bluuebunny/crossref_metadata_2025 huggingface dataset:
On Optimum Propulsion by Means of Small Periodic Motions of a Rigid Profile II. Optimization of the Period and Numerical Results
Authors: Urbach H. P.
Published: September 2015

We present some numerically calculated optimum thrust generating small amplitude periodic motions of a rigid profile in an inviscid imcompressible fluid. The motions considered consist in general of both a heaving and a pitching part and have common period. Apart from the prescribed thrust, the motions have to satisfy the demand that the contribution to the total thrust of the suction at the leading edge not exceed a given number and are furthermore subjected to a constraint on their amplitude. Solutions of an analogous optimization problem for pure heaving motions are also discussed. Furthermore, the problem of optimizing the period of the motions is considered.
Two‐Dimensional Advection‐Diffusion Equations with Constant Limiting Solutions
Authors: Howes F. A.
Published: September 2015

We study the asymptotic behavior of solutions of two‐component systems of equations in two dimensions that are related to the steady streamfunction‐vorticity equations for large values of the Reynolds number. In particular, we determine the value of the constant limiting “vorticity” under certain circumstances.
Modulated High‐Frequency Waves
Authors: Almgren Robert F.
Published: September 2015

We construct asymptotic approximations to solutions of nonlinear hyperbolic conservation laws, when the initial data is small‐amplitude high‐frequency waves with modulated wave number. We show that the nonlinear multiwave interaction terms approach zero in the asymptotic limit, so that the wave components satisfy decoupled Burgers equations, provided a certain nonresonance condition holds. This extends previous results on more strictly nonresonant or everywhere resonant waves, to permit modulated high frequencies to pass through resonant and nonresonant values. We show how these results apply to high‐frequency wave propagation in a nonhomogeneous medium or on a nonuniform base state. We illustrate our conclusions with numerical examples and discuss a phenomenon of localized resonance.
Tiling Bricks with Bricks
Authors: Damiani E. , D'Antona O. , Naldi G. , Pavarino L.
Published: September 2015

The problem addressed by the paper is the filling of a large brick with replicas of smaller bricks of different sizes. We solve this problem by reducing it to an algebraic problem about polynomials. As a by‐product, we obtain new combinatorial interpretations of the connection constants linking some classical polynomial sequences of combinatorics.
Homoclinic Bifurcations in n Dimensions
Authors: Fowler A. C.
Published: September 2015

Bifurcations near homoclinic orbits in n dimensions are described. Depending on the eigenvalues of the Jacobian at the fixed point whose real parts are closest to zero, a strange invariant set of periodic and aperiodic orbits can be produced, which can be described by a Bernoulli shift on a finite set of symbols. These results generalize earlier ones of Shil'nikov, Gaspard, Tresser, Glendinning, and Sparrow, amongst others.
The Effect of Profile Symmetry on the Nonlinear Stability of Mixing Layers
Authors: Djordjevic V. D. , Redekopp L. G.
Published: September 2015

The nonlinear stability of arbitrary mixing‐layer profiles in an incompressible, homogeneous fluid is studied in the high‐Reynolds‐number limit where the critical layer is linear and viscous. The type of bifurcation from the marginal state is found to depend crucially on the symmetry properties of the basic‐state profile. When the vorticity profile of the mean flow is perfectly symmetric, the bifurcation is stationary. When the symmetry of the profile is broken, the bifurcation is Hopf. The nonsymmetry of the mixing layer also introduces some changes in the critical layer and the matching of flow quantities across it.
Homoclinic Bifurcations for Partial Differential Equations in Unbounded Domains
Authors: Fowler A. C.
Published: September 2015

The connection between low‐dimensional chaos in ordinary differential equations, and turbulence in fluids and other systems governed by partial differential equations, is one that is in many circumstances not clear. We discuss some examples of turbulent fluid flow, and consider ways in which they may be related to much simpler sets of ordinary differential equations, whose behavior can be reasonably well understood. (We are not advocating drastic Fourier truncation.) The generation of aperiodic solutions through the occurrence of homoclinic orbits is briefly analysed for ordinary differential equations, and the same kind of heuristic analysis is sketched for partial differential equations (in one space dimension). It is suggested that such an analysis can explain certain features of chaos, which have been observed in real fluids.
Phase Shift Modulations for Stable, Oscillatory, Traveling, Strongly Nonlinear Waves
Authors: Haberman Richard
Published: September 2015

Oscillatory dispersive waves propagating in a slowly varying medium are analyzed
for Klein‐Gordon equations with perturbations. The method of multiple
scales is extended to include two fast scales, the usual traveling‐wave phase and
time, in order to allow initial conditions not usually permitted. An exact
wave‐action equation is introduced if the traveling wave is stable, involving
averages over the periodic wave as well as time. This is equivalent to an
extended averaged Lagrangian principle. The equation for the slow modulations
of the phase shift of the traveling wave is derived from the higher order terms in
the exact action equation and is shown to be the same as in earlier more
restrictive studies.
A Frequency Selection Criterion in Spatially Developing Flows
Authors: Chomaz Jean‐Marc , Huerre Patrick , Redekopp Larry G.
Published: September 2015

