Displacement of a viscous fluid from a Hele-Shaw cell with a sink

It is found that the radial geometry does not stabilize the evolution of instability in the displacement of a more viscous fluid by a less viscous fluid from a circular Hele-Shaw cell with a sink. A linear analysis shows the absolute instability of the radial displacement front. The appearance of isolated fingers is observed during numerical simulations.     

Atomization of a Powder by a Swirling Flow with a Recirculation Zone

The problem of the motion of a swirling flow in an axisymmetric channel with permeable walls is investigated numerically. Various flow regimes including those with the formation of recirculation zones are obtained. The problem of atomization of a powder by a swirling flow for the purpose of obtaining a finely dispersed mixture is considered. Particle concentration distributions in the flow are calculated, the formation of characteristic deposition zones is demonstrated, and the unsteady process of particle transfer is investigated with allowance for deposition on the lateral surface of the channel.     

Interaction of a screw dislocation with a notch in a piezoelectric bi-material

Summary The electro-elastic interaction of a screw dislocation and a notch in a piezoelectric bi-material is analyzed. The electro-elastic fields induced by the dislocation are derived using the conformal mapping and the image-dislocation approach, where the solution for a piezoelectric bi-material without a notch is used as a base. The stress and the electric displacement intensity factors of the notch and the image force on the dislocation are given explicitly. We find that intensity factors are expressed in terms of the effective material constants, while the radial component of the image force is independent of the notch angle and the angular position of the dislocation in the polar coordinate system. Numerical results for the image force are provided for the use when one of the two media is purely elastic. They illustrate the behavior of the dislocation in the neighborhood of the notch.     

Gravity-induced spreading of a drop of a viscous fluid over a surface

The unsteady-state nonlinear problem of spreading of a drop of a viscous fluid on the horizontal surface of a solid under the action of gravity and capillary forces is considered for small Reynolds numbers. The method of asymptotic matching is applied to solve the axisymmetrical problem of spreading when the gravity exerts a significant effect on the dynamics of the drop. The flow structure in the drop is determined at large times in the neighborhood of a self-similar solution. The ranges of applicability of the quasiequilibrium model of drop spreading with a dynamic edge angle and a self-similar solution are found. It is shown that the transition from one flow model to another occurs at very large Bond numbers. Institute of Mechanics of Multiphase Systems, Siberian Division, Russian Academy of Sciences, Tyumen’ 625000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 59–67, May–June, 1999.     

Stability in a case of motion of a paraboloid over a plane

We solve a nonlinear orbital stability problem for a periodic motion of a homogeneous paraboloid of revolution over an immovable horizontal plane in a homogeneous gravity field. The plane is assumed to be absolutely smooth, and the body–plane collisions are assumed to be absolutely elastic. In the unperturbed motion, the symmetry axis of the body is vertical, and the body itself is in translational motion with periodic collisions with the plane.The Poincare´ section surfacemethod is used to reduce the problemto studying the stability of a fixed point of an area-preserving mapping of the plane into itself. The stability and instability conditions are obtained for all admissible values of the problem parameters.