Orthogonal Curvilinear Coordinates Research Articles

The Conservation Equations for Multicomponent Gas Mixtures in Arbitrary Coordinate Systems

The conservation equations for multicomponent reacting gas mixtures are generally given only in Cartesian or orthogonal curvilinear coordinate systems. Actually, the conservation equations are easily expressed in an arbitrary coordinate system. We present the general equations in tensor notation and then indicate the simplifications which arise for orthogonal curvilinear coordinates.

On the general form of the Beltrami equation and Papkovich's solution of the axially symmetrical problem of the classical theory of elasticity

The paper shows the connection between the general form of the Beltrami equation of compatibility, which had already been derived by the author, and the form given for it in orthogonal curvilinear coordinates by Lurye (para 2, 3).Some of the properties of Papkovich's general solution of the axially symmetrical problem (without body forces) of the theory of elasticity and its relation to the Finzi-Krutkov solution are discussed so as to supplement to a certain extent the paper of Trenin (para 4, 5, 6).

The Stresses in a Plate of Infinite Size Having Two Contacted Circular Holes

This paper deals with the stresses in a plate of infinite size having two contacted circular holes of equall diameter under 1) tension in all direction, 2) tension or 3) shear in the arbitrary direction. This plate was shaped with the orthogonal curvilinear coordinates used for inversion, and Airy function was determined from the edge conditions with the aid of Fourier transform and the stresses along the rim of holes were derived from it. The stress distributions under tension of shear in various directions were calculated. The maximum factor of stress concentration was 3.869 (under tension), and 4.843 (under shear).

The Equations of Mean Motion of an Incompressible Viscous Fluid in Orthogonal Curvilinear Coordinates

The transformation of the equations of motion of a viscous fluid to orthogonal curvilinear coordinates has been discussed by Jeffery (1915). The transformation of the vorticity equations to orthogonal curvilinear coordinates has been treated by the present author (1950). But analogous equations for the mean motion of a viscous fluid do not appear to have attracted the same attention. The first section of this paper deals with the transformation of the equations of mean motion. In the second section, the theory is illustrated by applications to cylindrical and spherical polar coordinates. In the attempt to transform the equations of mean motion, the authorwill employ the same notations as the cited paper by Jeffery.

On the Definition of Stress in the Finite Deformation Theory

In the present paper, three kinds of stresses, i.e. the stress tensor, the true and nominal stress in the finite deformation theory being considered, taking orthogonal curvilinear coordinates as the independent variables, the interrelations of each stress are obtained. The equilibrium equations corresponding to each definition of stress are given in the form convenient to the statical and dynamical problems respectively. The tensorial equation is more simple formally, but not so substancially.

Stress Distribution of a thick Tube, subjecting to an Internal Pressure, which has a Cross Section of Regular Polygon and a Circular Hole in its Center

(This Paper was read before the Meeting of the Japan Society of Mechanical Engineers and Applied Mechanics held in Nagoya on the 26th. Nov. 1945) The solution for an internal pressure acting at a circular hole in the center of a regular polygonal tube can be obtained from the following complex function z=C∫ (1-tn)n^^2 dt, z=x+iy, t=ρ (cos θ+i sin θ) using the orthogonal curvilinear coordinate. The differential equation for Airy's stress function F in this case becomes [numerical formula] where h2=c-2 (1-2ρncosnθ+ρ2n) n^^2 The boundary conditions for this problem are σρ=0, τρθ=0 for ρ=1 σρ=-p, τρθ=0 for ρ=ρ0, where σρ, σθ, represent a normal stress acting a curve of ρ=const. and θ=const. on the x, y plane, and τρθ a tangential stress acting along the both curves. Then σρ, σθ, τρθ and F can be write in the following form : [numerical formula] where Ao, As, Bs, Co, Cs and Ds are constants and they are determined by the boundary conditions. The details are descibed in the original paper.

