Laplace Equation In Cylindrical Coordinates Research Articles

Image charges from boundary value problems

The examples usually solved by means of the method of images are revisited solving directly the Laplace equation. We also give a simple derivation of the axially symmetric solutions of the Laplace equation in spherical coordinates and of the translationally symmetric solutions of the Laplace equation in cylindrical coordinates.

Vacuum gravitational fields with a null Killing vector

Vacuum gravitational fields admitting a light-like Killing field were systematically studied starting around 1960. Besides the already known plane waves, a second class of gravitational wave fields was found. In contrast to plane waves, their wave surfaces were not flat, but had a negative Gaussian curvature. Recently, such solutions found attention again as "twisted gravitational waves". In the paper we review and extend the earlier results. In suitable coordinates, the metric assumes a simple shape. The waves are then determined by a single function that satisfies a Laplace equation in cylindrical coordinates. The "twisted waves" prove to be a special case.

The Proof of the Riemann Hypothesis and an Application to Physics

In this manuscript, a proof for the age-old Riemann hypothesis is delivered, interpreting the Riemann Zeta function as an analytical signal, and using a signal analyzing affine model used in radar technology to match the warped Riemann Zeta function on the time domain with its conjugate pair on the warped frequency domain (a Dirichlet series), through a scale invariant composite Mellin transform. As an application of above, since the Navier Stokes system solution’s Dirichlet transforms are also Dirichlet series, a minimal general solution of the 3d Navier Stokes differential equation for viscid incompressible flows is constructed through a fractional derivative Fourier transform of the found begin-solutions preserving the geometric properties of the 2d version assuming that the solution is an analytic solution that suffices the Laplace equation in cylindrical coordinates, which is the momentum equation for both the 2d and the 3d Navier Stokes systems of differential equations.

Axially symmetric relativistic thin disks immersed in spheroidal matter haloes

Two infinite families of axially symmetric relativistic thin disks of dust immersed in spheroidal matter haloes are presented. The disks are obtained from solutions of the Einstein equations for an axially symmetric conformastaticspacetimein which the metric tensor is characterised only by one metric function. By introducing a finite discontinuity on the first derivatives of the metric tensor, solutions with a singularity of the delta function type are obtained, so describing thin disks. The nonzero components of the energy-momentum tensor, both for the disk and the halo, are obtained from the Einstein equations. In this way, the energy densities and pressures of the sources are determined. By imposing the fulfilment of all the energy conditions we obtain a constraint over the solutions, in such a way that the metric function can be properly expressed in terms of a solution of the Laplace equation. By using the solution to Laplace equation in cylindrical coordinates we find infinite disks and by using the solution to Laplace equation in oblate spheroidal coordinates we find finite disks. In both cases we obtain particular solutions with energy densities and pressures well behaved everywhere. We also show that the masses of de disks and the haloes are finite. Finally we solve the geodesic equation for circular orbits in the plane of the disk.

Evidence for the importance of particle distribution in effective medium theory: a model study

This study investigates the fundamental effects of particle distribution in a heterogeneous composite, which has never been revealed before. This work is based on the Laplace equation in cylindrical coordinates and shows that the effective thermal conductivity has a different value even at the same volume fraction depending on the particle distribution configuration. Consequently, applying the effective medium approximation without using detailed information about the particle spatial distribution can result in considerable error. This is especially true for the case of augmented conduction by insertion particles.

Fluid Pressures on Unanchored Rigid Flat-Bottom Cylindrical Tanks Under Action of Uplifting Acceleration

To protect flat-bottom cylindrical tanks against severe damage from uplift motion, accurate evaluation of accompanying fluid pressures is indispensable. This paper presents a mathematical solution for evaluating the fluid pressure on a rigid flat-bottom cylindrical tank in the same manner as the procedure outlined and discussed previously by the authors (Taniguchi, T., and Ando, Y., 2010, “Fluid Pressures on Unanchored Rigid Rectangular Tanks Under Action of Uplifting Acceleration,” ASME J. Pressure Vessel Technol., 132(1), p. 011801). With perfect fluid and velocity potential assumed, the Laplace equation in cylindrical coordinates gives a continuity equation, while fluid velocity imparted by the displacement (and its time derivatives) of the shell and bottom plate of the tank defines boundary conditions. The velocity potential is solved with the Fourier–Bessel expansion, and its derivative, with respect to time, gives the fluid pressure at an arbitrary point inside the tank. In practice, designers have to calculate the fluid pressure on the tank whose perimeter of the bottom plate lifts off the ground like a crescent in plan view. However, the asymmetric boundary condition given by the fluid velocity imparted by the deformation of the crescent-like uplift region at the bottom cannot be expressed properly in cylindrical coordinates. This paper examines applicability of a slice model, which is a rigid rectangular tank with a unit depth vertically sliced out of a rigid flat-bottom cylindrical tank with a certain deviation from (in parallel to) the center line of the tank. A mathematical solution for evaluating the fluid pressure on a rigid flat-bottom cylindrical tank accompanying the angular acceleration acting on the pivoting bottom edge of the tank is given by an explicit function of a dimensional variable of the tank, but with Fourier series. It well converges with a few first terms of the Fourier series and accurately calculates the values of the fluid pressure on the tank. In addition, the slice model approximates well the values of the fluid pressure on the shell of a rigid flat-bottom cylindrical tank for any points deviated from the center line. For the designers’ convenience, diagrams that depict the fluid pressures normalized by the maximum tangential acceleration given by the product of the angular acceleration and diagonals of the tank are also presented. The proposed mathematical and graphical methods are cost effective and aid in the design of the flat-bottom cylindrical tanks that allow the uplifting of the bottom plate.

