Dynamic interaction of a spherical radiator in a fluid-filled cylindrical borehole within a poroelastic formation

Harmonic acoustic radiation from a modally oscillating spherical source positioned at the center of a fluid-filled cylindrical cavity embedded within a fluid-saturated porous elastic formation is studied in an exact manner. The formulation utilizes the Biot theory of dynamic poroelasticity along with the cylindrical to spherical wave-field transformations, and the pertinent boundary conditions to obtain a closed-form series solution. The analytical results are illustrated with a numerical example in which the spherical source, with its polar axis oriented along the main axis of a water-filled borehole and embedded within a water-saturated Ridgefield sandstone formation, is excited in vibrational modes of various orders. The magnitude of the reflected component of acoustic pressure along the axis of the borehole for a pulsating (n = 0), an oscillating (n = 1), and also a multipole (n = 0–3) spherical source as a function of the excitation frequency is calculated and discussed for representative values of the parameters characterizing the system. Special attention is paid to the effects of source excitation frequency, size, surface velocity profile, and internal impedance as well as borehole interface permeability condition on the reflected pressure magnitudes. Limiting cases are considered and fair agreements with well-known solutions are obtained.     

Dynamic interaction of an eccentric multipole cylindrical radiator suspended in a fluid-filled borehole within a poroelastic formation

Acoustic radiation and the dynamic field induced by a cylindrical source of infinite extent, undergoing angularly periodic and axially-dependent harmonic surface vibrations, while eccentrically suspended in a fluid-filled cylindrical cavity embedded within a fluid-saturated porous elastic formation, are analyzed in an exact manner. This configuration, which is a realistic idealization of an acoustic logging tool suspended in a fluid-filled borehole within a permeable surrounding formation, is of practical importance with a multitude of possible applications in seismo-acoustics. The formulation utilizes the novel features of Biot dynamic theory of poroelasticity along with the translational addition theorem for cylindrical wave functions to obtain a closed-form series solution. The basic dynamic field quantities such as the resistive and the reactive components of the modal acoustic radiation impedance load on the source in addition to the radial and transverse stresses induced in the surrounding formation by an eccentric pulsating/oscillating cylinder in a water-filled borehole within a water-saturated Ridgefield sandstone medium are evaluated and discussed. Special attention is paid to the effects of source eccentricity, excitation frequency, and mode of surface oscillations on the modal impedance values and the dynamic stresses. Limiting cases are considered and good agreements with available solutions are obtained.     

Nonaxisymmetric deformation of open spherical shells with a curvilinear hole

A number of qualitative and quantitative mechanical effects are revealed in solving two-dimensional problems. Compression zones can occur in a thin shell with an oblong elliptical hole under internal pressure. The external edge has a strong effect on their position and size. In some cases, the fixed outer edge may stiffen the stress state near the hole __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 92–99, May 2008.     

Nonaxisymmetric elastoplastic impact of a parabolic body on a spherical shell

A method is proposed to calculate a spherical shell under nonaxisymmetric impact of a massive body. The motion of the shell is described by momentless equations, which are solved using the Laplace transformation and an asymptotic expansion of the required quantities in a small parameter. The contact interaction force P(t) was determined for the elastoplastic model of local bearing deformation for a parabolic impactor. Plots of the solution are given. The validity of the results is confirmed by good agreement between the solution and the limiting cases — an axisymmetric impact and an impact on a half-space.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 181–186, January–February, 2005.     

Acoustoelastic effects on mode waves in a fluid-filled pressurized borehole in triaxially stressed formations

Based on the nonlinear theory of acoustoelasticity, considering the triaxial terrestrial stress, the fluid static pressure in the borehole and the fluid nonlinear effect jointly, the dispersion curves of the monopole Stoneley wave and dipole flexural wave propagating along the borehole axis in a homogeneous isotropic formation are investigated by using the perturbation method. The relation of the sensitivity coefficient and the velocity-stress coefficient to frequency are also analyzed. The results show that variations of the phase velocity dispersion curve are mainly affected by three sensitivity coefficients related to third-order elastic constant. The borehole stress concentration causes a split of the flexural waves and an intersection of the dispersion curves of the flexural waves polarized in directions parallel and normal to the uniaxial horizontal stress direction. The stress-induced formation anisotropy is only dependent on the horizontal deviatoric terrestrial stress and independent of the horizontal mean terrestrial stress, the superimposed stress and the fluid static pressure. The horizontal terrestrial stress ratio ranging from 0 to 1 reduces the stress-induced formation anisotropy. This makes the intersection of flexural wave dispersion curves not distinguishable. The effect of the fluid nonlinearity on the dispersion curve of the mode wave is small and can be ignored.The project supported by the National Natural Science Foundation of China (10272004) and The Special Science Foundation of the Doctoral Discipline of the Ministry of Education of China(20050001016) The English text was polished by Keren Wang.     