The possible existence of global modes or self‐excited linear resonances in spatially developing systems is explored within the framework of the WKBJ approximation. It is shown that the existence and properties of the dominant global mode may be deduced from the variations of the local absolute frequency ω0 with distance X. The main results are summarized in two theorems: (1) A system with no region of absolute instability does not sustain temporally growing global modes with an O(1) growth rate. (2) If the singularity X, closest to the real X‐axis of the complex function ω0(X) is a saddle point, the most unstable global mode has, to leading order in the WKBJ approximation, a complex frequency ω0(Xs). Thus, it will be temporally growing only if ω0(Xs) is positive.
On the Nonlinear Development of Small Three‐Dimensional Perturbations in Plane Poiseuille Flow: Single‐Mode Approach
Authors: Shanthini Rajaratnam
Published: September 2015

Nonlinear aspects of developing three‐dimensional perturbations in plane Poiseuille flow have been elucidated at the primary, instead of the conventional secondary, level. Three‐dimensional perturbation velocities generate normal vorticity by stretching and tilting the basic‐flow vorticity. The amplitude of the induced normal vorticity, and hence that of the streamwise perturbation velocity, can grow temporally to significant peak values before the exponential decay predicted by the linear theory sets in. These growths, according to the linear theory, do not influence the amplitudes of the normal perturbation velocity that are monotonically decaying with time. It is shown in this study that the normal velocity continues to be oblivious to the development of induced normal vorticity, even in the nonlinear regime, if the perturbation velocities are described by waves traveling in a single oblique direction. Also, the Reynolds number dependence of the amplitude of the normal vorticity is discussed.
Asymptotic Analysis of Reacting Materials with Saturated Explosion. I. Low‐Frequency Waves
Authors: Almgren Robert F. , Majda Andrew , Rosales Rodolfo R.
Published: September 2015

We derive and study asymptotic equations describing the behavior of low‐frequency disturbances in a chemically reacting gas, whose amplitude is inversely proportional to the large activation energy of the reaction. We make an assumption of small heat release, so that fuel depletion is included in the model at leading order. For homogeneous combustion, introducing this assumption removes the thermal‐runaway singularity which always forms in models previously used. We demonstrate that for spatially varying solutions, wave‐propagation effects permit a singularity still to form, if the initial data has a step discontinuity with amplitude larger than a critical value. We suggest that appearance of this singularity in the model always signifies the birth of a detonation wave, and we describe the evolution of the wave beyond the singularity time.
Asymptotic Analysis of Reacting Materials with Saturated Explosion. II. High‐Frequency Waves
Authors: Almgren Robert F. , Majda Andrew , Rosales Rodolfo R.
Published: September 2015

The interaction of small‐scale material inhomogeneities with high‐frequency acoustic waves is known to have a prominent role in accelerating the heat‐release rate in liquid and solid explosive materials. In the present paper, simplified asymptotic equations are studied which incorporate the above interaction, and which include reactant depletion at leading order. Because fuel may be completely exhausted, singularities do not always form in the model equations; it is conjectured that when a singularity does form, the material has initiated. The detailed mechanisms by which shock formation and resonant wave interaction can either enhance or retard reaction are explored. In a realistic model for inhomogeneous condensed‐phase reaction, with pressure‐dependent reaction rate and nonconstant initial fuel concentration, initiation of the material depends on correct placement of the fuel relative to the acoustic waves.
Nonlinear Eigenvalue Problems under Strong Localized Perturbations with Applications to Chemical Reactors
Authors: Ward Michael J. , Keller Joseph B.
Published: September 2015

Nonlinear eigenvalue problems are considered for partial differential equations and boundary conditions of the for
urn:x-wiley:00222526:media:sapm19918511:sapm19918511-math-0001
The problems are perturbed by deleting a small subdomain Dε from D and imposing a boundary condition on the surface of the resulting hole, or else by changing the constant b in the boundary condition to a different constant ε−1k on a small part of ∂D. In both cases the perturbed solution is constructed for ε small by the method of matched asymptotic expansions. Particular attention is paid to the calculation of λc(ε), the critical value of λ at which the number of solutions changes by two, for example from two to none. This value of λ represents a simple fold point in the bifurcation diagram of ‖u‖ versus λ. The analysis is applied to chemical reactors with F(x,u)=eu, in which case λc is the critical value of the Frank‐Kamenetskii parameter. The asymptotic results are compared with exact and numerical results for some special cases, and fair agreement is found. Some previous asymptotic calculations are found to be incorrect.
Numerical Study of Oscillatory Solutions of the Gas‐Dynamic Equations
Authors: E Weinan , Yang Huanan
Published: September 2015

Solutions of the one‐dimensional gas‐dynamic equations with oscillatory initial data are studied numerically. A fourth‐order ENO scheme is used, together with an artificial compression method to effectively eliminate the smearing of contact discontinuities. The numerical results show that the oscillations die out in velocity and pressure, and persist in density. A system of equations governing the propagation of these oscillations is derived.
A Computational Method for Solitary Internal Waves in a Continuously Stratified Fluid
Authors: Turkington Bruce , Eydeland Alexander , Wang Sheng
Published: September 2015

