The large deformation of nonlinearly elastic rings in a two-dimensional compressible flow

The nonlinear nonlocal system of the equilibrium equations ofan elastic ring under the action of an external two-dimensionaluniformly subsonic potential barotropic steady-state gas flowis considered. The configurations of the elastic ring are identifiedby a pair of functions (, ). The simple curve represents theshape of the ring and the real-valued function identifies theorientation of the material sections of the ring. The pressurefield on the ring depends nonlocally on , and on two parametersU and P which represent the pressure and the velocity at infinity.The system is shown to be equivalent to a fixed-point problem,which is then treated with continuation methods. It is shownthat the solution branch ensuing from certain equilibrium states((₀, ₀), 0, P₀) in the solution-parameter space of ((₀, ₀),0, P₀) either approaches the boundary of the admissible ((,), U,p)'s in a well-defined sense, or is unbounded, or is homotopicallynontrivial in the sense that there exists a continuous map from the branch to a two-dimensional sphere which is not homotopicin the sphere to a constant, while restricted to the branchminus ((₀, ₀), 0, P₀) is homotopic to a constant in the sphere.Furthermore, by fixing the pressure parameter at P₀ and by consideringthe one-parameter problem in ((, ), U), the following holds.Every hyperplane in the solution-parameter space of the ((,), U)'s which contains the equilibrium state ((₀, ₀), 0) anddoes not include a welldetermined one-dimensional subspace intersectsthe solution branch above at a point different from ((₀, ₀),0).     

A Criticality Criterion for Thermal Explosions in Well-stirred Systems with Reactant Consumption

The autonomous differential equations for the temperature andreactant consumption in a first-order well-stirred exothermicreaction are considered. An examination of the phase-plane solutionsallows the qualitative behaviour of the Semenov number as afunction of maximum temperature rise ^* to be established. Inthe limit of infinite adiabatic temperature rise (B) and zeroactivation energy parameter ( = 0), the relationship between and stationary temperature _s is known to be e_¹ = _s. Criticalityarises at the maximum of (_s) and leads to the critical Semenovvalues (_s)_cr = 1, _cr = e^–1. For sufficiently large B,it is shown that the (^*) curve has a bifurcation at ^* = 1, withthe upper branch monotonically increasing and the lower branchmonotonically decreasing for ^* > 1. In the limit B thesebecome respectively the straight line = e^–1, _s 1 andthe unstable branch of = _se^–1, _s 1 and the unstablebranch of = _s e^–. Criticality for finite B is definedas occurring at the bifurcation, namely ^*_cr = 1, with _cr(B)the value of at this point. Values of these Semenoy numbersare obtainable from the numerical calculations of Boddingtonet al. [Proc. R. Soc. Lond. (1983), 390, 13–30]. The newcriterion is applied to an approximate phase-plane solution.The corresponding critical parameter is found to be _cr = e^–1[1+B^–(2–e^–1)+O(B^–1)].     

Elliptic Problems Involving Phase Boundaries Satisfying a Curvature Condition

Two theorems related to equilibrium free-boundary problems arepresented. One arises as a time-independent solution to thephase-field equations. The other is the relevant time-independentproblem for the Stefan model, modified for the surface tensioneffect. It also serves as a preliminary result for the phase-fieldformulation. Under appropriate conditions, we prove that, givenan appropriate positive constant and a smooth function u: R;,where is an annular domain in R², there exists a curve suchthat u(x)=—K(x) for all x , where K is the curvature.Using this result, we prove the existence of solutions to O=²+ ?(—³) + 2u that have a transition layer behaviour (from=—1 to =+1) for small and make the transition on thecurve . This proves there exist solutions to the phase fieldmodel that satisfy a Gibbs-Thompson relation.     

