Single Partial Differential Equation Research Articles

Side wall effects in centrifugal separation of mixtures

The flow of a centrifugally separating mixture of particles (droplets) and fluid is considered, with emphasis on the azimuthal no-slip requirement on the side wall of the cylindrical container. In the framework of the ‘‘mixture’’ averaged equations—when the Ekman number E, particle Taylor number β, and relative density difference ε are small—a single partial differential nonhomogeneous parabolic equation is obtained for the angular velocity ω(r,t) (relative to the rotating vessel). Analytical and numerical solutions are presented for the dilute limit and for the more general case, respectively. It is shown that, during the separation process, ω develops a quasisteady boundary layer of thickness δ(λ/(λ−1))1/2 on r=1, only if λ>1, where δ=H1/2E1/4 is the typical Stewartson layer scale, λ=E1/2/εβH, and H is the dimensionless height of the container. If λ≤1, however, the side wall affects ω in an unsteady, diffusing viscous domain. Correspondingly significant variations in the axial flux are induced by the Ekman layers. Comparisons to numerical results (produced by a code discussed elsewhere) of the full two-fluid equations display a favorable agreement.

Numerical Prediction of Cavitation in Noncircular Journal Bearings

A numerical investigation of gaseous cavitation in noncircular journal bearings of finite length is presented in this paper. Four bearing profiles are considered: circular, worn-circular, two-lobe (lemon) and elliptical. The alternating direction implicit (ADI) method is used to numerically solve a single partial differential equation that is valid throughout the bearing, i.e., both in full film and cavitated regions. The equation is part of the Elrod algorithm which is used to simulate bearing cavitation. This procedure, which conserves mass throughout the bearing, automatically implements boundary conditions at rupture and reformation that are thought to model moderately to heavily loaded bearings. Performance predictions are compared to results obtained from a variety of earlier papers. Presented at the 43rd Annual Meeting in Cleveland, Ohio May 9–12, 1988

A singular dispersion relation arising in a caricature of a model for morphogenesis.

In 1983 Oster et al. proposed a model for morphogenesis consisting of a system of partial differential equations in which the dispersion relation for the problem linearised about the zero solution has a singularity. That is, the initial growth rate sigma of a small perturbation of wave number k from the zero solution tends to positive or negative infinity as k tends to some critical value kc from above or below respectively. We consider here as a caricature of the model a single partial differential equation with a similar dispersion relation in a bounded one-dimensional domain. The wave number, or equivalently the domain size, may be thought of as a bifurcation parameter. For the Neumann problem a phenomenon arises in which, as the domain size l increases past a critical value ln, the linear stability of the n-th mode jumps from one solution to a remote solution. That is, for l less than ln the trivial solution is unstable and a certain non-trivial solution is stable to perturbations of mode n, whereas for l greater than ln the opposite is true. For the Dirichlet or the Robin problem a linear stability change in the trivial solution occurs, but no corresponding change in any other solution has been found. The corresponding initial boundary value problems are then considered. An asymptotic analysis is performed in the weakly nonlinear limit in the particular case in which only one mode is unstable and gives an asymptotic solution for two classes of nonlinearity, one symmetric and the other asymmetric about u = 0. A development of the method of harmonic balance is then used to obtain approximate solutions in the strongly nonlinear case and when more than one mode may be unstable.

A class of semi-analytical solutions for a rotating toroidal plasma

In this paper, the problem of stationary MHD flow for a rotating toroidal plasma is investigated by assuming that the entropy is a surface quantity. Then, the system of ideal MHD equations is reduced to a single second-order elliptic partial differential equation known as the modified Grad-Shafranov (or Maschke-Perrin) equation. Under the assumption that both the function,P s andf2 are quadratic polynomials of the flux function, a class of semi-analytical solutions is obtained for a plasma contained in a perfectly conducting toroidal boundary with a rectangular cross section. The flux function, poloidal current and the generalized pressure are obtained and discussed for relevant values of the parameters.

Multi-scaling expansion in the systems integrable by the inverse scattering transform

The conformal Galilei algebra (cga) and the exotic conformal Galilei algebra (ecga) are applied to construct partial differential equations (PDEs) and systems of PDEs, which admit these algebras. We show that there are no single second-order PDEs invariant under the cga but systems of PDEs can admit this algebra. Moreover, a wide class of nonlinear PDEs exists, which are conditionally invariant under cga. It is further shown that there are systems of nonlinear PDEs admitting ecga with the realisation obtained very recently in [D. Martelli, Y. Tachikawa, Comments on Galilei conformal field theories and their geometric realisation, preprint, arXiv:0903.5184v2 [hep-th], 2009]. Moreover, wide classes of nonlinear systems, invariant under two different 10-dimensional subalgebras of ecga are explicitly constructed and an example with possible physical interpretation is presented.

