Numerical solutions of a model for the propagation of a surface-catalysed flame in a tube

Numerical simulations of a surface-catalysed flame in a tubeare performed, corresponding to an experiment where a premixedfuel is fed into a tube whose inner surface is coated with acatalyst. In these experiments, subsequent to ignition, a reactionwave can be seen as a red-hot region which propagates back alongthe tube towards the inlet, and is due to low temperature combustionoccurring only on the inner surface of the tube where the catalystis present. The solutions of a mathematical model for this behaviourshow that initial-value problems do indeed result in such steadilypropagating waves. The numerically obtained wave speeds andsteady solution are compared to a previous large Damköhlernumber (D_a) asymptotic analysis using a simple reaction ratemodel, and agreement is very good even for moderately largevalues of D_a. However, for such Damköhler numbers, thewave speeds are found to be much larger than observed experimentally.Indeed, the simulations show that O(1) values of D_a are requiredto obtain the lower experimental wave speeds. Nevertheless,the wave speeds as a function of flow rate through the tubedo not agree well with the preliminary experimental resultsfor any choice of the parameters. A more realistic, Arrheniusreaction rate model is then considered. The Arrhenius modelpredicts a rapid change in temperature at the wave front, inmuch better agreement with the experiments than for the simplerreaction model.     

Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients

^** Email: todor{at}math.ethz.ch^*** Corresponding author. Email: schwab{at}math.ethz.ch A scalar, elliptic boundary-value problem in divergence formwith stochastic diffusion coefficient a(x, ) in a bounded domainD ^d is reformulated as a deterministic, infinite-dimensional,parametric problem by separation of deterministic (x D) andstochastic ( ) variables in a(x, ) via Karhúnen–Loèveor Legendre expansions of the diffusion coefficient. Deterministic,approximate solvers are obtained by projection of this probleminto a product probability space of finite dimension M and sparsediscretizations of the resulting M-dimensional parametric problem.Both Galerkin and collocation approximations are considered.Under regularity assumptions on the fluctuation of a(x, ) inthe deterministic variable x, the convergence rate of the deterministicsolution algorithm is analysed in terms of the number N of deterministicproblems to be solved as both the chaos dimension M and themultiresolution level of the sparse discretization resp. thepolynomial degree of the chaos expansion increase simultaneously.     

Transient diffusion-controlled smoulder propagation: a similarity solution approach

One-, two- and three-dimensional time-dependent smoulder propagationthrough solid reactants with plane bounding surfaces is considered.Propagation is maintained by the diffusion of oxidizer fromthe boundaries to the smouldering reaction zone. The resultingburnt solid is assumed to be porous and the unreacted solidis taken to be sufficiently dense for no oxidizer to be present.The activation energy of the reaction is taken to be so largethat the exothermic reaction term has a delta-function behaviour.This enables the reaction zone to be approximated by a narrowreaction front and results in the equations of heat and masstransfer being decoupled away from the front. The assumption,based on experimental observations, that the reaction frontspropagate with a speed proportional to t^–1/2, where tis the time, permits the introduction of similarity variables.The resulting intermediate asymptotic equations, lying betweenthose for very small and very large times, are solved and theequation of the smouldering fronts determined for each geometryis considered.     

The density of prime divisors in the arithmetic dynamics of quadratic polynomials

Let f [x], and consider the recurrence given by a_n = f(a_{n –1}), with a₀ . Denote by P(f, a₀) the set of prime divisorsof this recurrence, that is, the set of primes dividing at leastone non-zero term, and denote the natural density of this setby D(P(f, a₀)). The problem of determining D(P(f, a₀)) whenf is linear has attracted significant study, although it remainsunresolved in full generality. In this paper, we consider thecase of f quadratic, where previously D(P(f, a₀)) was knownonly in a few cases. We show that D(P(f, a₀)) = 0 regardlessof a₀ for four infinite families of f, including f = x² + k,k \{–1}. The proof relies on tools from group theoryand probability theory to formulate a sufficient condition forD(P(f, a₀)) = 0 in terms of arithmetic properties of the forwardorbit of the critical point of f. This provides an analogy toresults in real and complex dynamics, where analytic propertiesof the forward orbit of the critical point have been shown todetermine many global dynamical properties of a quadratic polynomial.The article also includes apparently new work on the irreducibilityof iterates of quadratic polynomials.     