Comparison of boundary collocation methods for singular and non-singular axisymmetric heat transfer problems

This paper presents a computational study of some boundary collocation solution methods for the Laplace equation in cylindrical coordinates with axisymmetry. The methods compared are (i) the direct boundary element method (BEM), (ii) the method of fundamental solutions (MFS) with fixed sources and (iii) the Trefftz method. Relative accuracy of these methods are compared for two test problems. The first problem is a simple problem of heat transfer through a cylindrical rod which is a standard benchmark problem in this field. The second problem deals with heat transfer in silicon melt for Czochralski (CZ) process which involves a singularity in the boundary conditions at the corner of the crystal–melt interface. All the three methods indicated above are highly successful for the simple (first) problem with MFS and Trefftz being simpler to implement than the BEM. However, the Trefftz method was not effective for the second problem due to the boundary singularity and the MFS showed oscillations near the singularity point. Hence the use of higher order non-conforming elements with accurate Gauss–Kronrod integration schemes in the direct BEM method was investigated. It was found that the boundary singularity does not deteriorate the accuracy of the results if this improved numerical integration procedure is used in the direct BEM. Hence higher order elements with Gauss–Kronrod integration schemes can be used for the solution of many free interface problems encountered in crystal growth.

Study of the concentration distribution of SiC vapour in the crystal growth zone

In the present work the gas dynamics in the growth zone of SiC crystals is investigated. it is shown that the propagation of SiC vapour from the growth cavity walls towards the lids is effected by diffusion. On this basis the calculation of the concentration distribution of SiC vapour ( n), the equilibrium vapour concentration ( n s) and the supersaturation (α) in the crystal growth zone at different radial and axial gradients is carried out by solving the Laplace equation in cylindrical coordinates for a stationary case corresponding to the conditions of crystal growth. The received results are compared with the available experimental data which gives a possibility for explaining some of the observed peculiarities during SiC crystal growth from vapour phase by the sublimation method.

Study of the concentration distribution of SiC vapour in the crystal growth zone

In the present work the gas dynamics in the growth zone of SiC crystals is investigated. It is shown that the propagation of SiC vapour from the growth cavity walls towards the lids is effected by diffusion. On this basis the calculation of the concentration distribution of SiC vapour, the equilibrium vapour concentration n s and the supersaturation a=[(n−n s )/ n s ] × 100% in the crystal growth zone at different radial and axial temperature gradients is carried out by solving the Laplace equation in cylindrical coordinates for a stationary case corresponding to the conditions of crystal growth. The results obtained are compared with the available experimental data which gives a possibility for explaining some of the observed peculiarities during SiC crystal growth from the vapour phase by the sublimation method.

Study of the concentration distribution of SiC vapour in the crystal growth zone

In the present work the gas dynamics in the growth zone of SiC crystals is investigated. It is shown that the propagation of SiC vapour from the growth cavity walls towards the lids is effected by diffusion. On this basis the calculation of the concentration distribution of SiC vapour (n), the equilibrium vapour concentration (ns) and the supersaturation ( alpha =((n-ns)/ns)*100%) in the crystal growth zone at different radial and axial temperature gradients is carried out by solving the Laplace equation in cylindrical coordinates for a stationary case corresponding to the conditions of crystal growth. The results obtained are compared with the available experimental data; this makes it possible to explain some of the peculiarities observed during SiC crystal growth from the vapour phase by the sublimation method.

Computation of primary current distribution and cell resistance at model electrodes with circular perforations

A model of a gas-evolving production-type cell with a circularly perforated anode is described. A unit of the model was composed of a disk cathode, a separator and a ring anode in turn. These were located in a cylindrical cell filled with solution. Primary current and potential distributions in the unit cell were computed by solving the Laplace equation in cylindrical coordinates by the finite element method. Geometric parameters determining the distributions were primarily the interelectrode distance and the percentage open area. Current distribution in the open part was larger than that in a rectangular cell with the same geometric parameters because of the cylindrically concentrated supply of the current from the inner part of the ring and the back side of the anode. The unit cell resistance was evaluated as a function of the geometric parameters. It exhibited a linear variation of the interelectrode distance and the square of the percentage open area. There was, however, a slight dependence of the percentage open area on the unit cell resistance and hence it is concluded that circularly perforated electrodes provide higher performance than louvre-type electrodes.

Certain Properties of Electrostatic Fields Encountered in Electron Lenses

The equipotential lines in a bipotential electron lens exhibit a fine structure which is revealed by application of the relaxation method to the solution of the Laplace equation in cylindrical coordinates The asymmetry of the potential field between the two edges of the cylinders is confirmed by the same method and is established to be of the order of 2% in one particular case