Jet flow around an elastic shell

In [1] an investigation was made of jet flow around an elastic plate. Below, in an exact nonlinear statement, a study is made of the problem of jet flow around an elastic cylindrical shell, fastened at one end and having the second end free. With certain limitations on the form of the shell, the single-valued solvability of the problem is demonstrated, and a method for its solution is proposed. Some results of calculations are given. A statement and a solution of the inverse problem of static hydroelasticity are also given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 24–32, March–April, 1977.     

Nonlinear Antiplane Deformation of an Elastic Body

The antiplane elastic deformation of a homogeneous isotropic prestretched cylindrical body is studied in a nonlinear formulation in actual–state variables under incompressibility conditions, the absence of volume forces, and under constant lateral loading along the generatrix. The boundary–value problem of axial displacement is obtained in Cartesian and complex variables and sufficient ellipticity conditions for this problem are indicated in terms of the elastic potential. The similarity to a plane vortex–free gas flow is established. The problem is solved for Mooney and Rivlin—Sonders materials simulating strong elastic deformations of rubber–like materials. Axisymmetric solutions are considered.     

Stability of an elastoplastic sphere under external pressure

The stability of ideally plastic, elastoplastic, and reinforced elastoviscoplastic bodies subjected to large subcritical strains was investigated in [1–4], The problems solved in these papers were related to the stability of systems in which homogeneous stress and strain fields arise in the initial state. The stability of an elastic thick-walled spherical shell subjected to external pressure leading to large subcritical strains was investigated in [5]. The stability of an axisymmetric sphere of elastoplastic material subjected to large plastic strains is examined below.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 155–159, September–October, 1977.     

A two-dimensional effective model describing fluid–structure interaction in blood flow: analysis, simulation and experimental validation

We derive a closed system of effective equations describing a time-dependent flow of a viscous incompressible Newtonian fluid through a long and narrow elastic tube. The 3D axially symmetric incompressible Navier–Stokes equations are used to model the flow. Two models are used to describe the tube wall: the linear membrane shell model and the linearly elastic membrane and the curved, linearly elastic Koiter shell model. We study the behavior of the coupled fluid–structure interaction problem in the limit when the ratio between the radius and the length of the tube, , tends to zero. We obtain the reduced equations that are of Biot type with memory. An interesting feature of the reduced equations is that the memory term explicitly captures the viscoelastic nature of the coupled problem. Our model provides significant improvement over the standard 1D approximations of the fluid–structure interaction problem, all of which assume an ad hoc closure assumption for the velocity profile. We performed experimental validation of the reduced model using a mock circulatory flow loop assembled at the Cardiovascular Research Laboratory at the Texas Heart Institute. Experimental results show excellent agreement with the numerically calculated solution. Major applications include blood flow through large human arteries. To cite this article: S. Čanić et al., C. R. Mecanique 333 (2005).     

Solution of certain variational problems of thermal resilience for thin shells considering the selection of optimum conditions for localized heat treatment

One of the possible ways of stating and solving the selection problem for optimum temperature fields for localized axisymmetric heating of shells is investigated. The minimum of shell elastic energy is taken as the optimization criterion. An infinite cylindrical shell was considered in a similar formulation in [1], The corresponding variational problem is formulated for the functional of elastic energy with additional limitations imposed on the function of twist angle at specified shell sections. The variational problem is reduced to an isoperimetric by the use of singular functionals of the -function kind. The related Euler equation is obtained, and this together with the problem resolvent equation constitute a complete set of equations for determining the extremum temperature field with related stress-strain state of the shell. Cylindrical, conical, and spherical shells are considered separately. A numerical analysis is made for the simplest conditions of localized heating of cylindrical and conical shells.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 4, pp. 47–54, July–August, 1968.     

Thermal buckling and post-buckling of pinned–fixed Euler–Bernoulli beams on an elastic foundation

In this article, both thermal buckling and post-buckling of pinned–fixed beams resting on an elastic foundation are investigated. Based on the accurate geometrically non-linear theory for Euler–Bernoulli beams, considering both linear and non-linear elastic foundation effects, governing equations for large static deformations of the beam subjected to uniform temperature rise are derived. Due to the large deformation of the beam, the constraint forces of elastic foundation in both longitudinal and transverse directions are taken into account. The boundary value problem for the non-linear ordinary differential equations is solved effectively by using the shooting method. Characteristic curves of critical buckling temperature versus elastic foundation stiffness parameter corresponding to the first, the second, and the third buckling mode shapes are plotted. From the numerical results it can be found that the buckling load-elastic foundation stiffness curves have no intersection when the value of linear foundation stiffness parameter is less than 3000, which is different from the behaviors of symmetrically supported (pinned–pinned and fixed–fixed) beams. As we expect that the non-linear foundation stiffness parameter has no sharp influence on the critical buckling temperature and it has a slight effect on the post-buckling temperature compared with the linear one.     

