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1.
This paper examines the effect of compressibility on the flowin the boundary layer on a semi-infinite, thermally insulatedflat plate placed at zero incidence to a uniform stream of electricallyconducting gas, with an aligned magnetic field at large distancesfrom the plate. The present discussion is limited to small values of the conductivityparameter = 4µv, and the Prandtl number is taken to beunity. The latter assumption permits a simplification of theanalysis, and the former allows the dependence of the flow onthe parameters ß = µH2/4U2 and M = U/cto beadequately illustrated without excessive computation. A seriessolution valid for small values of the conductivity parameterand for Mach numbers not too large is derived. Values of ß = 0.3 and 0.5, = 0.01 and 0.1 are consideredand for those values the skin friction decreases with increasingMach number, similar to the case when ß = 0. The analysissuggests that for larger values of ß the skin frictionmight even increase with the Mach number initially. This iscertainly the case with the tangential component of the magneticfield, which for ß = 0.5 exhibits a maximum at approximatelyunit Mach number. The reason for this behaviour lies in thefact that, in view of the temperature changes taking place inthe flow, the electrical conductivity and thereby the localvalue of can change by more than an order of magnitude. Thishas the effect of giving results which are akin to those forarbitrary large in incompressible flow even though the valueof based on the main stream gas properties remains low.  相似文献   

2.
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 e1 = 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)].  相似文献   

3.
Numerical results are reported for the computation of periodicsolution paths for a suspension bridge model represented bythe equation. un + EIuxxxx + ut, + Ku+ = W(x) + sin x sin µt. with hinged-end boundary conditions, as the forcing amplitude and frequency µ are varied. The term Ku+ models the factthat there is restoring force due to the cables only when theyare being stretched. It is found that an S-shaped curve is obtainedwhen the displacement amplitudes are plotted against the forcingamplitudes for some frequency regimes. As a two-parameter problem,it appears that the solution set resembles a cusp-like surfacewith the singular point near linear resonance. While the effectof strengthening the cable (i.e. increasing K) will enhancethe occurrence of the multiple solutions, the effect of increasingthe damping coefficient gives the opposite effect.  相似文献   

4.
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 Cn, dependingonlyon n and such that where  相似文献   

5.
A method is developed for evaluating Fourier integrals of theform A() = 1–1f(x) efax 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 functionsJr+(), r= 0, 1, 2,.... A partialsum AN() 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 AN() for all , and alsothe truncation error A()–AN() 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.  相似文献   

6.
Blow up of solutions of a generalized Boussinesq equation   总被引:2,自引:0,他引:2  
Consider the Cauchy problem utt = (f(u))xx + uxxtt x R, t 0, u(x,0) = u0(x), ut(x,0) = u1(x),7rcub; where f : R R C, f(0) = ). After treatment of the local existenceproblem, we show the blow up of the solution of the equation(1) under the folowing assumptions. Let > 0 be real, such that 2(l + 2)F(u) uf(u), (v0, Pv0)l2 + - F(u0)dx < 0 where P = 1 - d2/dx2, F'(s), and v0 is given by u1(x,0) = (v0(x))x. Then we focus on various perturbations of the question. We alsostudy the vectorial case in the same way, and finally we giveexamples.  相似文献   

7.
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.  相似文献   

8.
On hearing the shape of a bounded domain with Robin boundary conditions   总被引:2,自引:0,他引:2  
The asymptotic expansions of the trace of the heat kernel (t)= [sum ]j=1 exp(-tj) for small positive t, where {j} j=1 arethe eigenvalues of the negative Laplacian -n = -[sum ]nk=1 (/xk)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 Yi (i = 1,...,m) are assuming the Robin boundary conditions(ni + Yi) = 0 on i. Here /ni 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.  相似文献   

9.
Exact Remainders for Asymptotic Expansions of Fractional Integrals   总被引:1,自引:0,他引:1  
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()xo(x-)x-1 f(t)dt, >. Heref(t) is a locally integrable function on [0, ) and satisfies f(t) ast-5–0(ó >0), s=0 as   相似文献   

10.
The sector of analyticity of the Ornstein–Uhlenbeck semigroupis computed on the space := Lp (RN; µ) with respect to its invariant measure µ.If A= + Bx· denotes the generator of the Ornstein–Uhlenbecksemigroup, then the angle 2 of the sector of analyticity in is /2 minus the spectral angleof BQ, Q being the matrix determining the Gaussian measure µ.The angle of analyticity in is then given by the formula   相似文献   

11.
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.  相似文献   

12.
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.  相似文献   

13.
This paper examines the norms of the projections from C[–1,1] to PM[–1, 1] which are obtained by truncating Chebyshevand ultraspherical expansions after n terms. The norms are evaluatednumerically for n = 1, 2, 3 , ... , 10 and –0.1 5 wherethe ultraspherical weight function is (1 – x2)x.In particular the data show that the norm of the Chebyshev projectionis not the smallest in this range for 1 n 10.  相似文献   

