General Three-dimensional Solution Research Articles

General three-dimensional equilibrium for stationary inviscid fluids in the presence of a gravitational potential

Abstract In this paper, we study the properties of the general three-dimensional equilibrium solutions for steady-state equations of inviscid fluids. Incompressible, homogeneous, inhomogeneous and compressible flows motion in a gravitational potential are considered. General three-dimensional formulas for the gas pressure and the gravitational potential are obtained. For incompressible flows, the vector and scalar potentials of the velocity field are used to derive general formulas for general three-dimensional solutions. To verify our results, some examples are presented. For compressible flows, a class of three-dimensional solutions is constructed.

Analytical Solution of the Problem of Symmetric Thermally Stressed State of Thick Plates Based on the 3D Elasticity Theory

An important place among thermoelasticity problems is occupied by the plane elasticity problem obtained from the general three-dimensional problem after using plane stress state hypotheses for thin plates. In the two-dimensional formulation, this problem has become widespread in the study of the effect of temperature loads on the stress state of thin thermosensitive plates. The article proposes a general three-dimensional solution of the static problem of thermoelasticity in a form convenient for practical application. To construct it, a particular solution of the inhomogeneous equation, the thermoelastic displacement potential, was added by us to the general solution of Lamé's equations, the latter solution having been previously found by us in terms of three harmonic functions. It is shown that the use of the proposed solution allows one to satisfy the relation between the static three-dimensional theory of thermoelasticity and boundary conditions, and also to construct a closed system of partial differential equations for the introduced two-dimensional functions without using hypotheses about the plane stress state of a plate. The thermoelastic stress state of a thick or thin plate is divided into two parts. The first part takes into account the thermal effects caused by external heating and internal heat sources, while the second one is determined by a symmetrical force load. The thermoelastic stresses are expressed in terms of deformations and known temperature. A three-dimensional thermoelastic stress-strain state representation is used and the zero boundary conditions on the outer flat surfaces of the plate are precisely satisfied. This allows us to show that the introduced two-dimensional functions will be harmonic. After integrating along the thickness of the plate along the normal to the median surface, normal and shear efforts are expressed in terms of three unknown two-dimensional functions. The three-dimensional stress state of a symmetrically loaded thermosensitive plate was simplified to the two-dimensional state. For this purpose, we used only the hypothesis that the normal stresses perpendicular to the median surface are insignificant in comparison with the longitudinal and transverse ones. Displacements and stresses in the plate are expressed in terms of two two-dimensional harmonic functions and a particular solution, which is determined by a given temperature on the surfaces of the plate. The introduced harmonic functions are determined from the boundary conditions on the side surface of the thick plate. The proposed technique allows the solution of three-dimensional boundary value problems for thick thermosensitive plates to be reduced to a two-dimensional case.

General three-dimensional equilibria for gravitating ideal magnetohydrodynamics of field-aligned steady incompressible flows with an application to solar prominences

This paper investigates the motion of three-dimensional ideal magnetohydrodynamics with incompressible flows. The governing equation is performed at steady state, with the magnetic field parallel to the plasma flow. The equations of stationary equilibrium are derived and described mathematically in Cartesian space. Two approaches for derivation of general three-dimensional solutions for Alfvénic and non-Alfvénic flows at constant and variable fluid densities are constructed. The general vector and scalar potentials of the velocity field are used to derive general formulas of general three-dimensional solutions for Alfvénic and non-Alfvénic flows. To verify the general results we have obtained, some examples are presented. An application that may be of interest for coronal loops and solar prominences is presented.

Three-dimensional nonlinear ideal MHD equilibria with field-aligned incompressible and compressible flows

The equilibrium properties of three-dimensional ideal magnetohydrodynamics (MHD) are investigated. Incompressible and compressible flows are considered. The governing equations are taken in a steady state such that the magnetic field is parallel to the plasma flow. Equations of stationary equilibrium for both of incompressible and compressible MHD flows are derived and described in a mathematical mode. For incompressible MHD flows, Alfvénic and non-Alfvénic flows with constant and variable magnetofluid density are investigated. For Alfvénic incompressible flows, the general three-dimensional solutions are determined with the aid of two potential functions of the velocity field. For non-Alfvénic incompressible flows, the stationary equilibrium equations are reduced to two differential constraints on the potential functions, flow velocity, magnetofluid density, and the static pressure. Some examples which may be of some relevance to axisymmetric confinement systems are presented. For compressible MHD flows, equations of the stationary equilibrium are derived with the aid of a single potential function of the velocity field. The existence of three-dimensional solutions for these MHD flows is investigated. Several classes of three-dimensional exact solutions for several cases of nonlinear equilibrium equations are presented.