An iterative algorithm is presented to compute steady, translational nonlinear waves in an incompressible fluid with a stable density stratification. The method yields solitary internal waves as exact solutions of the governing Euler equations without the restrictions on wave amplitude or density profile involved in the various approximate models of weakly nonlinear long waves. Computed examples of large‐amplitude solutions in shallow and deep fluid regimes are given. A natural variational principle derived from the structure of the underlying dynamical equations forms the basis of the method. The construction of the numerical algorithm and the analysis of the properties of solutions are both developed from the same principle.
On the Laplace Transform of Radial Functions
Authors: Trione Susana Elena
Published: September 2015

Let ϕ(t) (t ∈ Rn be a radial function. Let f(z) be the Laplace transform of ϕ(t). Then a theorem due to A. Gonzá Domínguez shows that f(z) can be expressed as a Hankel transform. I prove two representation formulae which express the Laplace transform of radial functions by means of the mth‐order derivative of the Hankel transform of order 0 and − ½.
The Signaling Problem in Nonlinear Hyperbolic Wave Theory
Authors: He Yuanping , Moodie T. B.
Published: September 2015

This work is devoted to investigating weakly nonlinear hyperbolic waves arising from the action of small‐amplitude, high‐frequency boundary disturbances. By directly introducing a nonlinear phase variable corresponding to the leading wavefront and specifying a single‐wave‐mode boundary disturbance, we are able to construct an asymptotic solution. Furthermore, our result shows that, by properly arranging the relation of small amplitude to high frequency, a systematic procedure can be provided for constructing weakly nonlinear wave solutions with interior shocks and determining the shock initiation position (and time) when there is a local linear degeneracy at the leading wavefront.
The Nonlinear Instability Characteristics of Waves in Free Shear Layers
Authors: Benney D. J. , Zhou Y. F.
Published: September 2015

A more general mean flow first harmonic theory is derived and some consequences are discussed. The theory is applied to the Bickley jet and tanh mixing layer profiles.
A Simple Algorithm for the Evaluation of a Class of Generalized Hypergeometric Series
Authors: Srivastava H. M.
Published: September 2015

Making use of a remarkably simple algorithm, the author evaluates the sum of a certain class of generalized hypergeometric series of unit argument. It is also shown how closely this main summation formula is related to other (known or new) evaluations of lower‐order hypergeometric series. Finally, the author highlights the connection of his main result with a well‐known sum which happens to have attracted considerable attention in many recent works demonstrating the usefulness of the Riemann‐Liouville operator of fractional calculus.
Inverse Spectral Transform for the Modified Kadomtsev‐Petviashvili Equation
Authors: Konopelchenko B. G. , Dubrovsky V. G.
Published: September 2015

The 2 + 1‐modified Kadomtsev‐Petviashvili (mKP) equation is studied by the inverse‐spectral‐transform method. The initial‐value problems for the mKP‐1 and mKP‐11 equations are solved by the nonlocal Riemann‐Hilbert and
techniques for initial data decaying sufficiently rapidly at infinity. The lump solutions for the mKP‐I equation are found explicitly. Wide classes of the exact solutions for the mKP equation—namely, the rational solutions, including the plane lumps for the mKP‐I equation; solutions with functional parameters; the plane solitons; and breathers—are constructed by the use of the
method based on the nonlocal
. The Miura transformation between the mKP and KP equations is discussed.
Bifurcation to Mean Flows in Convection
Authors: Lu Ya Yan
Published: September 2015

Convection with a strip plate in the middle is studied in this paper. Simultaneous instability of two convection modes of different vertical structures with a same horizontal wave number is possible in this system. It is found that the interaction of these two modes generates mean flows similar to those observed by Krishnamurti and Howard [9] in a turbulent convection experiment. Coupled nonlinear equations are derived for the amplitudes of the two modes. Traveling wave solutions and more complicated time‐dependent solutions are also found near the onset of convection.
Some Closed‐Form Solutions of Burgers’ Equation
Authors: McAsey M. , Rubel L. A.
Published: September 2015

This paper shows how to use the method of quasisolutions to construct exact solutions to Burgers’ equation. A function υ=υ(x, y) is called a quasisolution of a PDE in case there exists a function φ (not a constant function) of one variable so that u(x, y)=φ(υ(x, y)) is a solution of the equation. We prove a theorem giving necessary and sufficient conditions for υ to be a quasisolution to Burgers’ equation. A function φ can then be found explicitly so that u=φ(υ) is an actual solution. Combining this technique with similarity methods, we find a continuum of solutions to Burgers' equation.
Two‐Wave Interactions for Weakly Nonlinear Hyperbolic Waves
Authors: He Yuanping , Moodie T. Bryant
Published: September 2015

This paper is devoted to studying the weakly nonlinear interaction of two waves whose propagation is governed by n × n hyperbolic systems of conservation laws. Our method of approach involves introducing two nonlinear phase variables and carrying out a perturbation analysis. This extended version of our previous single‐wave‐mode theory [5] is then applied to the equations of gas dynamics to study interacting sound waves. Numerical results for the wave‐wave interaction are presented graphically in a set of figures.
On Torsion of Prismatic Bodies
Authors: Wan Frederic Y. M.
Published: September 2015