14.
A fluid is injected from a slot into a stream of another fluid.In a simple model this leads to a two-phase two-free-boundaryproblem with the jump relation |u|2 – |u+|2 = on the free boundary {u=0}, and |u| = 1 on the free boundary{u > – Q}, where u is the stream function and Q isthe flux of the injected fluid. Using the variational theoryof Alt, Caffarelli & Friedman, we prove existence of (,1, u) such that there is a smooth fit for both free boundaries.  相似文献   

15.
Decomposition of weighted Triebel-Lizorkin and Besov spaces on the ball   总被引:1,自引:0,他引:1  
Weighted Triebel–Lizorkin and Besov spaces on the unitball Bd in d with weights wµ(x)=(1–|x|2)µ–1/2,µ0, are introduced and explored. A decomposition schemeis developed in terms of almost exponentially localized polynomialelements (needlets) {}, {} and it is shown that the membershipof a distribution to the weighted Triebel–Lizorkin orBesov spaces can be determined by the size of the needlet coefficients{f, } in appropriate sequence spaces.  相似文献   

16.
We study the asymptotic behaviour of blow-up interfaces of thesolutions to the one-dimensional nonlinear filtration equationin inhomogeneous media where m>1 isa constant and (x) = |x| (for |x| 1, with > 2) isa bounded, positive, smooth, and symmetric function. The initialdata are assumed to be smooth, bounded, compactly supported,symmetric, and monotone. It is known that due to the fast decayof the density (x) as |x| the support of the solution increasesunboundedly in a finite time T. We prove that as tT theinterface behaves like O((Tt)b), where the exponentb > 0 (which depends on m and only) is given by a uniqueself-similar solution of the second kind satisfying the equation|x| ut = (um)xx. The corresponding rescaled profilesalso converge. We establish the stability of the self-similarsolution of the second kind for the exponential density (x)=e–|x|for |x| 1. We give a formal asymptotic analysis of the blow-upbehaviour for the non-self-similar density (x) = e–|x|2.Several exact self-similar solutions and their correspondingasymptotics are constructed.  相似文献   

17.
Let G be a separable locally compact group and let be its dualspace with Fell's topology. It is well known that the set P(G)of continuous positive-definite functions on G can be identifiedwith the set of positive linear functionals on the group C*-algebraC*(G). We show that if is discrete in , then there exists anonzero positive-definite function associated with such that is a w*-strongly exposed point of P(G)0, where P(G)0={f P(G):f(e)1. Conversely, if some nonzero positive-definite function associatedwith is a w*-strongly exposed point of P(G)0, then is isolatedin . Consequently, G is compact if and only if, for every ,there exists a nonzero positive-definite function associatedwith that is a w*-strongly exposed point of P(G)0. If, in addition,G is unimodular and , then is isolated in if and only if somenonzero positive-definite function associated with is a w*-stronglyexposed point of P(G)0, where is the left regular representationof G and is the reduced dual space of G. We prove that if B(G)has the Radon–Nikodym property, then the set of isolatedpoints of (so square-integrable if G is unimodular) is densein . It is also proved that if G is a separable SIN-group, thenG is amenable if and only if there exists a closed point in. In particular, for a countable discrete non-amenable groupG (for example the free group F2 on two generators), there isno closed point in its reduced dual space .  相似文献   

18.
A compressible fluid in a two-dimensional half-space (y >0) is bounded by a plane surface (y = 0) which is acousticallyhard except for a set of periodically arranged strips Sn givenby nda < x < nd + a, y = 0 with n = 0, 1, 2,....The velocity potential Re {(x, y)exp(–it)} satisfies theHelmholtz wave equation in the fluid region y>0, with /y= 0 on the plane y = 0, x Sn. The boundary condition on thepistons Sn is taken to have the form where the prescribed forcing function V(x) is the same on eachstrip, so that V(x + nd) = V(x), and the operators L and M arepolynomial functions of the second derivative 2/x2. This boundarycondition includes the possibilities of an elastic plate, amembrane, or an impedance surface for Sn. When the separationdistance d is much greater than the strip width 2a and wavelength2/k, the problem is reduced to that of finding the potentialp due to a single piston So set in a rigid baffle, togetherwith a potential c subject to a similar condition with forcingfunctions exp (ikx) in place of V(x). The problem is generalizedto allow for the possibility of a phased forcing function V(x),such that V(x + nd) = exp (ißnd)V(x), where ßis a given constant.  相似文献   

19.
Optimal order H1 and L error bounds are obtained for a continuouspiecewise linear finite element approximation of an obstacleproblem, where the obstacle's height as well as the contactzone, c, are a priori unknown. The problem models the indentationof a membrane by a rigid punch. For R2, given ,g R+ and an obstacle defined over E we consider the minimization of |v|21,+over (v, µ) H10() x R subject to v+µ on E. In additionwe show under certain nondegeneracy conditions that dist (c,hc)Ch ln 1/h, where hc is the finite element approximation toc. Finally we show that the resulting algebraic problem canbe solved using a projected SOR algorithm.  相似文献   

20.
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 constantsn and eddy-current distributions in(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.  相似文献   

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