The Decay of Eddy-currents in Thin Sheets and Related Water-wave Problems

The decay of the eddy-currents that are induced in a thin, uniform,imperfectly-conducting sheet by switching off the source ofan external magnetic field is investigated. For the two-dimensionalproblem of an infinite strip the (non-dimensional) decay constants_n and eddy-current distributions i_n(x) are the eigenvalues andeigenfunctions of the integral equation with the constraint. For the circular disc the corresponding equation is where and K and E are complete elliptic integrals. For both problemsthe initial eddy-currents have inverse-square-root singularitiesat the edges but during their decay the eddy currents are finiteat the edges and the normal magnetic fields have logarithmicsingularities there. Numerical results are given for variousinitial-value problems. The eddy current problems are closely related to water-waveproblems in which there is a strip-shaped or circular aperturein a horizontal rigid dock. If _n and _n are the decay constantsand magnetic scalar potentials for the strip and _n and _n theangular frequencies and velocity potentials for the normal modesin the strip-shaped aperture, then _n =_n² and _n and _n are thereal and imaginary parts respectively of a holomorphic function.The velocities in the normal modes are deduced from the solutionof the eddy-current problem and are found to agree with resultsgiven in Miles (1972). For circular geometries the eigenvaluesand eigenfunctions of the axisymmetric eddy-current problemare the same as those of the water-wave problem that has angularvariation eⁱ; where (, , z) are cylindrical polar co-ordinateslocated at the centre of the basin.     

Optimal Parameter Choice in the {theta}-Method for Parabolic Equations

The general first-order method, known as the -method, is appliedto the semi-discrete form of a parabolic equation. It is shownthat to every required local accuracy there corresponds a valueof the parameter that is optimal in the sense of allowing thelargest step for which the error remains bounded below . Anasymptotic formula for in terms of is obtained, showing thatthe maximum step-size for the optimal -method is more than twiceas large as that for the Crank-Nicolson method. A numericalexample is given, showing good agreement between theory andpractice.     

The Model Plasma Problem: Existence of a Solution Branch

We are interested in the model plasma problem –u = u⁺in ,u = –d on , _au⁺ dx=j where is a bounded domain in with boundary ; here, j isa given positive number, the function u and the positive number are the unknowns of the problem, and d is a real parameter.Using a variant of the implicit function theorem, we can provethe existence of a global solution branch parametrized by d.The method has the advantage that it can be used for analysingthe approximation of the above problem by a finite-element method.     

An analogue of Berry's conjecture for the phase in fractal obstacle scattering

In this paper, the authors consider the high-frequency asymptoticsof the phase s() of acoustic waves scattered by an obstacleRⁿ with fractal boundary. Under certain conditions, it is provedthat if is –Minkowski measurable with –Minkowskimeasure µ then there exists a positive constant C_n, dependingonlyon n and such that where     

The Critical Conditions in a Reactive Slab with Slowly Varying Surface Temperature

Liouville's non-linear partial differential equation is consideredfor an infinite rectangular strip domain with a slowly varyingboundary condition. The equation describes a layer of chemicallyreactive material under conditions where the resistance to surfaceheat transfer is negligible and the ambient temperature variesslowly along the surface. Symmetrical heating by a zero orderexothermic reaction is assumed. If is a small dimensionlesstemperature difference between regions where the surface temperatureis effectively constant, a perturbation series solution in may be determined provided the Frank-Kamenetskii parameter satisfies _c(). It is shown that a plausible value for thecritical parameter is _c() = _c(0) e^–e,where _c(0) = 0.878.The corresponding critical temperature distribution is shownto have a dependence on different from that for subcriticalcases.     