Extended internal geometry description in modelling of dynamics of fluid-filled permeable media

The starting point of our considerations are balance equations obtained by the space-averaging method. These equations contain the explicit consequences of the admitted inhomogeneities of microscopic fields. The proposed postulates relating relative velocities of phases at micro- and macro-levels and properties of internal geometry, allow one to find the form of a few constitutive functions apparently dependent on the internal geometry of the medium.     

Motion of a permeable shell in a spherical container filled with non-Newtonian fluid

This paper presents an analytical study of creeping motion of a permeable sphere in a spherical container filled with a micro-polar fluid. The drag experienced by the permeable sphere when it passes through the center of the spherical container is studied.Stream function solutions for the flow fields are obtained in terms of modified Bessel functions and Gegenbauer functions. The pressure fields, the micro-rotation components,the drag experienced by a permeable sphere, the wall correction factor, and the flow rate through the permeable surface are obtained for the frictionless impermeable spherical container and the zero shear stress at the impermeable spherical container. Variations of the drag force and the wall correction factor with respect to different fluid parameters are studied. It is observed that the drag force, the wall correction factor, and the flow rate are greater for the frictionless impermeable spherical container than the zero shear stress at the impermeable spherical container. Several cases of interest are deduced from the present analysis.     

Nonaxisymmetric equilibrium shapes of a drop on a plane

A nonuniform temperature distribution, the presence of surface-active substances and impurities, and also other factors lead to a change in the wetting angle along a plane. A study is made of the influence of a small perturbation of the equilibrium contact angle on the shape of the free surface of the liquid. Two cases are considered: a surface of small slope in a gravity field and a nearly spherical shape under conditions of weightlessness. The equilibrium shapes of a liquid drop on an inclined plane under conditions of hysteresis of the wetting are also obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 164–167, July–August, 1983.I thank I. E. Tarapov and I, I, Ievlev for constant interest in the work and valuable comments.     

Submerged fluid-filled cylindrical shell subjected to a shock wave: Fluid–structure interaction effects

A submerged fluid-filled circular cylindrical shell subjected to a shock wave propagating in the external fluid is considered. The study focuses on a number of acoustic and structural effects taking place during the interaction. Specifically, the influence of the acoustic phenomena in the fluid on the stress–strain state of the shell is analysed using two different visualization techniques. The effect that the parameters of the shell have on the internal acoustic field is addressed as well, and the ‘shock transparency’ of various shells is discussed. Special attention is paid to the analysis of the contribution of the terms in the shell equations representing bending stiffness, and the limits of applicability of the membrane theory of thin shells are discussed in the fluid–structure interaction context. The possibility of cavitation in the internal fluid is investigated, and the effect that cavitation could have on the structural dynamics of the shell is discussed. The present paper is a follow-up of the author's earlier study of the interaction between fluid-filled cylindrical shells and external shock waves.     

Tangential interaction of elastic spherical particles in contact

The elastic interaction of spherical particles is studied. The distribution of the stress, normal to the contact plane, is determined by the rod model suggested recently, which is applicable in the more wide range of deformations as compared with the classical Hertz law. In the rod model context an inner part of compressed particles is regarded as an elastic cylindrical rod, which radius is equal to the contact radius. The rod reaction is added to the normal particle interaction corresponding with the Hertz solution. The resulting normal force passes into the Hertz solution for infinitesimal deformations and gives stronger particle repulsion for finite deformations. Here we solve the Mindlin problem for the rod model, i.e., derive the tangential interaction of initially compressed particles when a relative displacement takes place. The analytical expressions, which determine the total displacement of the sphere’s centers and the corresponding tangential force, are derived. So, the generalization of the classical Mindlin law is obtained for the rod model.     