The method of decaying residual solution is applied to obtain an approximate interior solution for the torsion of slender prismatic elastic bodies under different end conditions. The approximate solution is generally accurate up to terms that are exponentially small in the length‐to‐cross‐sectional‐width ratio. For stress end conditions, the result is identical to the classical Saint‐Venant torsion solution. Similar types of simple solutions, not known previously, are obtained for different types of mixed end conditions. For displacement conditions at both ends, the corresponding Saint‐Venant type result requires an accurate solution of a canonical problem for a semi‐infinite prismatic body that is to be obtained once and for all. The solution of the canonical problem is elementary for a circular cross section. The approximate interior solution in that case is identical to the known exact interior solution.
On Solutions of Ordinary Differential Equations Arising from a Model of Crystal Growth
Authors: Byatt‐Smith J. G. B.
Published: September 2015

The properties of the solutions of the equations φ′″ + φ′ = −eφ and φ′″ + φ′ = 1/(2φ) in the complex plane are discussed. Both equations have Stokes phenomena as the imaginary axis is crossed. The Stokes multiplier of the subdominant term in each case is accurately calculated by transforming the equation into an integral equation by using a Laplace integral. The existence and uniqueness of the solutions of these integral equations are also discussed.
The Effect of Mean Flows on Enhanced Diffusivity in Transport by Incompressible Periodic Velocity Fields
Authors: Majda Andrew J. , McLaughlin Richard M.
Published: September 2015

Avellaneda and one of the authors ([1], [3]) have recently established that an upper bound for the enhanced diffusivity in the large scale, long time advection‐diffusion with periodic steady incompressible velocity fields has the form where Pe is the Peclet number and is the reciprocal of the Prandtl number. In this paper, flow fields with maximal and minimal enhanced diffusion are studied. Maximal enhanced diffusion requires that in some directions the enhanced diffusion tensor also has the lower bound . For minimal enhanced diffusion, the effect of the velocity field is to boost the enhanced diffusivity by a negligible amount that is bounded by a fixed constant times the bare diffusivity regardless of Peclet number. Stieltjes measure formulas are used to develop a simple, necessary, and sufficient condition for maximal enhanced diffusion and also to characterize minimal enhanced diffusion.

It is established here that constant mean flows can have a dramatic effect on maximal and minimal enhanced diffusion. In particular, for flows in two space dimensions, an explicit criterion is developed that guarantees the surprising fact that mean flows with rational ratios typically generate maximal enhanced diffusion through interaction with an arbitrary steady periodic incompressible flow with zero mean. In contrast, a simple criterion for flows without stagnation points is developed here that guarantees that the effect of mean flows with irrational ratios on advection‐diffusion in two dimensions creates minimal enhanced diffusion. The theory for the phenomena mentioned above is elementary yet mathematically rigorous. Examples are emphasized throughout this work including a discussion of enhanced diffusivity for a class of flows recently introduced by Childress and Soward [8]. The theory developed here is also supplemented by a series of numerical experiments that both verify the theoretical predictions and display interesting crossover phenomena at rather large but finite Peclet numbers.
Explicit Solutions of the Two‐Dimensional Navier Stokes Equations
Authors: Ranger K. B.
Published: September 2015

Explicit solutions are found for the stream function satisfying the Navier Stokes equations representing the steady two‐dimensional motion of a viscous incompressible liquid. The solutions contain two arbitrary analytic functions and in general are confined to certain regions of the x, y plane.
The Mixed Volumes and Geissinger Multiplications of Convex Sets
Authors: Chen Beifang
Published: September 2015

This paper introduces Geissinger multiplication on the vector space generated by indicator functions of closed convex sets. Minkowski's mixed volume for compact convex sets is naturally represented in terms of the volume of the Geissinger multiplication of their indicator functions. Some properties of mixed volumes and new results are obtained by this representation, including a polynomial identity.
Metastable Patterns, Layer Collapses, and Coarsening for a One‐Dimensional Ginzburg‐Landau Equation
Authors: Ward Michael J.
Published: September 2015

The internal layer motion associated with the Ginzburg‐Landau equation urn:x-wiley:00222526:media:sapm199491151:sapm199491151-math-0001 is analyzed, in the limit , for various boundary conditions on x = ±1. The nonlinearity Q(u) results either from a double‐well potential or a periodic potential, each having wells of equal depth. Using a systematic asymptotic method, some previous work in deriving equations of motion for the internal layers corresponding to metastable patterns is extended. The effect of the various types of boundary conditions and nonlinearities will be highlighted. A dynamical rescaling method is used to numerically integrate these equations of motion. Using formal asymptotic methods, certain canonical problems describing layer collapse events are formulated and solved numerically. A hybrid asymptotic‐numerical method, which incorporates these layer collapse events, is used to give a complete quantitative description of the coarsening process associated with the Ginzburg‐Landau equation. For the Neumann problem with a double‐well potential, the qualitative description of the coarsening process given in Carr and Pego [4] will be confirmed quantitatively. In other cases, such as for a periodic potential with Dirichlet boundary conditions, it will be shown that, through layer collapse events, a metastable pattern can tend to a stable equilibrium solution with an internal layer structure.
The Construction of Solutions to the Two‐Dimensional Navier Stokes Equations
Authors: Ranger K. B.
Published: September 2015