Exact Remainders for Asymptotic Expansions of Fractional Integrals

This is a continuation of work begun in an earlier paper inwhich we used the theory of distributions to derive explicitexpressions for the remainder terms associated with the asymptoticexpansions of the Stieltjes transform. In this paper similarresults are obtained for the fractional integral of order definedby 1f(x)=1/f()^x_o(x-)^x-1 f(t)dt, >. Heref(t) is a locally integrable function on [0, ) and satisfies f(t) a_st-5–0(ó >0), s=0 as      

Computer extended series solution for unsteady flow in a contracting or expanding pipe

Unsteady flow in a semi-infinite contracting or expanding pipeis reinvestigated using long series analysis. The proposed seriesmethod is useful in analysing the problem for a moderately largeconstant ( = aa/, where a = a(t), the radius of the pipe isa function of time, a(t) is the velocity of the wall, and iskinematic viscosity). For positive values of (expansion ofthe pipe) accuracy of the series representing shear stress andpressure gradient is increased from = 2.89 to = 6.0 by extractingthe singularity followed by completion of the series. For negativevalues of (contraction of the pipe), we revert the series whichresults into the increase of the region of validity of the transposedseries from = -25.0 to = -2.89. Later we use Padé approximantsfor summing them. Also, the asymptotic solution for large valuesof is obtained and it agrees closely with pure numerical valuesof shear stress at the wall and pressure gradient.     

On hearing the shape of a bounded domain with Robin boundary conditions

The asymptotic expansions of the trace of the heat kernel (t)= [sum ]_j=1 exp(-t_j) for small positive t, where {_j} _j=1 arethe eigenvalues of the negative Laplacian -_n = -[sum ]ⁿ_k=1 (/x^k)²in Rⁿ (n = 2 or 3), are studied for a general multiply connectedbounded domain which is surrounded by simply connected boundeddomains _i with smooth boundaries _i (i = 1,...,m), where smoothfunctions Y_i (i = 1,...,m) are assuming the Robin boundary conditions(n_i + Y_i) = 0 on _i. Here /n_i denote differentiations along theinward-pointing normals to _i (i = 1,...,m). Some applicationsof an ideal gas enclosed in the multiply connected bounded containerwith Neumann or Robin boundary conditions are given.     

On a Penalty-Perturbation Theory for Plate Problems

A penalty-perturbation method previously proposed by Westbrook(J. Inst. Maths Applics (1974) 14, 79–82) for the solutionsof static bending problems for elastic plates is analysed here.The method replaces the single fourth-order biharmonic equationby a system of three second-order equations which is "singularly"perturbed with respect to a small penalty parameter . The existenceof solutions of the perturbed problem for each > 0 is establishedand the behaviour of these solutions as 0 0 is studied. Inparticular, the results show that while these solutions arecontinuous in at = 0, analyticity in at = 0 is lost exceptin special cases.     

The onset of thermal runaway in partially insulated or cooled reactors

The conditions for the onset of thermal runaway in partiallyinsulated or cooled reactors are investigated. The temperaturein the reactor is taken to satisfy a nonlinear elliptic equationand the reaction is modelled by an Arrhenius heat generationterm with finite activation energy. To determine the onset ofthermal runaway, the method of matched asymptotic expansionsis used to derive expressions for the critical Frank-Kamenetskiiparameter _c() for reactors containing either a small coolingrod or having a small cooling patch on their boundary. The theoryused to determine _c() is an extension of the results of Wardand Keller (1991). These previous results of Ward and Kellerare also extended to the case of finite activation energiesby using a numerical scheme to evaluate the coefficients inthe asymptotic results for _c(). In some special cases, the asymptoticexpansions for _c() are compared with numerical results for _c(),and clear agreement is found.     

Numerical Evaluation of Fourier Integrals

A method is developed for evaluating Fourier integrals of theform A() = ¹_–1f(x) e^fax dx, 0. The method consists of expanding the function f in a seriesof Chebyshev polynomials and expressing the integral A() asa series of the Bessel functionsJ_r+(), r= 0, 1, 2,.... A partialsum A_N() of the series provides an approximant to A(). The principalfeature of the method is that one set of N+1 evaluations off(x) suffices for the calculation of A_N() for all , and alsothe truncation error A()–A_N() is essentially independentof . Numerical tests show that the method is accurate, economicaland reliable. An application to the inversion of Fourier andLaplace transforms is briefly described.     

