One-dimensional Singular Equations Research Articles

The method for calculating singular integrals in problems of axially symmetric Stokes flows

The flow of a viscous fluid at small Reynolds numbers (Stokes flow) in a three-dimensional formulation is investigated. In this case, the inertial terms in the equations of motion can be neglected. Such flows can occur in nanotubes that can be considered as inclusions in representative volume elements of nanomaterials. By using the fundamental solution of Ossen, an integral representation of the velocity is proposed. This representation is used to receive an integral equation for an unknown density. The solution of the resulting equation makes it possible to calculate the fluid pressure on the walls of the shell. The case of axially symmetric flows is investigated. For this, an integral representation of the unknown velocity in cylindrical coordinates is obtained. By integrating over the circumferential coordinate, the two-dimensional singular integral equation is reduced to one-dimensional one. It has been proved that the components of the kernels in singular operators are expressed in terms of elliptic integrals of the first and second kind. It has been proved that the singularities of the kernels of one-dimensional singular integral equations have a logarithmic character. To calculate elliptic integrals, the Gaussian algorithm based on the use of the arithmetic-geometric mean value is proposed. This procedure allows us to obtain logarithmic singular components with high accuracy, which makes it possible to use special quadrature formulas to calculate such integrals. An algorithm with usage of the boundary element method for the numerical solution of the obtained singular integral equations is proposed. The method for solving one-dimensional singular equations, where the kernels contain elliptic integrals with logarithmic singularities (i.e logarithmic singularity is not expressed explicitly) has been tested. The obtained numerical results have been compared with the well-known analytical solutions. The data obtained indicate the high efficiency of the proposed numerical method.

Method of singular integral equations in liquid vibration problems for coaxial shells

The paper deals with the problem of free vibrations of an ideal incompressible fluid in coaxial shells of revolution. It is assumed that the motion of the fluid is irrotational that allows us to introduce the velocity potential. In these suppositions the potential is satisfied to Laplace equation. The boundary conditions are formulated on the wetted surfaces of the shells and on the free liquid surface. The non-penetration conditions are applied to the wetted surfaces. On the free surface we consider dynamical and kinematical boundary conditions. The dynamical condition consists in equality of the liquid pressure on the free surface to the atmospheric one. The kinematic condition requires that total time derivative of the free surface elevation will be equal to zero at any instant. Regarding the potential of velocities, a boundary value problem is formulated that is further reduced to the eigenvalue problem. To solve the boundary value problem for the Laplace equation, the boundary element method is used in a direct formulation. The axial symmetric form of the shells allows us to reduce the obtained system of singular equations to one-dimensional equations. The kernels in singular operators of obtained integral equations are expressed on terms of elliptical integrals of the first and second kinds, and have the logarithmic singularities. The special numerical technique is elaborated to treat with such kind integral equations. The resulting one-dimensional singular equation is solved by the method of discrete singularities. The integration region contains the free surface of the fluid that in the case of coaxial shells is a ring. So, the possibility of using the boundary integral equation approach coupled with application of the discrete singularities method is established to solution of the singular integral equation with incoherent boundaries. A numerical study has been carried out that made it possible to determine the frequencies and modes of the liquid sloshing in the shells for different ratios of the inner and outer radii of cylindrical coaxial shells. The obtained modes of natural vibrations will be used for numerical simulation of forced liquid vibrations in the tanks and reservoirs.

Analytic investigation of features of stresses in plane nonstationary contact problems with moving boundaries

We propose a technique for the analytic investigation of features of contact stresses in the vicinity of the nonstationary moving boundary of a contact region in plane nonstationary contact problems with moving boundaries, which is based on the reduction of a boundary two-dimensional singular integral equation resolving the problem to a system of two one-dimensional singular equations. As tools of research, a method for the reduction of singular integral equations to an equivalent Riemann type problem for piecewise analytic functions and a technique of fractional integro-differentiation are used. It is shown that, on the moving boundary of the contact region, a power singularity, the order of which depends on the velocity of the boundary, takes place.