Showing papers in "The Journal of The Australian Mathematical Society. Series B. Applied Mathematics in 1983"

Water waves, nonlinear Schrödinger equations and their solutions

Abstract: Equations governing modulations of weakly nonlinear water waves are described. The modulations are coupled with wave-induced mean flows except in the case of water deeper than the modulation length scale. Equations suitable for water depths of the order the modulation length scale are deduced from those derived by Davey and Stewartson [5] and Dysthe [6]. A number of ases in which these equations reduce to a one dimensional nonlinear Schrodinger (NLS) equation are enumerated.Several analytical solutions of NLS equations are presented, with discussion of some of their implications for describing the propagation of water waves. Some of the solutions have not been presented in detail, or in convenient form before. One is new, a “rational” solution describing an “amplitude peak” which is isolated in space-time. Ma's [13] soli ton is particularly relevant to the recurrence of uniform wave trains in the experiment of Lake et al.[10].In further discussion it is pointed out that although water waves are unstable to three-dimensional disturbances, an effective description of weakly nonlinear two-dimensional waves would be a useful step towards describing ocean wave propagation.

Non-linear optimization of the material constants in Ogden's stress-deformation function for incompressinle isotropic elastic materials

Abstract: In previous papers, three terms have been included in Ogden's stress-deformation function for incompressible isotropic elastic materials. The material constants have been calculated by elementary methods and the resulting fits to sets of experimental data have been moderately good.The purpose of the present paper is to improve upon established correlation between theory and experiment by means of a systematic optimization procedure for calculating material constants. For purposes of illustration the Levenberg-Marquardt non-linear least squares optimization algorithm is adapted to determine the material constants in Ogden's stress-deformation function.The use of this algorithm for three-term stress-deformation functions improves somewhat on previous results. Calculations are also carried out in respect of a four-term stress-deformation function and further improvement in the fit is achieved over a large range of deformation.

On a two lag differential delay equation

Abstract: Abstract The non-linear differential difference equation of the form is investigated. This equation, with constant coefficients, is used to model the population level, N, of a single species, and incorporates two constant time lags T2 > T1 > 0; for example, regeneration and reproductive lags. The linear equation is investigated analytically, and some linear stability regions are described. The special case in which the two delay terms are equally important in self damping, B = C, is investigated in detail. Numerical solutions for this case show stable limit cycles, with multiple loops appearing when T2/T1 is large. These may correspond to splitting of major peaks in population density observations.

Hearing the shape of an annular drum

Abstract: The asymptotic expansion for a spectral function of the Laplacian operator, involving geometrical properties of the domain, is demonstrated by direct calculation for the case of a doubly-connected region in the form of a narrow annular membrane. By utilizing a known formula for the zeros of the eigenvalue equation containing Bessel functions, the area, total perimeter and connectivity are all extracted explicitly.

A Galerkin-Petrov method for singular integral equations

Abstract: A Galerkin-Petrov method for the approximate solution of the complete singular integral equation with Cauchy kernel, based upon the use of two sets of orthogonal polynomials, is considered. The principal result of this paper proves convergence of the approximate solutions to the exact solution making use of a convergence theorem previously given by the author. In conclusion, some related topics such as a first iterate of the approximate solution and a discretized Galerkin-Petrov method are considered. The paper extends to a much more general equation many results obtained by other authors in particular cases.

Subsonic potential flow past a circle and the transonic controversy

Abstract: The Mach-number series expansion of the potential function for the two-dimensional flow of an inviscid, compressible, perfect, diatomic gas past a circular cylinder is obtained to 29 terms. Analysis of this expansion allows the critical Mach number, at which flow first becomes locally sonic, to be estimated as M * = 0.39823780 ± 0.00000001. Analysis also permits the following estimate of the radius of convergence of the series for the maximum velocity to be made: M c = 0.402667605 ± 0.00000005, though we have been unable to determine the nature of the singularity of M = M c . Since M c exceeds M * by some 1.1%, it follows that this particular “airfoil” can possess a continuous range of shock-free potential flows above the critical Mach number. This result hopefully resolves a 70-year old controversy.