Vibrated beds of powders: A new mathematical formulation

A new mathematical formulation was made to deal with the compressible gas model that represents the vibrated particle bed. A novel boundary condition, which incorporates the equation of motion of the bed and the equation of continuity in the air gap between the bed and the vessel base, was introduced. This made it possible to reduce three differential equations, hitherto treated as governing equations that characterize the vibrated particle bed, to a single partial differential equation with pertinent initial and boundary conditions. Experiments were also conducted to assess the validity of the model. Satisfactory agreements between the predicted and measured values have been observed within limited parameter ranges.

Double-null coordinates for the Vaidya metric.

Einstein's equations with spherical symmetry are formulated in double-null coordinates, and the high-frequency approximation to a unidirectional radial flow of unpolarized radiation (the Vaidya metric) is studied in detail. For this case the Einstein equations reduce to a single first-order nonlinear partial differential equation. Integration of this equation introduces an arbitrary function (of one null variable) which must be chosen so as to regularize the metric across horizons. Although the problem is, in general, not analytically solvable, we are able to extend the class of known analytic solutions from the constant-mass case (Kruskal-Szekeres metric) to linear and exponential mass functions. In the linear case we give the first explicit regular covering of a spacetime with a naked shell-focusing singularity.

Petrov type II perfect fluid spacetimes with vorticity

The authors present a stationary spacetime of Robinson-Trautman type containing a perfect fluid with vorticity and obeying the equation of state mu +3p=constant. Einstein's equations are reduced to a single fourth-order partial differential equation in two independent variables. A special case of the spacetime refers to a static sphere of perfect fluid and it is suggested that the solution may describe such a sphere disturbed in some way.

Solitons Generated by Pressure Gradients in Nematic Liquid Crystals

Abstract Equations of motion of director and velocity of nematic liquid crystals under pressure gradients are derived from the Ericksen-Leslie theory. Under suitable approximations, these coupled equations are reduced to a single (2 + 1) nonlinear partial differential equation for the director angle. Boundary effects are taken into account. Similarity analysis of this equation is presented. Properties of steady-state solutions are discussed. Numerical solutions are obtained. Traveling solutions of the (2 + 1) equation are given numerically showing the existence of solitons propagating along the direction of the pressure gradient (in both directions). Corresponding optical interference patterns of these solitons under both monochromatic and white lights are calculated. Good agreement between our theoretical results and recent experiments are obtained.

Theory of rectilinear flow of granular materials

Rectilinear flow of granular materials is of interest in connection with the bulk transport of granular solids in chutes and tubes, and also for the stability of slopes and behaviour of landslides. We formulate the theory of rectilinear flow, under constant body forces, of a frictional and cohesive material which satisfies the Coulomb-Mohr yield criterion. For flow parallel to a traction-free plane surface, the stress in deforming regions is governed by a single quasi-linear partial differential equation, whose characteristics are normal cross sections of the slip-surfaces, the surfaces on which the velocity is constant. The characteristic equations are solved to give explicit expressions for the characteristics. The theory is applied to the flow adjacent to a traction-free plane surface, and to the problem of flow in a channel between two vertical rough walls.

On Some Boundary Value Problems in Anisotropic Inhomogeneous Thermostatics and Elastostatics

This paper examines a class of anisotropic thermal and elastic problems which are governed by a single second order elliptic partial differential equation with variable coefficients. The general solution of this equation is obtained in terms of an infinite series. This general solution is used to investigate some particular boundary value problems.

Stretching of Plane Fluid Film

Equations governing the transient stretching of thin plane film of isothermal Newtonian fluids were introduced in both Cartesian and cylindrical coordinates. Using the equations in Cartesian form, it was shown that uniform film thickness is maintained only when the film is initially uniform in thickness and the deformation is uniform biaxial extension/contraction independent of position (x,y) describable by a stretch rate e11(t) in the x direction and e22(t) in the y direction where t is time. Furthermore, the stretching is free of wrinkling or three‐dimensional buckling when the conditions e11+2e11>0 and 2e11+e22>0 are both satisfied. In the one‐dimensional case of axisymmetrical stretching the governing equations in cylindrical coordinates reduce upon Lagrangian transformation to a single partial differential equation in f(u,t), where f is the present radial position of fluid element and u is its initial radial position. Numerical solutions of this equation show that deviation of deformation from unifor...

