Elliptic Cylinder Coordinates Research Articles

Photoacoustic waves of a fluidic elliptic cylinder: Analytic solution and finite element method study

In this study, we focused on the photoacoustic wave production of a fluidic elliptic cylinder for modelling blood vessels, where the consideration of the elliptic cross section can be important for some diagnosis of vascular diseases. First, under the condition of optically-thin absorption, the analytic solution based on the photoacoustic Helmholtz equation in elliptic cylinder coordinates by using Mathieu functions was derived. Then, the finite element method (FEM) model was established to verify the analytic solution. In addition to photoacoustic waves and corresponding photoacoustic power spectra, both near- and far-field photoacoustic amplitude angular distributions were compared. The results revealed that the angular dependent photoacoustic power spectra are critical indicators of the ellipse shape variation. This finding can provide a considerable insight into the photoacoustic diagnosis of the blood vessel changes not only in terms of sizes but also shapes.

Small Péclet-number mass transport to a finite strip: An advection–diffusion–reaction model of surface-based biosensors

AbstractWe consider two-dimensional mass transport to a finite absorbing strip in a uniform shear flow as a model of surface-based biosensors. The quantity of interest is the Sherwood number Sh, namely the dimensionless net flux onto the strip. Considering early-time absorption, it is a function of the Péclet number Pe and the Damköhler number Da, which, respectively, represent the characteristic magnitude of advection and reaction relative to diffusion. With a view towards modelling nanoscale biosensors, we consider the limit Pe«1. This singular limit is handled using matched asymptotic expansions, with an inner region on the scale of the strip, where mass transport is diffusively dominated, and an outer region at distances that scale as Pe-1/2, where advection enters the dominant balance. At the inner region, the mass concentration possesses a point-sink behaviour at large distances, proportional to Sh. A rescaled concentration, normalised using that number, thus possesses a universal logarithmic divergence; its leading-order correction represents a uniform background concentration. At the outer region, where advection by the shear flow enters the leading-order balance, the strip appears as a point singularity. Asymptotic matching with the concentration field in that region provides the Sherwood number as$${\rm{Sh}} = {\pi \over {\ln (2/{\rm{P}}{{\rm{e}}^{1/2}}) + 1.0559 + \beta }},$$whereinβis the background concentration. The latter is determined by the solution of the canonical problem governing the rescaled inner concentration, and is accordingly a function of Da alone. Using elliptic-cylinder coordinates, we have obtained an exact solution of the canonical problem, valid for arbitrary values of Da. It is supplemented by approximate solutions for both small and large Da values, representing the respective limits of reaction- and transport-limited conditions.

Asymptotic analytical solution for horizontal shear waves in a piezoelectric elliptic cylinder shell

In this study, an asymptotic analytical solution for horizontal shear waves in a confocal piezoelectric elliptic cylinder shell, with poling direction parallel to the axis, is obtained. On the basis of tensor analysis and elastodynamic equations, the field-governing equation in elliptic cylinder coordinates, which is expressed by displacement and electric potential, is deduced. By the separation of variables, the partial differential equations are translated into ordinary differential equations. Considering electrically open boundary conditions, the ordinary differential equations are solved, and the wave functions of horizontal shear waves (SH waves) in a piezoelectric elliptic cylinder are obtained by the Wentzel–Kramers–Brillouin and power series methods. In a numerical example, the relation between the correction coefficient of the wave number and frequency of SH waves in different piezoelectric elliptic cylinder shells is discussed. The waves’ structures for the first three modes are also illustrated. The results reveal that one or more mode SH waves can propagate along the circumference in a confocal piezoelectric elliptic cylinder shell. The correction coefficient of the wave number of the first mode, which depends on the size of the piezoelectric elliptic cylinder shell, is approximately a constant. The number of modes increases with the frequency and thickness of the shell. For the first mode, the wave energy should concentrate on the convex surface at high frequency. These results should provide theoretical guidance not only for non-destructive evaluation of curved structures but also for the design of novel acoustic devices based on curved structures.