Axisymmertic Stokes flow images in spherical free surfaces with applications to rising bubbles

Abstract: A theorem is derived for the hydrodynamic image of an axially symmetric slow viscous (Stokes) flow in a sphere which is impermeable and free of shear stress. A second theorem establishes a sense in which such a flow past an arbitrary rigid surface or shear-free sphere becomes, on inversion in an arbitrary sphere with its centre on the axis of symmetry, a flow past the rigid or shear-free inverse of that surface or sphere. The theorems are used to simplify the proofs of a number of known results for images of point singularities in plane and spherical rigid and free boundaries, and for a pair of bubbles rising steadily in line in a viscous fluid. They also give for the first time accurate numerical solutions for the velocities of each of a larger number of spherical bubbles rising quasi-steadily in line. These enable one to assess the accuracy of simple approximations to those velocities.

Steady states of the reaction-diffusion equations. Part 1: Questions of existence and continuity of solution branches

Abstract: When material is undergoing an exothermic chemical reaction which is sustained by the diffusion of a reactant, the steady-state regime is governed by a coupled pair of nonlinear elliptic partial differential equations with linear boundary conditions. In this paper we consider questions of existence of solutions to these equations. It is shown that, with the exception of the special case in which the mass-transfer is uninhibited on the boundary, a solution always exists, whereas in this special case a solution exists only for sufficiently low values of the exothermicity. Bounds are established for the solutions and the occurrence of minimal and maximal solutions is shown for some cases. Finally the behaviour of the solution set with respect to one of the parameters is studied.

A tournament problem

Abstract: It is shown that, for any round-robin tournament, one can find a pairing of the teams and allocate home and away matches so that only one member of each pair plays at home in each round.

Existence theorems for Segal quantization via spectral theory in Krein space

Abstract: Segal's unitarizing complex structure J is shown, in the Fermi-Dirac ca

Cyclic gravity waves in deep water

Abstract: Numerical evidence is presented for the existence of unsteady periodic gravity waves of large height in deep water whose shape changes cyclically as they propagate. It is found that, for a given wavelength and maximum wave height, cyclic waves with a range of cyclic periods exist, with a steady wave of permanent shape being an extreme member of the range. The method of solution, using Fourier transforms of the nonlinear surface boundary conditions, determines the irrotational velocity field in the water and the water surface displacement as functions of space and time, from which properties of the waves are demonstrated. In particular, it is shown that cyclic waves are closer to the point of wave breaking than are steady permanent waves of the same wave height and wavelength.

Field of a charged particle in the presence of scalar meson fields in general relativity

Abstract: Field equations for coupled gravitational and zero mass scalar fields in the presence of a point charge are obtained with the aid of a static spherically symmetric conformally flat metric. A closed from exact solution of the field equations is presented which may be considered as describing the field of a charged particle at the origin surrounded by the scalar meson field in a flat space-time.

Steady states of the reaction-diffusion equations. Part II: Uniqueness of solutions and some special cases

Abstract: and G. C. WAKE(Received 9 July 1982)AbstractIn an earlier paper (Part I) the existence and some related properties of the solution to acoupled pair of nonlinear elliptic partial differential equations was considered. Theseequations arise when material is undergoing an exothermic chemical reaction which issustained by the diffusion of a reactant. In this paper we consider the range of parametersfor which the uniqueness of solution is assured and we also investigate the conversequestion of multiple solutions. The special case of the one dimensional shape of theinfinite slab is investigated in full as this case admits to solution by integration.

Inversion of the struve transform of half integer order

Abstract: Defining a spherical Struve functio n (/) =n h ^jr/2rH n+ | /2 (') we show that the Struvetransform of half integer order, or spherical Struve transform,where n is a non-negative integer, may under suitable conditions be solved for/(r): 7 = 0 where 2" J=o a n ,JX' 2J is the sum of the first n + 1 terms in the asymptotic expansion of4> n (x) as x — oo. The coefficients in the asymptotic expansion are identified asIt is further shown tha 4>t function n {x) which ars e representable as spherical Struvetransforms satisfy n + 1 integral constraints, which in turn allow the construction ofmany equivalent inversion formulae.'School of Physics, University of Melbourne, Parkville, Victoria 3052 (permanent address) andTheoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, U.S.A. 2 School of Physics , University New South Wales Kensington 2033 N.S.W.'Department of Mathematics, University of Melbourne, Parkville, Victoria 3052.© Copyright Australian Mathematical Society 1983, Serial-fee code 0334-2700/83.161

