Equivalent Two-dimensional Problem Research Articles

Oblique derivative problems for elliptic equations on conical domains

We study the oblique derivative problem for uniformly elliptic equations on cone domains. Under the assumption of axi-symmetry of the solution, we find sufficient conditions on the angle of the oblique vector for Hölder regularity of the gradient to hold up to the vertex of the cone. The proof of regularity is based on the application of carefully constructed barrier methods or via perturbative arguments. In the case that such regularity does not hold, we give explicit counterexamples. We also give a counterexample to regularity in the absence of axi-symmetry. Unlike in the equivalent two-dimensional problem, the gradient Hölder regularity does not hold for all axi-symmetric solutions, but rather the qualitative regularity properties depend on both the opening angle of the cone and the angle of the oblique vector in the boundary condition.

Wave scattering by a three-dimensional submerged horizontal rectangular plate in a channel: experiments and numerical computations

Experiments are conducted in a wave tank to investigate wave scattering by a three-dimensional submerged horizontal rectangular plate in a channel. The free-surface elevation around the plate is presented for various water depths and depths of submergence of the plate. The wave forces and moments are obtained using an underwater load measuring system. The numerical simulations are performed with a parallelized three-dimensional boundary element method. The numerical set-up follows closely the set-up of the laboratory experiments. The numerical results are compared with the experimental results for non-breaking waves. In most cases a good agreement is found for the free-surface elevation, the vertical force and the moment. A physical interpretation of the flow around the plate is provided. The pressure distribution on the lower surface of the plate differs from the linear distribution from the leading edge to the trailing edge that is obtained in the equivalent two-dimensional problem. The reflection by the lateral walls is investigated.

Method for calculation of helical-waveguide eigenmodes on the basis of solving the equivalent two-dimensional problem by field expansion in circular-waveguide modes

We present a method for calculating the dispersion characteristics of eigenmodes of metal waveguides with helical corrugations on the inner surface, which is based on the transition to a new nonorthogonal system of coordinates. In the new coordinate system, the problem of finding helical-waveguide modes is rigorously equivalent to the problem of finding the modes of a circular unitradius waveguide which has an anisotropic filling and is homogenous in the longitudinal direction. To solve the equivalent problem, we expand the field of the desired eigenmode in the modes of an empty circular unit-radius waveguide. As a result, the problem is reduced to solving a generalized algebraic eigenvalue problem. The comparison with the results of earlier three-dimensional calculations shows that the developed method allows one to determine characteristics of helical-waveguides with sufficient accuracy for many important applications and requires much lower calculation costs.

Evaluation of diffraction loss in prism light guides by finite-difference time-domain field modeling

Prism light guides are hollow dielectric tubes that use prismatic facets to guide light by means of total internal reflection. An unresolved problem has been to determine the magnitude of loss caused by diffraction in prism light guides. Neither experimental measurement nor an analytical solution has yet been achieved, so we attacked the problem numerically, in two steps. First, we found a way to represent such a transitionally invariant three-dimensional system as an equivalent two-dimensional problem. Second, we employed the finite-difference time-domain algorithm, with periodic boundary conditions, to yield a computation problem of manageable size. We found that the diffraction-induced transmissivity of a prism light guide wall is of the order of the wavelength divided by the prism size-a result that has encouraging practical implications.

A short note on Squire's theorem for interfacial instabilities in a stratified flow of two superposed fluids

In a recent paper Schaflinger (1994) investigated interfacial instabilities in a stratified flow of two superposed fluids. The paper focused on a viscous resuspension flow (Schaflinger et al., 1990) in which a sediment of non-buoyant, solid particles is resuspended and driven by a clear fluid flow. At the suggestion of one reviewer, Schaflinger (1994) mentions that "[a]ccording to Squire (1933) three-dimensional disturbances in the classical, inviscid problem can be transformed into an equivalent two-dimensional problem, i.e. to each unstable three-dimensional disturbance there corresponds a more unstable two-dimensional one (Drazin and Reid, 1981, p. 129). Even though it seems to be imaginative to carry over Squire's theorem to the current study, the foundation supporting Squire's statement may not hold in the present problem. No studies dealing with this specific issue have been found in the literature. Therefore, this matter is subject to further, thorough investigations." Here we show that Squire's transformation indeed reduces the equations and the boundary conditions for a stratified flow to an equivalent two-dimensional problem.