Eigenfunction expansions for a fundamental solution of Laplace’s equation on R3 in parabolic and elliptic cylinder coordinates

A fundamental solution of Laplace’s equation in three dimensions is expanded in harmonic functions that are separated in parabolic or elliptic cylinder coordinates. There are two expansions in each case which reduce to expansions of the Bessel functions J0(kr) or K0(kr), r2 = (x − x0)2 + (y − y0)2, in parabolic and elliptic cylinder harmonics. Advantage is taken of the fact that K0(kr) is a fundamental solution and J0(kr) is the Riemann function of partial differential equations on the Euclidean plane.

Slip Flow in Elliptic Microducts with Constant Heat Flux

This paper outlines a numerical model for determining the dynamic and thermal performances of a rarefied fluid flowing in a microduct with elliptical cross-section. A slip flow is considered, in laminar steady state condition, in fully developed forced convection, with Knudsen number in the range 0.001−0.1, in H1 boundary conditions. The velocity and temperature distributions are determined in the elliptic cross-section, for different values of both aspect ratio γ and Knudsen number, resorting to the Comsol Multiphysics software, to solve the momentum and energy equations. The friction factors (or Poiseuille numbers) and the convective heat transfer coefficients (or Nusselt numbers) are calculated and presented in graphs and tables. The numerical solution is validated resorting to data available in literature for continuum flow in elliptic cross-sections (Kn = 0) and for slip flow in circular ducts ([Formula: see text]). A further benchmark is carried out for the velocity profile for slip flow in ellipticalcross-sections, thanks to a recent analytical solution obtained using elliptic cylinder coordinates and the separation of variables method. The Poiseuille and Nusselt numbers for elliptic cross-sections are discussed. The results may be used to predict pressure drop and heat transfer performance in metallic microducts with elliptic cross-section, produced by microfabrication for microelectromechanical systems (MEMS).

Time Harmonic Two-Dimensional Cavity Scar Statistics: Convex Mirrors and Bowtie

This article examines the localization of time harmonic high-frequency modal fields in two-dimensional cavities along periodic paths between opposing sides of the cavity. The cases where these orbits lead to unstable localized modes are known as scars. This article examines the enhancements for these unstable orbits when the opposing mirrors are convex, constructing the high-frequency field in the scar region using elliptic cylinder coordinates in combination with a random reflection phase from the outer chaotic region. The enhancements when the cavity is symmetric as well as asymmetric about the orbit are examined.

Slip flow in elliptic microchannels

Microscale fluid dynamics has received intensive interest due to the emergence of Micro-Electro-Mechanical Systems (MEMS) technology. When the mean free path of the gas is comparable to the channel's characteristic dimension, the continuum assumption is no longer valid and a velocity slip may occur at the duct walls. The elliptic cross-section is one useful channel shape that may be produced by microfabrication. The elliptic microchannels have potential practical applications in MEMS. Slip flow in elliptic microchannels has been examined and a detailed theoretical analysis has been performed. A solution is obtained using elliptic cylinder coordinates and the separation of variables method. A simple model is developed for predicting the Poiseuille number in elliptic microchannels for slip flow. The developed model may be used to predict mass flow rate and pressure distribution of slip flow in elliptic microchannels.

Computation of the homogeneous and forced solutions of a finite length, line-driven, submerged plate

A formulation is developed to predict the vibration response of a finite length, submerged plate due to a line drive. The formulation starts by describing the fluid in terms of elliptic cylinder coordinates, which allows the fluid loading term to be expressed in terms of Mathieu functions. By moving the fluid loading term to the right-hand side of the equation, it is considered to be a force. The operator that remains on the left-hand side is the same as that of the in vacuo plate: a fourth-order, constant coefficient, ordinary differential equation. Therefore, the problem appears to be an inhomogeneous ordinary differential equation. The solution that results has the same form as that of the in vacuo plate: the sum of a forced solution, and four homogeneous solutions, each of which is multiplied by an arbitrary constant. These constants are then chosen to satisfy the structural boundary conditions on the two ends of the plate. Results for the finite plate are compared to the infinite plate in both the wave number and spatial domains. The theoretical predictions of the plate velocity response are also compared to results from finite element analysis and show reasonable agreement over a large frequency range.