Low Reynolds number flow in a heated tube of varying section

Abstract: In this paper we study the effect of forced and free convection heat transfer on flow in an axisymmctric tube whose radius varies slowly in the axial direction. Asymptotic series expansions in terms of a small parameter ∈, which is a measure of the radius variation, are obtained for the velocity components, pressure and temperature on the assumption that the Reynolds number ( R ) is of order one. The effect of the free convection parameter or Grashof number ( G ) on the axial velocity, temperature distribution, shear stress and heat flux at the wall are discussed quantitatively for a locally constricted tube.

Bounds for the growth of a perturbation in some-double diffusive convection problems

Abstract: On presente les limites pour le module de la vitesse de croissance complexe d'une perturbation oscillatoire arbitraire neutre ou instable dans certains problemes de double diffusion concernant l'oceanographie, l'astrophysique et la mecanique des fluides non newtoniens

Scattering analysis and synthesis of wave trains

Abstract: Abstract The Zakharov-Shabat scattering transform is an exact solution technique for the nonlinear Schrödinger equation, which describes the time evolution of weakly nonlinear wave trains. Envelope soliton and uniform wave train solutions of the nonlinear Schrödinger equation are separable in scattering transform space. The scattering transform is a potential analysis and synthesis technique for natural wave trains. Discrete versions of the direct and inverse scattering transform are presented, together with proven algorithms for their numerical computation from typical ocean wave records. The consequences of discrete resolution are considered.

The existence of Hadley convective regimes of atmospheric motion

Abstract: The solutions of the equations describing deep global convection on a rotating planet are discussed. The existence of generalized steady axisymmetric solutions is established. It is then shown that these are classical solutions when the heat source is sufficiently smooth. The solutions are shown to be unique when the heating is sufficiently weak and asymptotically stable when the shear is sufficiently small. Finally, the application of these results to earth's atmosphere is discussed, with eddy viscosity replacing molecular viscosity.

Insulating circular rugs

Abstract: A thin, partially insulating circular rug is placed on a uniform half space up through which a steady heat flow passes. The corresponding dual integral equations are solved using Tranter's method, finite Legendre transforms and Mellin-Bames contour integrals. An untabulated Bessel (or Stieltjes) transform similar to the discontinuous WeberSchafheitlin integral is evaluated, and a simple expression derived for the rug's surface temperature.

Stability of plane Couette flow for high Reynolds number

Abstract: Abstract Experimental evidence shows that plane Couette flow becomes unstable when the Reynolds number R reaches certain critical values. Linear stability theory does not predict these observations and has been unable to locate these instabilities. A Chebyshev/QR numerical technique is used to investigate much higher values of R than those previously tested. In particular, values of R up to 108 are confidently tested, whereas previously values of R up to only 2 × 104 have been considered.

Interaction of internal waves in a continuous thermocline model

Abstract: Weak nonlinear interactions are studied for systems of internal waves when the Brunt-Vaisala frequency is proportional to sech z , where z = 0 is the centre of the thermocline. Explicit results expressed in terms of gamma functions have been obtained for the interaction coefficients appearing in the amplitude evolution equations. The cases considered include resonant triads as well as second and third harmonic resonance. In the non-resonant situation, the Stokes frequency correction due to finite-amplitude effects has been computed and the extension to wave packets is outlined. Finally, the effect of a mean shear on resonant interactions is discussed.

Thermal ignition in a reactive slab with unsymmetric boundary temperatures

Abstract: The problem of thermal ignition in a reactive slab with unsymmetric temperatures equal to 0 and T is considered. Steady state upper and lower solutions are constructed. It is found that T plays a critical role. Results similar to the case with symmetric boundary temperatures are expected when T is small. When T is sufficiently large, there is only one steady state upper or lower solution. The time dependent problem is then considered. Phenomena suggested by studying the upper and lower steady state solutions are confirmed.