A method is described for the determination of general solutions to the equations governing the steady two‐dimensional motion of a viscous liquid. Solutions are found both implicitly defined and explicitly containing two arbitrary complex functions. In general the solutions are confined to specific regions of the complex plane depending on the arbitrary functions.
On the Equation of Motion of a Thin Layer of Uniform Vorticity
Authors: Dhanak M. R.
Published: September 2015

A higher order extension to Moore's equation governing the evolution of a thin layer of uniform vorticity in two dimensions is obtained. The equation, in fact, governs the motion of the center line of the layer and is valid for consideration of motion whereby the layer thickness is uniformly small compared with the local radius of curvature of the center line. It extends Birkoff's equation for a vortex sheet. The equation is used to examine the growth of disturbances on a straight, steady layer of uniform vorticity. The growth rate for long waves is in good agreement with the exact result of Rayleigh, as required. Further, the growth of waves with length in a certain range is shown to be suppressed by making this approximate allowance for finite thickness. However, it is found that very short waves, which are quite outside the range of validity of the equation but which are likely to be excited in a numerical integration of the equation, are spuriously amplified as in the case of Moore's equation. Thus, numerical integration of the equation will require use of smoothing techniques to suppress this spurious growth of short wave disturbances.
Slow Evolution in Perturbed Hamiltonian Systems
Authors: Lebovitz Norman R. , Neishtadt Anatoly
Published: September 2015

For parametrized Hamiltonian systems with an arbitrary, finite number of degrees of freedom, it is shown that secularly stable families of equilibrium solutions represent approximate trajectories for small (not necessarily Hamiltonian) perturbations of the original system. This basic result is further generalized to certain conservative, but not necessarily Hamiltonian, systems of differential equations. It generalizes to the conservative case a theorem due, in the dissipative case, to Tikhonov, to Gradshtein, and to Levin and Levinson. It justifies the use of physically motivated approximation procedures without invoking the method of averaging and without requiring nonresonance conditions or the integrability of the unperturbed Hamiltonian.
Integrable Equations with a Forcing of a Distribution Type
Authors: Fokas A. S. , Its A. R. , Novokshenov V. Yu.
Published: September 2015

We consider the nonlinear Schrodinger equation in the variable q(x,t) with the forcing iu(t)δ(x)+iu1(t)δ′(x). We assume that q(x,0),u(t),u1(t) are given and that these functions as well as their first two derivatives belong to L1∩L2(ℝ+). We show that the solution of this problem can be reduced to solving two Riemann—Hilbert (RH) problems in the complex k‐plane with jumps on Im(k2) = 0. Each RH problem is equivalent to a linear integral equation that has a unique global solution. These linear integral equations are uniquely defined in terms of certain functions (scattering data) b(k), c(k), d(k), and f(k). The functions b(k) and d(k) can be effectively computed in terms of q(x,0). However, although the analytic properties of c(k) and f(k) are completely determined, the relationship between c(k), f(k), q(x,0), u(t), and u1(t) is highly nonlinear. In spite of this difficulty, we can give an effective description of the asymptotic behavior of q(x,t) for large t. In particular, we show that as t → ∞, solitons are generated moving away from the origin. It is important to emphasize that the analysis of this problem, in addition to techniques of exact integrability, requires the essential use of general PDE techniques.
Euler‐Maclaurin Summation of Trigonometric Fourier Series
Authors: MacDonald D. A.
Published: September 2015

Trigonometric Fourier series are, in general, difficult to sum to high accuracy. An example is given by the series
urn:x-wiley:00222526:media:sapm1994923213:sapm1994923213-math-0001
in which α and β(>0) are rational numbers satisfying 0<β/α≤1, where λ is an independent variable and j is a positive integer or zero. This paper presents a method for the efficient evaluation of the sum of such series. Fourier series which are the real or the imaginary part of , but which are not explicitly expressible as simple polynomials in λ, are obtained as the sum of a logarithic term and an infinite series in powers of λ, whose expansion is valid when 0<λ≤(2π/α) and is exact. When the Fourier series is expressible as a polynomial in λ, the method identifies that polynomial.
Representations of Sheffer Polynomials
Authors: Di Bucchianico A.
Published: September 2015

In this paper we investigate which Sheffer polynomials can be represented as moments of convolution semigroups of probability measures. We also obtain general integral representations for shift‐invariant operators and for umbral operators. As a corollary, we obtain new proofs for representation theorems for Sheffer polynomials due to Sheffer and Thorne.
Surjections, Differences, and Binomial Lattices
Authors: Wagner Carl G.
Published: September 2015

The elementary problem of counting surjections from an n‐set to a k‐set is generalized to that of enumerating solutions of a1 ∨ ⋯ ∨ an = y, with each ai an atom of the k‐interval [x, y] in a binomial lattice L. When L is modular, the number of such solutions is representable as a q‐difference and satisfies a simple recurrence.
The Unconditional Sojourn Time Distribution in the Queue GI/M/1‐K with Processor Sharing Service
Authors: Yang Yongzhi , Knessl Charles
Published: September 2015

