Perturbation solution to the nonlinear problem of oblique water exit of an axisymmetric body with a large exit-angle

In this paper,a nonlinear,unsteady3-D free surface problem of the oblique water exitof an axisymmetric body with a large water exit-angle was investigated by means of theperturbation method in which the complementary angle a of the water exit angle waschosen as a small parameter.The original3-D problem was solved by expanding it into apower series of a and reduced to a number of2-D problems.The integral expressions forthe first three order solutions were given in terms of the complete elliptic functions of thefirst and second kinds.The zeroth-order solution didn‘t turn out to be a linear problem asusual but a nonlinear one corresponding to the vertical water exit for the same body.Computational results were presented for the free surface shapes and the forces exerted upto the second order during the oblique water exit of a series of ellipsoids with various ratiosof length to diameter at different Froude numbers.     

Determining the shape of an axisymmetric body in a viscous incompressible flow on the basis of the pressure distribution on the body surface

A method is developed for determining the shape of an axisymmetric body on the basis of the pressure coefficient distribution specified along the meridional section of the body. Viscosity is taken into account within the framework of the boundary layer model. The method is based on an iterative process, which involves the solutions of the inverse problem in the plane case and of the direct problem for an axisymmetric body. A code implementing the iterative process is written, and examples of numerical results are given.     

Numerical solution of a viscous incompressible flow problem through an orifice

A steady flow problem of a viscous, incompressible fluid through an orifice is widely applicable to many physical phenomena and has been studied previously by many researchers. A problem of such type has been solved by applying LAD method given by Roache [1]. The resulting system of linear equations is solved by Hockney's method [2].     

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.     

One solution of an axisymmetric problem of the elasticity theory for a transversely isotropic material

A numerical-analytical method based on approximation of the sought solution by a system of basis functions is proposed to solve the boundary-value problem of axisymmetric deformation of articles made of a transversely isotropic material. An algorithm for constructing polynomial functions on the basis of invariant-group solutions is described.     

Nonuniqueness of equilibrium states for axisymmetric elastic shells in tension

The problem of nonuniqueness of static axisymmetric solutions for a non-linearly elastic cylindrical shell in which the ends are pulled apart with a constant traction while retaining the radii of its ends fixed is studied. In the elastic case, we prove the existence of buckled states and the possibility of necking. In the hyperelastic case a global existence and nonuniqueness theorem is proved, via the energy criterion.     

An elasticity solution for axisymmetric problem of finite circular cylinder

In this paper,the complete double-series in the closed region expressing the double-variable functions and their partial derivatives are derived by the H-transformation and Stockes’transformation.Using the double-series,a series solution for the axisymmetric boundary value problem of the elastic circular cylinder with finite length is presented.In a numerical example,the cylinder subjected to the axisymmetric tra(?)s with various loaded regions is investigated and the distributions of the displacements and stresses are obtained.It is possible to solve the axisymmetric boundary value problems in the cylinderical coordinates for other scientific fields by use of the method presented in this paper.     

Nonlinear and viscous effects on wave propagation in an elastic axisymmetric vessel

In this paper, a power series and Fourier series approach is used to solve the governing equations of motion in an elastic axisymmetric vessel with the assumption that the fluid is incompressible and Newtonian in a laminar flow. We obtain solutions for the wave speed and attenuation coefficient, analytically where possible, and show how these differ under a number of different conditions. Viscosity is found to reduce the wave speed from that predicted by linear wave theory and the nonlinear terms to increase the wave speed in comparison to the linear solution. For vessels with a wall stiffness in the arterial range, the reduction in the wave speed due to the viscous terms is approximately 10% and the increase due to the nonlinear terms is approximately 5%. This difference between the linear and nonlinear wave speeds was found to be largely constant irrespective of the number of terms considered in the power series for the velocity profile. The linear wave speed was found to vary weakly with stiffness, whilst the nonlinear wave speed was found to vary significantly with the stiffness, especially at low values of stiffness. The 10% variation in the wave speed due to the viscous terms was found to be constant with wall stiffness whilst the 5% variation due to the nonlinear terms was found to vary with wall stiffness. The importance of the number of terms considered in the power series is discussed showing that only a relatively small number is required in the viscous case to obtain accurate results.     