The physical interaction of spherical particles in suspensions

Summary An experimental examination of the interaction between a falling sphere and an aqueous suspension of nonsettling spheres through which it is passing is described. It is shown that if theReynolds number is small the falling speed of a sphere is unaffected by the size of the suspended spheres, provided that the volume concentration of the suspension is unchanged. This is shown to be true even when the falling sphere is of the same size as the suspended spheres. At highReynolds numbers, however, the falling speed of a sphere diminishes as its size approaches that of the suspended particles. The significance of these results to the theory of the viscosity of suspensions is discussed.     

Dynamics of a fluid-filled curvilinear pipeline

A mathematical model is presented, and numerical experiments are performed to describe the mechanics of the slow movement of a pipeline. The problem reduction algorithm to one-dimensional formulation is offered. Results of numerical experiment for the model problem are adduced. The proposed mathematical model is found to adequately describe the dynamics of known phenomena of pipes. The cross-sections of the extended curvilinear thin-walled pipeline are numerically demonstrated to experience warping, which has experimental confirmation in the literature.     

Localization of wave motions in a fluid-filled cylinder

The properties of harmonic surface waves in a fluid-filled cylinder made of a compliant material are studied. The wave motions are described by a complete system of dynamic equations of elasticity and the equation of motion of a perfect compressible fluid. An asymptotic analysis of the dispersion equation for large wave numbers and a qualitative analysis of the dispersion spectrum show that there are two surface waves in this waveguide system. The first normal wave forms a Stoneley wave on the inside surface with increase in the wave number. The second normal wave forms a Rayleigh wave on the outside surface. The phase velocities of all the other waves tend to the velocity of the shear wave in the cylinder material. The dispersion, kinematic, and energy characteristics of surface waves are analyzed. It is established how the wave localization processes differ in hard and compliant materials of the cylinder __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 4, pp. 72–86, April 2008.     

Effect of momentum transfer condition at the interface of a model of creeping flow past a spherical permeable aggregate

Creeping flow past an isolated, spherical and permeable aggregate has been studied adopting the Stokes equation to model the fluid external to the aggregate and the Brinkman equation for the internal flow. At the interface of the clear fluid and porous region stress jump boundary condition for tangential stresses is used along with the continuity of velocity components and continuity of the normal stress. Using Faxen’s laws, drag and torque are calculated for different flow conditions and it is observed that drag and torque not only change with the permeability of the porous region, but as stress jump coefficient increases, the rate of change in behavior of drag and torque increases.     

Temperature force of interaction between spherical particles

The force of interaction between small particles in a gas induced by a temperature difference between the particle surface and the gas far away from the particle is considered. The particle dimensions correspond to the free-molecular, transitional, and continuum heat transfer regimes. A Monte-Carlo numerical method of direct statistical simulation of the solution of the nonlinear Boltzmann equation and the results of asymptotic solutions are used. The force of interaction between two hot or cold spherical particles is investigated. The dependence of the temperature force on the particle size, i.e. on the flow regime (Knudsen number), and the distance between the particles is examined. Approximations for these dependences are constructed.     

Deformation of a fluid-filled compliant cylinder in a uniform flow

We investigated surface compliance effects of a fluid-filled object in flow on its shape and internal flow through numerical simulation. A two-dimensional compliant cylinder containing fluid in a flow is a simple model of a cell, e.g. an erythrocyte, leukocyte or platelet. The thin membrane of the cylinder consisted of a network of mass-spring-damper (MSD) systems, representing its mechanical characteristics. We assumed that the stiffness and damping coefficients were those of latex gum. The two-dimensional flow inside and outside the membrane was obtained by solving the two-dimensional Navier–Stokes equations by using the finite element scheme at Re=400, based on the external flow velocity and diameter of an initial circular cylinder. The deformation of the membrane was calculated by solving the equation of motion for an MSD system by using the fourth-order Runge-Kutta method. The compliant cylinder deformed more if its stiffness was smaller than that of latex gum. The initial circular section of the cylinder became oval, with a flat front and a convex rear. The aspect ratio of the lateral to streamwise axis length of the oval became larger than unity, and increased with decreasing stiffness. The drag coefficient of the oval cylinder became larger than that of the circular cylinder, and increased with decreasing stiffness. The partial vibration at the rear, caused by shedding vortices, induced oscillating internal flows between two antinodes of the vibrating membrane. Since the object with smaller stiffness had higher ductility, velocity fluctuations of the external flow influenced the internal flow of the compliant object through deformation of the membrane.