We consider a processor shared GI/M/1 queue which can accommodate a finite number K of customers. Using singular perturbation techniques, we construct asymptotic expansions for the distribution of a tagged customer's sojourn time. We assume that K is large and treat several different cases of the model parameters.
The Coupling of Gravity Waves and Convection: Amplitude Equations and Planform Selection
Authors: Pavithran S. , Redekopp L. G.
Published: September 2015

Convective motion in a layer of fluid heated from below is considered where the boundaries are stress free and the upper surface supports interfacial gravity waves. Inviscid, immiscible, constant density, ambient fluid is separated from the convecting layer below by a stable density jump. An important parameter in the problem is δ representing the ratio of the interfacial density jump to the density change across the convecting layer. Amplitude equations are derived describing convective motion in the plane and a planform selection analysis performed. It is demonstrated that the breaking of the translational and Galilean invariance of the problem allows a strong coupling between a large‐scale interfacial mode and convection. The resulting phase dynamics is third order in time.
Exact Free Surface Flows for Shallow Water Equations I: The Incompressible Case
Authors: Sachdev P. L. , Vaganan B. Mayil
Published: September 2015

A comprehensive exact treatment of free surface flows governed by shallow water equations (in sigma variables) is given. Several new families of exact solutions of the governing PDEs are found and are shown to embed the well‐known self‐similar or traveling wave solutions which themselves are governed by reduced ODEs. The classes of solutions found here are explicit in contrast to those found earlier in an implicit form. The height of the free surface for each family of solutions is found explicitly. For the traveling or simple wave, the free surface is governed by a nonlinear wave equation, but is arbitrary otherwise. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed; in another case, the free surface is a horizontal plane while the flow underneath is a sine wave. The existence of simple waves on shear flows is analytically proved. The interaction of large amplitude progressive waves with shear flow is also studied.
The Nonlinear Critical‐Wall Layer in a Parallel Shear Flow
Authors: Finlay W. H. , Liu Chonghui , Maslowe S. A.
Published: September 2015

The nonlinear critical layer theory is developed for the case where the critical point is close enough to a solid boundary so that the critical layer and viscous wall layers merge. It is found that the flow structure differs considerably from the symmetric “eat's eye” pattern obtained by Benney and Bergeron [1] and Haberman [2]. One of the new features is that higher harmonics generated by the critical layer are in some cases induced in the outer flow at the same order as the basic disturbance. As a consequence, the lowest‐order critical layer problem must be solved numerically. In the inviscid limit, on the other hand, a closed‐form solution is obtained. It has continuous vorticity and is compared with the solutions found by Bergeron [3], which contain discontinuities in vorticity across closed streamlines.
Exact Free Surface Flows for Shallow Water Equations II: The Compressible Case
Authors: Sachdev P. L. , Vaganan B. Mayil
Published: September 2015

Exact free surface flows with shear in a compressible barotropic medium are found, extending the authors’ earlier work for the incompressible medium. The barotropic medium is of finite extent in the vertical direction, while it is infinite in the horizontal direction. The “shallow water” equations for a compressible barotropic medium, subject to boundary conditions at the free surface and at the bottom, are solved in terms of double psi‐series. Simple wave and time‐dependent solutions are found; for the former the free surface is of arbitrary shape while for the latter it is a damping traveling wave in the horizontal direction. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed. In the case of an isothermal medium, when γ = 1, we again find simple wave and time‐dependent solutions.
Weakly Nonlocal Solitary Waves in a Singularly Perturbed Nonlinear Schrödinger Equation
Authors: Grimshaw R.
Published: September 2015

We consider the nonlinear Schrödinger equation perturbed by the addition of a third‐derivative term whose coefficient constitutes a small parameter. It is known from the work of Wai et al. [1] that this singular perturbation causes the solitary wave solution of the nonlinear Schrödinger equation to become nonlocal by the radiation of small‐amplitude oscillatory waves. The calculation of the amplitude of these oscillatory waves requires the techniques of exponential asymptotics. This problem is re‐examined here and the amplitude of the oscillatory waves calculated using the method of Borel summation. The results of Wai et al. [1] are modified and extended.
Cyclic Tableaux and Symmetric Functions
Authors: Chen William Y. C. , Lih Ko‐Wei , Yeh Yeong‐Nan
Published: September 2015

We introduce the notion of cyclic tableaux and develop involutions for Waring's formulas expressing the power sum symmetric function pn in terms of the elementary symmetric function en and the homogeneous symmetric function hn. The coefficients appearing in Waring's formulas are shown to be a cyclic analog of the multinomial coefficients, a fact that seems to have been neglected before. Our involutions also spell out the duality between these two forms of Waring's formulas, which turns out to be exactly the “duality between sets and multisets.” We also present an involution for permutations in cycle notation, leading to probably the simplest combinatorial interpretation of the Möbius function of the partition lattice and a purely combinatorial treatment of the fundamental theorem on symmetric functions. This paper is motivated by Chebyshev polynomials in connection with Waring's formula in two variables.
A Uniformly‐Valid Asymptotic Solution to a Matrix System of Ordinary Differential Equations and a Proof of its Validity
Authors: Woodruff S. L.
Published: September 2015