The temporal evolution of a system in combustion theory

Abstract: Abstract A model governing the combustion of a material is considered. The model consists of two non-linear coupled parabolic equations with initial and boundary conditions. An approximation for the rate of reactant consumption is made to enable the system to the treated by laplace transform. Three simple geometries are considered; namely, an infinite slab, an infinite circular and a sphere. The results obtained are then compared with numerical solutions for spme specific values of the parameters. There is good agreement over time duration for which numerical work was performed.

A discrete Fourier analysis of coarse mesh rebalancing and some associated iterative methods

Abstract: Iterative methods for solving systems of linear equations may be accelerated by coarse mesh rebalance techniques. The iterative technique, the Method of Implicit Non-stationary Iteration (MINI), is examined through a local-mode Fourier analysis and compared to relaxation techniques as a potential candidate for such acceleration. Results of a global-mode Fourier analysis for MINI, relaxation methods, and the conjugate gradient method are reported for two test problems.

Stationary waves forced by topography in a vertically sheared, stratified, rotating fulid

Abstract: Abstract The effect of an isolated topographic bump in a two-layer fluid on a β-plane is investigated. An analytical solution is derived in terms of the appropriate Green's function for arbitrary topography of finite horizontal extent. It is found that the disturbances generated by the bump are composed of two fundamental modes which may be wave-like or evanescent. The wave-like modes are topographically induced Rossby waves which occur only when there is eastward flow in at least one of the layers. These waves are always confined to the downstream (eastward) side of the bump. Whereas previous studies of this type have concentrated on eastward flow over topography, the theory has been extended here to include a wide range of vertically sheared flows. Particularly important is the case of low level westward flow combined with upper level eastward flow, as it has direct application, for example, to the summertime atmospheric circulation over the sub-tropical regions of the continental land.masses. In this case a wave-like disturbance extends far downstream from the bump for sufficiently large shear, and is of smaller amplitude in the upper layer than in the lower layer because of the effects of the stratification. For small shears, the wave-like mode in the lower layer is small and the character of the disturbance is evanescent, confining it to the immediate neighbourhood of the bump. A stability analysis of the solutions shows that the disturbances may be baroclinically unstable for sufficiently large mean shear.

Some elementary exact channel flows

Abstract: Abstract New polynomial solutions of the Navier-Stokes equations for steady uni-directional flow of viscous incompressible fluid, with a free surface, down inclined channels of specialized cross-section are considered. An inverse method is uded to obtain the geometrical shape of the channel by equating the polynomials solution to zero (i.e. the no-slip condition) and thence determining the boundary shape.

Static plane-symmetric solution of a scalar-tensor theory of gravitation

Abstract: Vacuum field equations in a scalar-tensor theory of gravitation, proposed by Ross, are obtained with the aid of a static plane-symmetric metric. A closed form exact solution to the field equations in this theory is presented which can be considered as an analogue of Taub's empty space-time in Einstein's theory.

Surface-wave interaction with a deeply submerged circular duct

Abstract: On considere l'interaction d'une onde de surface de frequence angulaire ω avec une conduite circulaire profondement immergee, verticale et ouverte. Le probleme de valeur limite resultant est resolu par la technique de Wiener Hopf

The effect of an enclosed air cavity on a rectangular drum

Abstract: The effect of an enclosed air cavity on the natural vibration frequencies of a rectangular membrane is investigated. The modes specified by an even integer are not affected. For the odd-odd modes, the frequency equation is found via a Green's function formulation and is solved to first order in a parameter representing the effect of the cavity of the rectangular drum. The frequencies are raised, with the fundamental being most affected. In the case of degeneracies, each degenerate mode contributes to the frequency shift, but the degeneracy itself is not broken to first order.

On maximum principles for diffusion in the presence of three diffusion paths

Abstract: Abstract This note examines maximum principles for systems of parabolic partial differential equations describing diffusion in the presence of three diffusion paths. The particular system under consideration arises from a random walk model. For a more general system constraints on the various constants are given which guarantee maximum principles. Remarkably, the physical system arising from the random walk model automatically satisfies these constraints.