On the stability of one-dimensional stationary solutions of hyperbolic systems of differential equations containing points at which one characteristic velocity becomes zero

The stability of the stationary solutions of hyperbolic systems of partial differential equations containing a point at which one of the characteristic velocities becomes zero, is investigated. The functions sought are assumed to be time and coordinate dependent, and their number is arbitrary. The study of stability carried out below is based on the results obtained in /1, 2/, according to which the behaviour of the unsteady perturbations near the critical point is described by a single non-linear partial differential equation irrespective of the number of equations in the initial system. The equation is written in terms of a function analogous to the Riemann invariant connected with the vanishing characteristic velocity. The equation is used below to examine all possible cases of continuous solutions of an arbitrary hyperbolic system of equations with continuous and discontinuous right-hand sides, and conditions are formulated under which the growth of perturbations near the critical point at which one of the characteristic velocities becomes zero, leads to the instability of the whole solution in toto. The investigation is carried out taking into account the onset and development of the perturbations connected with other characteristic velocities which have a constant sign within the region considered.

An improved curvilinear finite difference (CFD) method for arbitrary mesh systems

This paper describes the automatic computer generation of finite difference approximations for partial derivatives in arbitrary mesh systems using an “Improved CFD” method. In the first part of the paper, the algorithm of the proposed Improved CFD method, used to solve a general second-order linear partial differential equation defined in a two-dimensional domain, will be presented. The performance of this numerical method will then be compared with the “Original CFD” method developed by Lau[1], and a so-called least square surface fit method developed by Liszka and Orkisz[2]. In the proposed method, extensive use is made of matrix algebra. The method is therefore highly systematic and can be easily implemented into computer programmes. Numerical examples tested in this paper indicate that, for irregular meshes, better numerical accuracy can be attained by the proposed method as compared with the Original CFD method. At the same time, the straightforward extension of the present method to the generation of higher-order-than-two finite difference approximations and to the solution of three-dimensional field problems will also be demonstrated. In the second part of the paper, the computational procedures for the numerical solution of a first boundaryvalue problem which is governed by a single second-order nonlinear partial differential equation will be derived.

Effective macroscopic description for heat conduction in periodic composites

Effective parameters for the macroscopic behaviour of periodic composites are determined, concerning static or transient heat conduction, when the wavelength is large compared to the length of the period. Different situations are analysed using the homogenization method, which lead to different macroscopic descriptions: single classical partial differential equations, single integral partial differential equations with memory effect, systems of partial differential equations, etc. Some simple examples are given where analytical results are possible.

Note on equivalent Lagrangians and symmetries

Shows that every ordered pair of equivalent Lagrangians determines an equivalence class of dynamical symmetries of non-Noether type. Its explicit construction merely requires a particular solution of a single partial differential equations. Various related aspects are discussed.

Generalized noninteracting vortices

The Abelian Higgs model allows noninteracting multivortex solutions for a special choice of coupling constant. We show that more general noninteracting vortices are permitted in models with a general scalar interaction V(f), provided that the kinetic term for the scalar fields in the Lagrangian is suitably modified. We discuss properties of the general system and show that the static field equations are solved by a single nonlinear partial differential equation.

Buckling of antisymmetric cross- and angle-ply laminated plates

A system of three well-known equations of equilibrium governing the buckling response of arbitrarily laminated composite plates is reduced to a single eighth order partial differential equation in terms of a displacement function. This equation is then solved in closed form to predict the buckling response of antisymmetric cross- and angle-ply plates for different boundary conditions. The effect of various plate parameters including the effect of coupling between inplane extension and out of plane bending upon the buckling response of composite plates is discussed. The results are presented in nondimensional graphical form.

The buckling of anti-symmetrically laminated angle-ply and cross-ply plates

The buckling analysis of anti-symmetrically laminated angle-ply and cross-ply plates with two opposite edges simply supported and the other two free has been carried out using essentially an energy approach. The resulting system of three governing equations has been reduced to a single eighth order partial differential equation which is then solved in closed form for natural boundary conditions. Results are presented for buckling under uniform uniaxial compression. The degrading effect of coupling between in-plane extension and out-of-plane bending is shown to be alleviated both by the choice of a suitable thickness ratio of the laminae and/or increasing the number of layers in a laminate of a given thickness.

Kerr-Schild-Vaidya fields with axial symmetry

This paper contains the Kerr-Schild-Vaidya fields with axial symmetry (all metric functions independent of Vaidya's coordinateβ) in closed form. The general problem of Kerr-Schild pure radiation fields without any symmetry can be reduced to a single partial differential equation by means of Kerr's theorem.