Interaction of a spherical shockwave and an elastic circular cylindrical shell

The paper studies the interaction of a spherical shock wave with an elastic circular cylindrical shell immersed in an infinite acoustic medium. The shell is assumed infinitely long. The wave source is quite close to the shell, causing deformation of just a small portion of the shell, which makes it possible to represent the solution by a double Fourier series. The method allows the exact determination of the hydrodynamic forces acting on the shell and analysis of its stress state. Some characteristic features of the stress state are described for different distances to the wave source. Formulas are proposed for establishing the safety conditions of the shell.Translated from Prikladnaya Mekhanika, Vol. 40, No. 9, pp. 94–104, September 2004.     

Drag of porous cylinders in a viscous fluid at low reynolds numbers

The problem of flow past a permeable cylinder at low Reynolds numbers is of interest for the solution of a number of problems in chemical technology in, for example, the design of porous electrodes and porous catalysts and in the calculation of nonstationary filtration of aerosols by fibrous filters. In the present paper, we solve the problem of transverse flow of a viscous fluid past a continuous cylinder in a porous shell and, in particular, in the case of a porous cylinder under conditions of constrained flow (system of cylinders) and an isolated cylinder at arbitrary permeability. The analogous problem of Stokes flow past permeable spheres has been solved in a number of papers [1–3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 122–124, November–December, 1979.     

Jet flow around an elastic plate

An iteration method to solve a certain class of nonlinear plane jet problems of static hydroelasticity is proposed. The convergence of simple iterations is proved. The problem of the symmetric flow around an elastic plate is solved by the Kirchhoff scheme as an illustration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 35–42, July–August, 1976.The authors are grateful to G. Yu. Stepanov who supervised the present research.     

Creep buckling and postbuckling of circular cylindrical shells under axial compression

An infinitely long, axially compressed, circular cylindrical shell with an imperfection in the shape of the axisymmetric classical buckling mode, undergoing steady or non-steady creep, is analyzed. The axisymmetric problem is solved incrementally using nonlinear shell equations The ratio of the applied stress to the classical buckling stress determines if the shell will collapse axisymmetrically or if it will bifurcate into a nonaxisymmetric mode, and whether or not bifurcation will result in instantaneous collapse. The bifurcation problem is formulated exactly and the initial postbuckling behavior is investigated via an asymptotic elastic analysis, based on Koiter's general theory Numerical results are compared with available experimental data.     

Induced anisotropy effect in the neighborhood of an operating well

The mathetical model of porous medium anisotropy induced by an intense flow is discussed. The model is based on the orientation and deformation of the particles of the elastic matrix. The governing law of flow with induced anisotropy is analyzed and the model is numerically investigated for plane-radial flow. Problems and boundary conditions are formulated. The results of the numerical solution explain the effects observed under field conditions.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 67–72, November–December, 1993.     

Approximate solution to the problem of the interaction of a soft shell with a viscous incompressible fluid

The method of finite differences on a nonuniform mesh is used to study the nonstationary flow of a viscous incompressible fluid generated by traveling axisyiametric elastic waves along the surface of a soft cylindrical shell. Expressions are found for the fields of the velocities, vorticities, flow functions, and hydrodynamic forces acting on the body, and also the displacements and velocities of the points of the shell under the influence of the internal driving load and the external hydrodynamic pressure. The boundary conditions of contact between the fluid and the shell are satisfied on the deformed and nondeformed surfaces of the shell.Translated from Izvestiya Akadeinii Nauk SSSR, Mekhanika Zhidkostl i Gaza, No. 3, pp. 132–137, May–June, 1980.     

Dynamic interaction of an oscillating sphere and an elastic cylindrical shell filled with a fluid and immersed in an elastic medium

The paper studies the interaction of a harmonically oscillating spherical body and a thin elastic cylindrical shell filled with a perfect compressible fluid and immersed in an infinite elastic medium. The geometrical center of the sphere is located on the cylinder axis. The acoustic approximation, the theory of thin elastic shells based on the Kirchhoff—Love hypotheses, and the Lamé equations are used to model the motion of the fluid, shell, and medium, respectively. The solution method is based on the possibility of representing partial solutions of the Helmholtz equations written in cylindrical coordinates in terms of partial solutions written in spherical coordinates, and vice versa. Satisfying the boundary conditions at the shell—medium and shell—fluid interfaces and at the spherical surface produces an infinite system of algebraic equations with coefficients in the form of improper integrals of cylindrical functions. This system is solved by the reduction method. The behavior of the hydroelastic system is analyzed against the frequency of forced oscillations.Translated from Prikladnaya Mekhanika, Vol. 40, No. 9, pp. 75–86, September 2004.     