Volume Estimate Via Total Curvature in Hyperbolic Spaces

Let D Hⁿ(–k²) be a convex compact subset of the hyperbolicspace Hⁿ(–k²) with non-empty interior and smooth boundary.It is shown that the volume of D can be estimated by the totalcurvature of D. More precisely, , where K denotes the Gauss–Kronecker curvature of D andVol(S^n–1) denotes the Euclidean volume of the sphere.2000 Mathematics Subject Classification 53C21.     

On an Operator Theory Approach to the Corona Problem

This paper deals with an operator theory approach to the coronaconjecture for H(Dⁿ). Treil gave a counter-example to this conjecturein the case where n = 1 for operator-valued functions; thusone might hope to use this to disprove the corona conjecturefor H(Dⁿ) (for n 2). This paper shows that this natural approachtowards a negative answer fails. On the other hand, the secondresult here shows that ‘commutant lifting’ cannotbe true for more than two contractions for any constant. Thisobstructs a natural attempted proof of the corona conjecturefor H(Dⁿ) (for n 2) by our previous result.     

An error bound for the modified successive overrelaxation method

In this paper we consider the modified successive overrelaxation(MSOR)methodto appropriate the solution of the linear system D^-1/2 Ax =D^-1/2b, where A is a symmetric, positive definite and consistentlyordered matrix and D is a diagonal matrix with the diagonalidentical to that of A. The main purpose of this paper is to obtain some theoreticalresults, namely a bound for the norm of _n = v –v_n in termsof the norms _n –v_n-1, _n+1 –v_n and their inner product,where v =D^-1/2 x and v_n is the nth iteration vector, obtainedusing the (MSOR)method.     

Flow Through Symmetrically Constricted Tubes

Fluid morion through a tube is discussed when there is a moderateor severe symmetric constriction at the wall, and the oncomingflow is fully developed. The Reynolds number R is assumed large.During a moderate constriction, where the typical slope isO(R^–), upstream separation can be provoked due to thedownstream pressure being transmitted back, through the inviscidcore flow. Separation can also occur after a point of maximumconstriction. Computations and analysis indicate that the upstreamseparation point is pushed increasingly ahead as the slope israised. The implication for a severe constriction, where isO(1), is that the flow separates at a large distance O(a lnR) upstream (a being the tubewidth) and produces a shear layerwhich, on nearing the constriction, reaches O(a) distances fromthe wall.     

Animals, Trees and Renewal Sequences: Corrigendum

IN SECTION 3 of the above we omitted to mention aperiodicity.The period p of the pseudo renewal sequence {a_n: n > 0} isgiven by p = g.c.d. {n > 1: a_n > 0}. We are only concernedwith aperiodic renewal sequences (i.e. where p = 1). As it standsTheorem 3.1 is incorrect and should be restated as: THEOREM 3.1 If a = (a_n: n = 0,1,...) is an aperiodic pseudo-renewalsequence its limit a satisfies g_na^–n > 1 where a^–1 is to be interpreted as; if a = 0.     

Isometries of the Space of Convex Bodies in Euclidean Space

The isometries of the space of convex bodies of E^d with respectto the Hausdorff-metric are precisely the mappings of the formC i(C) + D where i is a rigid motion of E^d and D a fixed convexbody.     

The Uniqueness of the Prime Markoff Numbers

Given the Diophantine equation a²+b²+c²=3abc, a solution tripleof natural numbers (a, b, c) can be arranged in ascending orderso that abc. Then, given the largest element c, one can askwhether this uniquely determines the triple. This is referredto as the Markoff conjecture. The paper proves that, if c isprime, then there is indeed only one triple that solves theequation with c as the largest element. The proof uses onlystandard algebraic number theory, but it was prompted by geometricconsiderations.     

A Functional Relation for Accessory Parameters for Genus 2 Algebraic Curves with an Order 4 Automorphism

Let C be a genus 2 algebraic curve defined by an equation ofthe form y² = x(x² – 1)(x – a)(x – 1/a). Asis well known, the five accessory parameters for such an equationcan all be expressed in terms of a and the accessory parameter b corresponding to a. The main result of the paper is thatif a' = 1 – a², which in general yields a non-isomorphiccurve C', then b'a'(a'² – 1) = – – ba(a²– 1). This is proven by it being shown how the uniformizing functionfrom the unit disk to C' can be explicitly described in termsof the uniformizing function for C.     