General temperature rise solution for a moving plane heat source problem in surface grinding

In the present study, the general solutions for a transient state as well as for the temperature rise formed everywhere in the workpiece due to a rectangular-shaped moving plane heat source arising at the grinding zone are derived. The present analysis starts from a point heat source solution by applying the method of separation of variables to a three-dimensional heat conduction problem. Because the workpiece moving velocity is quite small, the convective term related to the workpiece velocity is first excluded from the heat conduction equation. This workpiece velocity effect will be included in the model by slightly modifying the coordinate variable in the sliding direction shown in the solution of the point heat source. Therefore, the general three-dimensional solution of the stationary temperature rise can be expressed in an integral form as a function of the product value of the unknown initial condition and the particular solution of temperature rise. The unknown initial temperature rise in the solution can be replaced by the point heat source due to frictional that multiplying the product of the Dirac delta functions defined for three directions. Using the definition of the Dirac delta function, the temperature rise solution for a point heat source can thus be obtained. This solution is further extended to obtain the moving and uniform heat sources arising in a rectangular grinding zone. A comparison among the experimental result and the theoretical results predicted by the present model and Jaeger’s model [Jaeger JC (1942) Proc Roy Soc, NSW 76:203–224.] show that the present model is quite accurate and is generally superior to Jaeger’s model; it can be applied to predict the three-dimensional temperature rise distributions in the workpiece.

Mechanical modelling of grain boundary sliding of polycrystals

Using a variational method, a general three-dimensional solution to the problem of a sliding spherical inclusion embedded in an infinite anisotropic medium is presented in this paper. The inclusion itself is also a general anisotropic elastic medium. The interface is treated as a thin interface layer with interphase anisotropic properties. The displacements in the matrix and the inclusion are expressed as polynomial series of the cartesian coordinate components. Using the virtual work principle, a set of linear algebraic equations about unknown coefficients are obtained. Then the general sliding spherical inclusion problem is accurately solved. Based on this solution, a self-consistent method for sliding polycrystals is proposed. Combining this with a two-dimensional model of an aggregate polycrystal, a systematic analysis of the mechanical behaviour of sliding polycrystals is given in detail. Numerical results are given to show the significant effect of grain boundary sliding on the overall mechanical properties of aggregate polycrystals.

Three-dimensional scalar beam diffraction by a half plane

The complex source point technique greatly simplifies the inclusion of source directivity in scattering and diffraction solutions. By replacing its coordinates by appropriate complex coordinates, an omnidirectional source becomes a beam source. Any diffraction solution for the source becomes the corresponding solution for beam diffraction. Geometrical diffraction theory solutions for local beam sources can be most easily obtained in this way. Here the complex source point technique is applied to a canonical solution in geometrical diffraction theory: three-dimensional scalar beam diffraction by a half plane. First, a general three-dimensional solution for a scalar point source and its evolution to a beam source is reviewed. An exact uniform asymptotic solution for point and beam source diffraction by a rigid or soft half plane is then given for arbitrary incidence in three dimensions. A method for numerical evaluation of the diffraction integrals is shown. In this method the Fresnel integrals with complex arguments are converted to complementary error functions with complex arguments, for which efficient computer subroutines are available. The procedure is illustrated by a numerical example and three-dimensional numerical plots are presented. Electromagnetic (vector) field three-dimensional beam diffraction by half planes will yield to the same approach. So also will three-dimensional beam diffraction by wedges using the uniform geometrical theory of diffraction but the resulting expressions are more complicated.

Expressions for solution of the Laplace and Helmholtz equations in three-dimensional eddy-current problems

General three-dimensional solutions are given for the Laplace equation and Helmholtz equation in the cylindrical co-ordinate system, for processes which remain harmonic with time. The expressions for solution which are presented, not only for the magnetic and electrical field variables, but also for the higher-order vector potential, permit an analytical solution. The interrelationships between the field variables, the vector potential, and the higher-order vector potential are discussed. Several examples of engineering applications of the analytical calculation method are given.

Computation of steady and time-dependent temperature distribution for building elements: A general three-dimensional solution

A computational method of temperature distribution calculation for building elements in two- and three-dimensional time-dependent heat flow problems is developed with a general approximate solution using, as a numerical approach, the finite difference method. An analytical evaluation method of the accuracy of calculations is developed. The method described was compared with experimental values and good correspondence was obtained. The time-dependent study which is presented includes thermal resistance and inertia (heat capacity) of the components' materials and different geometric compositions.

A three-dimensional solution for plates and laminates

A general three-dimensional solution is presented for statics and dynamics of plates, homogeneous or laminated, of orthotropic materials. The solution is in series form. Using parts of the general solution a variety of problems, especially of rectangular configurations, can be solved. As Mindlin's approximate analysis for vibration of thick plates is often adequate for specific practical purposes, a general solution for Mindlin's analysis is also given.