A new multiple‐scale perturbation technique is employed to find the approximate solution to a fairly general matrix system of ordinary differential equations. This system includes a linear part given by a slowly‐varying matrix and a small nonlinear part. The general proof of the method given in previous work is used to show rigorously that the present approximate solution is indeed asymptotic to the solution of the differential system. Some typical special cases of the general solution are also given.
There Are a Lot of Magic Squares!
Authors: Böna Miklös
Published: September 2015

We prove a surprisingly high lower bound for the number of magic squares.
Solutions of the Stokes, Oseen, and Navier‐Stokes Equations in Three Dimensions
Authors: Ranger K. B.
Published: September 2015

A constructive analytical procedure is described for generating a class of solutions pertaining to the fluid velocity field satisfying the three‐dimensional Navier‐Stokes equations. The arbitrary functions occurring in the solution are related to those that arise in the general solution of the Stokes and Oseen equations.
A Nim‐like Game and Dynamic Recurrence Relations
Authors: Gan Boon‐Beng , Yeh Yeong‐Nan
Published: September 2015

The nim‐like game 〈n, f; X, Y〉 is defined by an integer n ≥ 2 a constraint function f, and two players and X and Y. Players X and Y alternate taking coins from a pile of n coins, with X taking the first turn. The winner is the one who takes the last coin. On the kth turn, a player may remove tk coins, where 1 ≤ t1 ≤ n − 1 and 1 ≤ tk ≤ max{1, f(tk−1) for k > 1. Let the set Sf = {1} ∪ {n| there is a winning strategy for Y in the nim‐like game 〈n, f; X, Y〉}. In this paper, an algorithm is provided to construct the set Sf = {a1, a2,…} in an increasing sequence when the function f(x) is monotonic. We show that if the function f(x) is linear, then there exist integers n0 and m such that an+1 = an + an−m for n > n0 and we give upper and lower bounds for m (dependent on f. A duality is established between the asymptotic order of the sequence of elements in Sf and the degree of the function f(x). A necessary and sufficient condition for the sequence {a0, a1, a2,…} of elements in Sf to satisfy a regular recurrence relation is described as well.
The Stability and Nonlinear Evolution of Edge Waves
Authors: Yang Jianke
Published: September 2015

The stability of steady‐state standing edge waves to large‐scale disturbances is studied. Both stable and unstable edge‐wave modes are found. These modes strongly affect the dynamics of edge‐wave evolution.
Divided Differences and Linearly Recursive Sequences
Authors: Verde‐Star Luis
Published: September 2015

We show that the theory of divided differences is a natural tool for the study of linearly recurrent sequences. The divided differences functional associated with a monic polynomial w on degree n + 1 yields a vector space isomorphism between the space of polynomials of degree at most equal to n and the space of linearly recurrent sequences f that satisfy the difference equation w(E)f=0 where E is the usual shift operator. Using such isomorphisms, we can translate problems about recurrent sequences into simple problems about polynomials. We present here a new approach to the theory of divided differences, using only generating functions and elementary linear algebra, which clarifies the connections of divided differences with rational functions, polynomial interpolation, residues, and partial fractions decompositions.
Nonlinear Stability of Nonstationary Cross‐Flow Vortices in Compressible Boundary Layers
Authors: Gajjar J. S. B.
Published: September 2015

The nonlinear evolution of long‐wavelength non stationary cross‐flow vortices in a compressible boundary layer is investigated; the work extends that of Gajjar [1] to flows involving multiple critical layers. The basic flow profile considered in this paper is that appropriate for a fully three‐dimensional boundary layer with O(1) Mach number and with wall heating or cooling. The governing equations for the evolution of the cross‐flow vortex are obtained, and some special cases are discussed. One special case includes linear theory, where exact analytic expressions for the growth rate of the vortices are obtained. Another special case is a generalization of the Bassom and Gajjar [2] results for neutral waves to compressible flows. The viscous correction to the growth rate is derived, and it is shown how the unsteady nonlinear critical layer structure merges with that for a Haberman type of viscous critical layer.
The Modulation Equations for the Asymptotic Suction Velocity Profile and the Ekman Boundary Layer
Authors: Dwarka Pankaj R. , Herron Isom H.
Published: September 2015

In this article two types of flows are considered, the asymptotic suction velocity profile, which is a nearly parallel flow, and the Ekman boundary layer, which is a nonparallel flow. The modified Orr‐Sommerfeld equation for the asymptotic suction velocity profile, which is the linearized stability equation for this flow, is analyzed and it is shown to have finitely many eigenvalues. In addition, the Ekman boundary layer is considered and the modulation equation for this nonparallel flow is derived for the first time.
The Interaction of Shocks with Dispersive Waves I. Weak Coupling Limit
Authors: Axel Ralph M. , Newton Paul K.
Published: September 2015