Equilibrium of an Elastic Spherical Shell Filled with a Heavy Fluid under Pressure

This paper considers the problem of equilibrium of a nonlinearly elastic spherical shell filled with a heavy fluid and resting on a smooth, absolutely rigid, flat surface. The weight of the shell is assumed to be negligible in comparison with the weight of the fluid filling it. The contact region with the supporting plane is one of the unknowns in the problem. Equilibrium equations for a membrane shell are obtained in an accurate nonlinear formulation. Stresses and strains of a shell made of a Mooney–Rivlin material are numerically investigated. The results are compared with calculation results for the case of inflation of a spherical shell ignoring the weight of the fluid filling. The effect of the fluid weight on shell strains and stresses is estimated.     

Nonlinear steady-state motion of an elastic plate floating on the surface of a liquid of infinite depth

The nonlinear problem of steady-state waves in an ideal fluid of infinite depth with a thin elastic plate floating on its surface is considered. The solution is found by a perturbation method. Three approximations are obtained. A case of branching of the solution is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 119–123, March–April, 1987.     

Diffraction of an elastic wave in a disc

A numerical solution is constructed for the axisymmetric problem of the diffraction of a plane longitudinal wave in a rigid disc (cylinder) of finite thickness. The disc is enclosed in an unbounded elastic medium; at the contact surface, the tangential stresses are limited by some constant. The incident wave moves along the axis of the cylinder and has the form of a semiinfinite washed-out step. At the same time, a solution is obtained to the corresponding static problem. A study was made of the dependence of the rate of motion of the cylinder and the stress field on the parameters of the problem. In particular, it is shown that the contact conditions have a considerable effect on the stress field only near the lateral surface. The results obtained can be useful for evaluating the errors in measurement of the stresses and velocities in an elastic medium, and possibly also in certain other cases.Deceased.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, No. 3, pp. 139–150, May–June, 1972.     

Displacement of oil by solutions of an active additive from a uniform stratum with allowance for gravity

Self-similar solutions describing the displacement of oil by solutions of an adsorbed active additive have been obtained and investigated [1–3] in the framework of a one-dimensional flow model with neglect of diffusion, capillary, and gravity effects. In the present paper, a self-similar solution is constructed for the problem of oil displacement by an aqueous solution of an active additive from a thin horizontal stratum with allowance for gravity under the assumption that there is instantaneous vertical separation of the phases. This makes it possible to estimate the effectiveness of flooding a stratum by solutions of surfactants and polymers in the cases when gravitational segregation of the phases cannot be ignored.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 87–92, January–February, 1984.     

Self-similar problems of propagation of an extended hydrofracture in a permeable medium

The propagation of an extended hydrofracture in a permeable elastic medium under the influence of an injected viscous fluid is considered within the framework of the model proposed in [1, 2]. It is assumed that the motion of the fluid in the fracture is turbulent. The flow of the fluid in the porous medium is described by the filtration equation. In the quasisteady approximation and for locally one-dimensional leakage [3] new self-similarity solutions of the problem of the hydraulic fracture of a permeable reservoir with an exponential self-similar variable are obtained for plane and axial symmetry. The solution of this two-dimensional evolution problem is reduced to the integration of a one-dimensional integral equation. The asymptotic behavior of the solution near the well and the tip of the fracture is analyzed. The difficulties of using the quasisteady approximation for solving problems of the hydraulic fracture of permeable reservoirs are discussed. Other similarity solutions of the problem of the propagation of plane hydrofractures in the locally one-dimensional leakage approximation were considered in [3, 4] and for leakage constant along the surface of the fracture in [5–7].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 91–101, March–April, 1992.     

Stability of laminar boundary layer on an elastic surface in an incompressible fluid

Interest in the present problem arose after the publication of the results of the experiments of Kramer [1–3]. In addition to the studies indicated in [4], the articles [5–8] are devoted to the question of the interaction of a flexible elastic surface with the boundary layer. In the present paper the problem of the interaction of an elastic surface with disturbances arising in the boundary layer is posed as in [4]. The approximate nature of the methods of solving the problem of the hydrodynamic stability of the laminar boundary layer leads to a difference in the final computational formulas even in the case when authors use the same Heisenberg-Tollmien-Schlichting-Lin scheme. Therefore, in what follows we present a comparison of the data on the stability of the boundary layer on a solid wall obtained by several authors with the calculations using the formulas, which are then generalized to the case of the elastic surface.The author wishes to thank G. I. Petrov and V. A. Medvedev for discussions of the present study.     

Flow around a flexible cylindrical shell by a plane stream of an ideal liquid

The flow by a plane stream of an ideal liquid around a cylindrical shell of zero flexural stiffness (a soft cylindrical shell), or a gas bubble on the boundary of which forces of tension act, was studied in [1–6]. The flow around an elastic plate in a linear formulation was considered in [7, 8]. We consider the flow, around a flexible cylindrical shell which possesses a flexural stiffness and at the same time admits large displacements, by a plane system of an ideal incompressible liquid. An application of methods of the theory of functions of a complex variable leads to an effective solution of the problem. The shape of the shell, the forces in it, the forces acting on the shell, and the field of velocities of the flow of the liquid are determined.