Nonstationary indentation of a blunt rigid body into an elastic layer: An axisymmetric problem

The nonstationary indentation of a blunt rigid body into an elastic layer is studied. The general formulation of the problem includes different boundary conditions in the contact area and on the free surface of the layer. The simplified nonmixed problem that arises at the early stage of interaction and allows obtaining approximate results for later times is solved exactly. The solution obtained is compared with that for the plane case __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 35–46, July 2008.     

A general solution of the axisymmetric contact problem for biphasic cartilage layers

A unilateral axisymmetric contact problem for articular cartilage layers is considered. The articular cartilages bonded to subchondral bones are modeled as biphasic materials consisting of a solid phase and a fluid phase. It is assumed that the subchondral bones are rigid and shaped like bodies of revolution with arbitrary convex profiles. The obtained closed-form analytical solution is valid over time periods compared with the typical diffusion time and can be used for increasing loading.     

Second approximation in the problem of the strong viscous interaction on slender three-dimensional bodies

We consider the hypersonic flow of a perfect gas past a slender three-dimensional body in a regime of strong viscous interaction. We give equations which make it possible to reduce the problem of determining the aerodynamic characteristics of a body which is not axisymmetric to the problem of computing the flow past an equivalent body of rotation at zero angle of incidence. The second approximation for the heat transfer and drag coefficients is found by the method of external and internal combinations of asymptotic expansions. The region in which this method can be applied and the accuracy of the asymptotic theory are estimated by comparison with exact numerical computations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti, i Gaza, No. 5, pp. 107–113, September–October, 1970.     

Nonstationary axisymmetric problem of the impact of a spherical shell on an elastic half-space (initial stage of interaction)

The supersonic stage of interaction (where the rate of expansion of the contact region is no less than the speed of compression waves) between a Timoshenko-type spherical shell (indenter) and an elastic half-space (foundation) is studied. The expansion of the desired functions in series in Legendre polynomials and their derivatives are used to construct a system of resolving equations. An analytical-numerical algorithm for solving this system is developed. A similar problem was considered in [1], where the original problem was replaced by a problem with a periodic system of indenters.     

Numerical solution to the problem of gas exhausting from planar and axisymmetric vessels

A study is made of a gas jet exhausting from an infinite vessel with planar or axisymmetric walls whose generator makes angles ± with the symmetry axis of the jet. The flow regime is determined by the pressure ratio _a = P_a/P_O, where P_O is the pressure in the vessel and P_a the pressure in the surrounding space.Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 174–176, March–April, 1982.We thank M. E. Deich for interest in the work and helpful discussions of the results.     

A general solution and the application of space axisymmetric problem in piezoelectric material

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Analytical solution in the problem of constructing axisymmetric noses with minimum wave drag

An analytical procedure for determining the axisymmetric nose shapes that ensure minimum wave drag is developed. The solution is constructed as an improving variation on the conical shape. The target function is built up in an approximate form on the basis of the assumption that the relationship between the geometric and gasdynamic parameters is local in nature. It is shown that the optimal bodies are truncated power-law bodies with an exponent equal to 2/3. The bodies thus obtained are compared with those constructed in accordance with an exact formulation of the problem.     

Bifurcation of the solution to the problem of steady traveling waves in a layer of viscous liquid on an inclined plane

Traveling waves in a viscous liquid flowing down an inclined plane can be described at small and moderate Reynolds numbers by an ordinary differential equation in the thickness of the layer [1, 2]. As the Reynolds number tends to zero, this equation goes over into an equation of third order with quadratic nonlinearity [3]. Periodic solutions of this last equation bifurcating from the plane-parallel solution have been investigated by Nepomnyashchii and Tsvelodub [3–6]. In the present paper, a study is made of the bifurcation of periodic solutions from periodic solutions, namely, an investigation is made of the values of the wave number for which a periodic solution is not unique; a bifurcation equation is derived, the number of bifurcating solutions is found, and their behavior near a bifurcation point is considered; and the bifurcating solutions are continued numerically with respect to a parameter (the wave number) from the neighborhoods of the bifurcation points.