The Vorticity Equations in Orthogonal Curvilinear Coordinates

The transformation of the equations of motion of a viscous fluid to orthogonal curvilinear coordinates has been discussed by G. B. JEFFERY, and that of the equations of motion in dynamical meteorology to orthogonal curvilinear coordinates by the present authoi, out the analogous equations for the vorticity do not appear to have attracted the same attention. The first section of this paper deals with the transformation of the vorticity equations to the orthogonal curvilinear coordinates. The second section is devoted to applications to cylindrical and spherical polar coordinates.

Transformation of the Equations of Motion in Dynamical Meteorology to Orthogonal Curvilinear Coordinates

Transformation of the Equations of Motion in Dynamical Meteorology to Orthogonal Curvilinear Coordinates

On the Solution of the Equilibrium Equations of Elasticity in General Curvilinear Coordinates

A system of stress functions consisting of a harmonic vector and a harmonic scalar potential is introduced into the equations of equilibrium of an isotropic elastic body in arbitrary curvilinear coordinates and tensor form. The components of displacement and of the strew tensor me expressed in terms of these potentials and it is shown that the total number of independent components of the potentials is reducible to three.The tensor equations am then specialized to orthogonal curvilinear coordinates and expressions derived for the " physical " components of displacement and stress in terms of the stress functions. Reference is &e to the relation between the present approach and previous ones.

A New Approximate Solution in Boundary Value Problems by the Use of Orthogonal Curvilinear Coordinates

A New Approximate Solution in Boundary Value Problems by the Use of Orthogonal Curvilinear Coordinates

Equivalent Circuits of Compressible and Incompressible Fluid Flow Fields

(1) Div pv — Q; curl v — V b6 b6 (2) Div a grad = & = fc-^ + b2 -^ + h + h bt bt b\fr bil/ (3) Curl c curl ip = d = di--^ + d t ~ + dz-ip + d* bt bt under transient, sinusoidal, or steady-state conditions, thereby allowing their solution either by a network analyzer or by numerical and analytic circuit methods. In fluid dynamics these equations represent the steady potential flow of a compressible fluid when expressed in the hodograph plane or the general nonviscous flow of an incompressible fluid when expressed in the physical plane or space. (In representing the flow of a compressible fluid in the physical space, the networks have variable units, since the equations are nonlinear.) The same equations and networks may be considered to represent static electromagnetic fields in current-carrying regions or in regions with stationary charges or poles. The second equation may also represent the conduction of heat (or diffusion) in a nonisotropic medium, or it may represent the general wave equation; All networks are expressed in curvilinear orthogonal coordinates to simplify the networks in case of curvilinear boundaries and to express three-dimensional problems with axial or other symmetry by means of a two-dimensional network. The coefficients p, a, and c are not scalars but elements of diagonal matrices, allowing special types of nonisotropic properties in the fields. All coefficients may be constants or functions of the independent space variables, thereby allowing nonhomogeneous fields. The fields may contain multiply connected regions, and a method showing how to represent Riemann-surfaces by equivalent circuits is given. Most transient and steady-state equivalent circuits require only resistances, inductances, and capacitors, while some transient networks also require ideal transformers. The blocks into which space is divided may have arbitrary lengths in different directions. General nonisotropic fields and nonorthogonal reference frames require an extension of the present equivalent circuits. Numerical methods of solving the equivalent circuits are given in another publication. In general the circuits may be used to check the consistency and accuracy of the results arrived at by other methods, approximate or exact. The unbalanced currents at the junctions (easily calculated) give a quantitative measure of the deviations from the correct answer.