A Criticality Criterion for Thermal Explosions in Well-stirred Systems with Reactant Consumption: Addendum

OUR attention has been drawn to the fact that the criticalitycondition ^* = 1 of Adler & Herbert (1985), for well-stirredreactive systems, has been derived previously (Gray, 1975).It arises from an examination of trajectories in the temperaturereactant phase plane when a tube stability argument is employed.Using the criterion ^* = 1, values of the critical Semenov numberhave also been obtained numerically (Gray & Jones, 1981). Our work on criticality for systems with reactant consumptioncame about by trying to reconcile the inflection criterion ofBoddington et al. (1983), for finite B, with the correspondingmaximum criterion in the limit B . Our contribution was to showthat the Semenov number versus maximum temperature ^* curvehas a bifurcation at ^* = 1 for all B. Both Gray's work and ourown are attempting to resolve the same problem; the approachesare, however, quite distinct and complement each other     

Numerical studies of the fourth Painlev equation

In this paper the authors investigate numerically solutionsof a special case of the fourth Painlev equation given by with a parameter, satisfying theboundary condition Equation (1a) arises asa symmetry reduction of the derivative nonlinear Schrdinger(DNLS) equation, which is a completely integrable soliton equationsolvable by inverse scattering techniques. Previous resultsconcerned with solutions of (1) are largely restricted to thecase when is an integer and very little has been proved when is a noninteger. Here a numerical approach to describing solutionsof (1) for noninteger is adopted, and information is obtainedcharacterizing connection formulae which describe how the asymptoticbehaviours of solutions as + relate to those as –.     

Exact Solutions of the Space Charge Equation Using the Hodograph Method

In this paper we show that a hodograph method first introducedby Budd & Wheeler (1987) to develop a new numerical algorithmto solve the space charge equation •()=0 can be employedto derive all the known exact solutions, which have recentlybeen found by Smith (1988). All these solutions are shown tobe separable solutions of the equations resulting from the applicationof the hodograph transformation. We also briefly describe thisnew numerical method and use one of the exact solutions to demonstrateits accuracy.     

Asymptotic Convergences in a One-Phase Continuous-Casting Stefan Problem with High Extraction Velocity

Asymptotic convergence for variational solutions and for thefree boundaries in a one-phase continuous-casting problem ofthe Stefan type are derived when the Péclet number ß+. This corresponds, in particular, to a model with high extractionvelocity, where an ultraparabolic problem is obtained from aparabolic one and, in the stabilized case (i.e. when t +), aparabolic problem results from an elliptic one. The asymptoticstabilization in time as t + is also discussed in the limitß = +.     

Existence of Solutions of a Two-Point Free-Boundary Problem Arising in the Theory of Porous Medium Combustion

The existence of solutions of a two-point free-boundary problemarising from the theory of travelling combustion waves in aporous medium is examined. The problem comprises a third-ordernonlinear ordinary differential equation posed on an unknowninterval of finite length; four boundary conditions are given,two at either end of the interval. The equations possess a trivialsolution for all values of the bifurcation parameter . A shootingtechnique is employed to prove the existence of a nontrivialsolution for 0 < < _c and nonexistence theorems are provedfor (0, _c).     

A Mathematical Theory of Full-wave Rectification

The results of a systematic asymptotic analysis of two basicfull-wave rectifier circuits are presented in this paper. Thephysical circuits are modelled by two systems of nonlinear differentialequations which are both singularly perturbed in the limit RC.Here is the angular frequency of the source, R is the resistanceof the load, and C is the capacitance of the circuit. The methodof multiple scales is used to construct the asymptotic behaviourof the solutions as RC. The results contain transient, ripple,and steady-state information. The latter two components simplifyand yield the standard results when the diodes' forward impedanceis small. The analysis of more complicated full-wave rectifiersfollows from the procedure described.