Smoothness Properties of the Unit Ball in a JB*-Triple

An element a of norm one in a JB^*-triple A is said to be smoothif there exists a unique element x in the unit ball A₁^* of thedual A^* of A at which a attains its norm, and is said to beFréchet-smooth if, in addition, any sequence (x_n) ofelements in A₁^* for which (x_n(a)) converges to one necessarilyconverges in norm to x. The sequence (a²ⁿ⁺¹) of odd powers ofa converges in the weak^*-topology to a tripotent u(a) in theJBW^*-envelope A^** of A. It is shown that a is smooth if andonly if u(a) is a minimal tripotent in A^** and a is Fréchet-smoothif and only if, in addition, u(a) lies in A.     

Two Diophantine Birds with One Stone

Integer Solutions are found to the equations t²–3(a²,b², (a + b)², (a–b)²) = p², q², r², s². These lead surprisinglyto solutions to the equations u² + (c², d², (c + d)², (c –d)²) = p², q², v², w², with the same values of p and q.     

On the mixed Cauchy problem with data on singular conics

We consider a problem of mixed Cauchy type for certain holomorphicpartial differential operators with the principal part Q_2p(D)essentially being the (complex) Laplace operator to a power,^p. We provide inital data on a singular conic divisor givenby P = 0, where P is a homogeneous polynomial of degree 2p.We show that this problem is uniquely solvable if the polynomialP is elliptic, in a certain sense, with respect to the principalpart Q_2p(D).     

Numerical Investigations into the Stokes Phenomenon. II

The Stokes phenomenon associated with the differential equationsW " = WZ (z²–a²) and W" = w(z² –a²)(x²–b²)isconsidered. As an application to the method introduced in paper I, somenumerical and analytical results concerning the Stokes constantsof these equations are presented.     

Herman Rings and Arnold Disks

For (,a) C* x C, let f_,a be the rational map defined by f_,a(z)= z² (az+1)/(z+a). If R/Z is a Brjuno number, we let D bethe set of parameters (,a) such that f_,a has a fixed Hermanring with rotation number (we consider that (e²ⁱ,0) D). Resultsobtained by McMullen and Sullivan imply that, for any g D, theconnected component of D(C* x (C/{0,1})) that contains g isisomorphic to a punctured disk. We show that there is a holomorphic injection F:DD such thatF(0) = (e²ⁱ ,0) and , where r is the conformal radius at 0 of the Siegel disk of the quadraticpolynomial z e²ⁱ z(1+z). As a consequence, we show that for a (0,1/3), if f_l,a has afixed Herman ring with rotation number and if m_a is the modulusof the Herman ring, then, as a0, we have e m_a=(r/a) + O(a). We finally explain how to adapt the results to the complex standardfamily z e^(a/2)(z-1/z).     

Differentiable Peak-Interpolation on Bounded Domains with Smooth Boundary

Let D be a bounded domain in Cⁿ with C² boundary, and, for 1 k , let A^k(D) be the algebra of functions holomorphic on Dand C^k on a neighbourhood of . It is shown that each peak-interpolation set for A^k(D) is afinite set.     

Boundary Accessibility of a Domain Quasiconformally Equivalent to a Ball

A boundary point of a domain D in Rⁿ is said to be broadly accessibleif it ‘almost lies’ on the boundary of a round ballcontained in D. If f is a quasiconformal mapping of the unitball Bⁿ onto D, then it is shown that broadly accessible boundarypoints on D correspond under f to a set of full measure on Bⁿ.2000 Mathematics Subject Classification 30C65.     

The effect of a changing diffusion coefficient in super-Case II polymer-penetrant systems

In certain polymer-penetrant systems, nonlinear viscoelasticeffects dominate those of Ficlcian diffusion. By introducinga dependence of the chemical potential on concentration history,this behaviour can be modelled by a memory integral. The mathematicalframework presented is a moving boundary-value problem wherethe boundary separates the polymer into two distinct states:glassy and rubbery. In each region, a different operator holdsat leading order. The problem which results is not solvableby similarity solutions, but can be solved by perturbation andintegral equation techniques. By introducing a new model wherethe diffusion coefficient changes with phase, asymptotic solutionsare obtained where sharp fronts initially move like t³/2. This‘super-Case II’ behaviour is found in various non-Fickianpolymer-penetrant systems.