Free convection boundary layer flow with uniform suction or injection over a cone

The effect of uniform suction or injection on free convection boundary layer over a cone was theoretically investigated. The non-linear ordinary differential equations were obtained by the difference-differential method after transforming it to an equivalent two-dimensional problem by Mangler's transformation. The solutions of the resulting equations can be expressed in the form of integral equations. Numerical calculations were performed solving the integral equations by the iterative numerical quadrature. The velocity profiles, temperature profiles, skin friction parameters and heat transfer parameters with constant wall temperature were computed for various values of suction/injection parameter and cone angle parameter.

Three-dimensional darcian free convection near a stagnation point on an isothermal surface

The unsteady free convection boundary-layer flow in a fluid-saturated porous medium near a general three-dimensional stagnation point is considered. It is shown that the problem can be reduced to an equivalent two-dimensional problem by a simple transformation of variables. The form of solution then depends only on the sign of the quantity a + b, where a and b are the principal curvatures of the body at the stagnation point.

Stability of Hartmann flows

The stability of Hartmann flows for arbitrary magnetic Reynolds numbers is investigated in the framework of linear theory. The initial three-dimensional problem reduces to the equivalent two-dimensional problem. Perturbation theory is used to find asymptotic expressions for the eigenvalues. Distinguishing two types of disturbances — magnetic and hydrodynamic — is shown to be advantageous in a number of cases. Simple features of the stability are considered for particular cases. The well-know Lundquist result is generalized. An energy approach is applied to the problem of stability. The results of simulations involving the solution of the linear stability problem are described. A distinctive picture of stability is developed. There are several types of instability and they can develop simultaneously. The hydrodynamic and magnetic phenomena interact with each other in a very complex fashion. The magnetic field can either enhance flow stability or reduce it.

The stability of thermally stratified plane Poiseuille flow

In studying the stability of a thermally stratified fluid in the presence of a viscous shear flow, we have a situation in which there is an important interaction between the mechanism of instability due to the stratification and the Tollmien-Schlichting mechanism due to the shear. A complete analysis has been carried out for the Bénard problem in the presence of a plane Poiseuille flow and it is shown that, although Squire's transformation can be used to reduce the three-dimensional problem to an equivalent two-dimensional one, a theorem of Squire's type does not follow unless the Richardson number exceeds a certain small negative value. This conclusion follows from the fact that, when the stratification is unstable and the Prandtl number is unity, the equivalent two-dimensional problem becomes identical mathematically to the stability problem for spiral flow between rotating cylinders and, from the known results for the spiral flow problem, Squire's transformation can then be used to obtain the complete three-dimensional stability boundary. For the case of stable stratification, however, Squire's theorem is valid and the instability is of the usual Tollmien—Schlichting type. Additional calculations have been made for this case which show that the flow is completely stabilized when the Richardson number exceeds a certain positive value.

Nonstationary Motion of Purely Supersonic Wings

A general theory is presented for the calculation of the total forces acting on purely supersonic wings. The method applies to wings having an arbitrary downwash distribution (stationary or non-stationary) and is valid whenever all of the wing edges are supersonic. The general three-dimensional non-stationary problem is reduced to an equivalent two-dimensional problem. In the case of harmonic oscillations the aerodynamic coefficients are expressed in terms of known or tabulated functions. The specific example of an oscillating delta wing is considered and values of the aerodynamic coefficients for plunging, pitching, and rolling oscillations are calculated for two Mach numbers.