NUMERICAL ANALYSIS OF THE EFFECT OF RADIATION ON LAMINAR STEADY NATURAL CONVECTION IN A TWO-DIMENSIONAL PARTICIPATING MEDIUM BETWEEN TWO HORIZONTAL CONFOCAL ELLIPTICAL CYLINDERS

Combined radiation and natural convection in a participating medium between two horizontal confocal elliptical cylinders is investigated numerically. The equations of steady, laminar two-dimensional natural convection are written by using an elliptic-cylinder coordinates system, the stream function, and the vorticity. The finite volume radiation solution method and the control volume technique are used to discretize the coupled equations of momentum, energy, and radiative transfer. Numerical solutions are obtained for Rayleigh numbers in the range 104 to 2x105 and the radiation-conduction parameter ranging from 0 to infinity. The special case corresponding to the convective flow within the annulus formed by an elliptical cylinder surrounding a flat plate is also considered.

Mathieu integral transforms

Orthogonality and completeness relations are presented for the quasiorthogonal (i.e., orthogonal with respect to a discontinuous weight function) eigenfunctions of a singular (in the sense of Sturm–Liouville theory) boundary-value problem involving the two-dimensional Helmholtz equation in elliptic–cylinder coordinates. These relations yield as special cases integral transforms whose kernels are products of periodic Mathieu functions and modified Mathieu functions of integral order. The new transforms are analogs of the Weber–Orr transform and of a recently published [J. Math. Phys. 30, 41 (1989)] generalized Hankel transform, and would be applicable to boundary-value problems with elliptical geometries. The proof of the orthogonality and completeness relations is surprisingly simple and is based on a novel application of the Sokhotski–Plemelj equations of distribution theory.

An acoustic waveguide with variable cross section

The modes of propagation and the allowed range of frequencies for an acoustic waveguide with variable cross section have been found by solving the wave equation in elliptic cylinder coordinates. This waveguide is formed by two parallel planes and two surfaces obtained by the translation of two branches of hyperbola perpendicular to the planes, so that the cross section is a rectangle with variable length. Numerical results for relevant physical quantities are given.

Diffraction of Cylindrical Electromagnetic Waves by Ribbons and Slits

The diffraction of cylindrical electromagnetic waves by a slit in an infinitely thin, perfectly conducting screen and also by a perfectly conducting ribbon is studied. The problems are formulated in elliptic-cylinder coordinates and solved in terms of Mathieu functions. Both Dirichlet and Neumann boundary conditions are investigated. The scattered intensity in the far field and total cross section are obtained. Typical numerical results are presented for both the long-wave and medium-wave regions.

Diffraction of Plane Waves by a Slit between Two Different Media

The diffraction of plane electromagnetic waves by a slit in an infinitely thin, perfectly conducting screen between two media with different dielectric constants is studied using rigorous diffraction theory. Both the case where the electric vector is parallel to slit and the case where the magnetic vector is parallel to slit are examined. The problem is formulated in elliptic cylinder coordinates and solved in terms of Mathieu functions. The explicit determination of the diffracted wave-expansion coefficients leads to solving infinite systems of complex linear equations. The long-wave (Rayleigh scattering) region is studied in detail. The scattered intensity at infinity, transmission coefficient, and backscatter coefficient are evaluated. Finally, numerical results are presented for some special cases.

The Diffraction of Sound by Ribbons and by Slits

The exact solution of the diffraction problem in elliptic cylinder coordinates is computed by the use of newly compiled tables of Mathieu functions. Curves are given for the distribution-in-angle and total intensity scattered, for the diffraction by a slit or the scattering by a thin ribbon, of waves whose length is the same order of magnitude as the slit or ribbon width. The curves are for different angles of incidence, and for different wave-lengths. Calculations can also be made of the effective area of strips of absorbing material placed on a reflecting wall.

The Diffraction of Waves by Ribbons and by Slits

The exact solution of the diffraction problem in elliptic cylinder coordinates is computed by the use of newly computed tables of Mathieu functions. Curves are given for the distribution-in-angle and total intensity scattered, for the diffraction by a slit or the scattering by a thin ribbon, of waves whose length is the same order of magnitude as the slit or ribbon width. The curves are for different angles of incidence, for different wave-lengths, and for two boundary conditions: zero value and zero normal gradient.