The extension of the heaviside expansion theorem to the equations of engineering and physics in curvilinear orthogonal coordinates

Ever since Heaviside's Expansion Theorem and his operational methods have been available for the solution of transient circuit problems, attempts have been made to extend similar operational methods to the solution of field problems. These methods have not as yet been developed to such a degree as to be employed in field problems that require partial differential equations with more than two independent variables. Such problems are those involving the propagation of electromagnetic or elastic waves, the conduction of heat through solids, the propagation of acoustic waves in air and the distribution of flux in the airgap of electrical machinery all of which may involve one, two, or three dimensions. This paper undertakes to solve such problems by operational methods similar to Heaviside's, in those types of coordinates of the greatest usefulness to engineers, namely curvilinear orthogonal coordinates. The result is given in formulae which are broader than the Expansion Theorem, being applicable to one, two, or three dimensions and include the Expansion Theorem as a special case. Four formulae are derived for the field due to a point source in a given space for the following cases 1. (1) La Place-Poisson equation for steady state conditions. 2. (2)The equation of conduction of heat. 3. (3) Maxwell's wave equation. 4. (4) The wave equation for dissipative medium. These results are very similar and have many elements in common. It is further shown that by restricting the type of problem to be solved to a certain form (the Green's Function) basic solutions of one, two, or three dimensional equations of the type specified can be expressed in the form of separate functions for each of the variables. These separate functions can be listed in tables similar to tables of integrals and will have a corresponding value in eliminating repetition in mathematical operations. From the solution given in such tables (for the Green's Function) the solution for other problems can be derived by well known theorems without the necessity of solving the differential equation again for these different conditions.

The equations of motion of a viscous fluid in tensor notation

The tensor notation, which is probably best known in its application in Einstein's four-dimensional theory of gravitation, can be used in its three-dimensional form in obtaining the general equations in many branches of mathematical physics. In the present paper the various steps in the proof of the equations of motion of a viscous fluid are translated into tensor form. An attempt has been made to develop the various formulae in outline from first principles, with references where necessary to a standard work, and also to give whenever possible a geometrical interpretation of the various vector quantities met with in the course of the argument. In particular the fundamental assumptions of Stokes, on which the viscous equations are based, are stated in the form of the relation between the stress vector across an element of area and the vector element of area. The equations of motion in tensor notation are afterwards translated into ordinary notation for the special case of orthogonal curvilinear coordinates. These can be at once reduced to the commonly useful forms of spherical polars, ellipsoidal coordinates, etc. In a final section it is shown how the tensor equations can be used to deduce the equations of viscous flow in terms of Stokes stream function for motion symmetrical about an axis. The stream function is shown to be identical with the surviving component of the covariant vector potential of which the other two components vanish.

Prize Subjects of the Paris Academy of Sciences

THE following subjects for prizes are announced by the Academy for 1897 and following years; prizes for years other than 1897 are specially indicated. In Geometry, the Grand Prize in the Mathematical Sciences (1898) will be awarded for the best memoir extending the part played in analysis by divergent series; the Bordin Prize (3000fr.) for a study of the questions relating to the determination, properties, and applications of systems of orthogonal curvilinear coordinates of n variables, indicating particularly the degree of generality of these systems (1898); the Francœur Prize (1000fr.), and the Poncelet Prize (2000 fr.), for work contributing to the progress of pure or applied mathematics. In Mechanics, the Extraordinary Prize of 6coo fr. is offered for work increasing the efficiency of the French naval forces; a Montyon Prize (700fr.) for the invention or improvement of instruments useful in agriculture or the mechanical arts; the Plumey Prize (2500fr.), for improvements in steam navigation; the Fourneyron Prize, for a complete discussion of the motion and stability of bicycles. In Astronomy, the Lalande Prize (540 fr.) will be given for the observation or work most useful to the progress of the science; the Damoiseau Prize (1500fr.), for calculations connecting the appearances of Halley's comet, taking into account the attraction of Neptune, and giving the exact time of its next appearance in 1910; also, for 1898, for a study of the perturbations of Hyperion, the satellite of Saturn, deducing the mass of Titan; the Valz Prize (460 fr.), for the most interesting astronomical observation made during the year; and the Janssen Prize (for 1898), for the most important discovery in Astronomical Physics.