We introduce and analyze a model for the interaction of shocks with a dispersive wave envelope. The model mimicks the Zakharov system from weak plasma turbulence theory but replaces the linear wave equation in that system by a nonlinear wave equation allowing the formation of shocks. This paper considers a weak coupling in which the nonlinear wave evolves independently but appears as the potential in the time‐dependent Schrodinger equation governing the dispersive wave. We first solve the Riemann problem for the system by constructing solutions to the Schrodinger equation that are steady in a frame of reference moving with the shock. Then we add a viscous diffusion term to the shock equation and by explicitly constructing asymptotic expansions in the (small) diffusion coefficient, we show that these solutions are zero diffusion limits of the regularized problem. The expansions are unusual in that it is necessary to keep track of exponentially small terms to obtain algebraically small terms. The expansions are compared to numerical solutions. We then construct a family of time‐dependent solutions in the case that the initial data for the nonlinear wave equation evolves to a shock as t → t* < ∞. We prove that the shock formation drives a finite time blow‐up in the phase gradient of the dispersive wave. While the shock develops algebraically in time, the phase gradient blows up logarithmically in time. We construct several explicit time‐dependent solutions to the system, including ones that: (a) evolve to the steady states previously constructed, (b) evolve to steady states with phase discontinuities (which we call phase kinked steady states), (c) do not evolve to steady states.
A Constructive Analytical Approach to the Calculation of Flows Representing the Steady Motion of a Viscous Incompressible Liquid
Authors: Ranger K. B.
Published: September 2015

A constructive analytical technique is described for the calculation of the fluid velocity components satisfying the Navier‐Stokes equations representing the steady motion of a viscous, incompressible liquid. These components are found in terms of the derivatives of two functions which in turn satisfy linear partial differential equations for which known solutions of general type are readily available.
Asymptotic Behavior of the Pollaczek Polynomials and Their Zeros
Authors: Rui Bo , Wong R.
Published: September 2015

In 1954, A. Novikoff studied the asymptotic behavior of the Pollaczek polynomials Pn(x; a, b) when , where t > 0 is fixed. He divided the positive t‐axis into two regions, 0 < t < (a + b)1/2 and t > (a + b)1/2, and derived an asymptotic formula in each of the two regions. Furthermore, he found an asymptotic formula for the zeros of these polynomials. Recently M. E. H. Ismail (1994) reconsidered this problem in an attempt to prove a conjecture of R. A. Askey and obtained a two‐term expansion for these zeros. Here we derive an infinite asymptotic expansion for , which holds uniformly for 0 < ε ≤ t ≤ M < ∞, and show that Ismail's result is incorrect.
Differential Relations and Recurrence Formulas for Representations of Lie Groups
Authors: Feinsiluer P. , Schott R.
Published: September 2015

General recurrence formulas for matrix elements of representations of Lie groups on enveloping algebras are given. In particular, we show the connection with the exponential of the adjoint representation which is an important feature of the construction. The methods are applicable for groups of arbitrary dimension and have been implemented using MAPLE. A three‐step nilpotent group and SU(3) provide examples of the theory.
On Wave Interactions II. Explosive Resonant Triads in Fluid Mechanics
Authors: Mahoney D. J.
Published: September 2015

This paper considers the occurrence of explosive resonant triads in fluid mechanics. These are weakly nonlinear waves whose amplitudes become unbounded in finite time. Previous work is expanded to include interfacial flow systems with continuously varying basic velocities and densities. The first paper in this series [10] discussed the surprisingly strong singular nature of explosive triads. Many of the problems to be studied here will be found to have additional singularities, and the techniques for analyzing these difficulties will be developed. This will involve the concept of a critical layer in a fluid, a level at which a wave phase speed equals the unperturbed fluid velocity in the direction of propagation. Examples of such waves in this context are presented.
On Wave Interactions III. Miscellaneous Results
Authors: Mahoney D. J.
Published: September 2015

In this paper, a collection of results pertaining to resonant triads in fluid flow systems is presented. The main topic is the extension of the analysis by Mahoney for wind‐driven shear flows to general, layered flows, that is to say, flows which have piecewise continuous velocity and density profiles. It is seen how the results for calculating the triad interaction coefficients for two‐layer flows generalize and carry over for multilayer flows.
Nonlinear Interaction between a Pair of Oblique Modes in a Supersonic Mixing Layer: Long‐Wave Limit
Authors: Gartside James , Balsa Thomas F.
Published: September 2015

The nonlinear interaction between a pair of symmetric, oblique, and spatial instability modes is studied in the long‐wave limit using asymptotic methods. The base flow is taken to be a supersonic mixing layer whose Mach number is such that the corresponding vortex sheet is marginally stable according to Miles’ criterion. It is shown that the amplitude of the mode obeys a nonlinear integrodifferential equation. Numerical solutions of this equation show that, when the obliqueness angle is less than π/4, the effect of the nonlinearity is to enhance the growth rate of the instability. The solution terminates in a singularity at a finite streamwise location. This result is reminiscent of that obtained in the vicinity of the neutral point by other authors in several different types of flows. On the other hand, when the obliqueness angle is more than π/4, the streamwise development of the amplitude is characterized by a series of modulations. This arises from the fact that the nonlinear term in the amplitude equation may be either stabilizing or destabilizing, depending on the value of the streamwise coordinate. However, even in this case the amplitude of the disturbance increases, although not as rapidly as in the case for which the angle is less than π/4. Quite generally then, the nonlinear interaction between two oblique modes in a supersonic mixing layer enhances the growth of the disturbance.