首页 | 本学科首页   官方微博 | 高级检索  
相似文献
 共查询到20条相似文献,搜索用时 31 毫秒
1.
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).  相似文献   

2.
The constitutive relations for the transport of heat, stress,electric charge, etc., in a continuum must be chosen so thatthe second law of thermodynamics is not violated; the constraintstake the form of inequalities, typically requiring the entropygenerated within a material element to be non-negative. Thepaper is concerned with this concept—its history, thephysical principles on which it depends, how to apply it whensecond-order or non-linear effects are important and how itis widely misused in modern continuum mechanics. The history is reduced to the contributions of five leadingthermodynamicists—Clausius, Maxwell, Gibbs, Boltzmannand Duhem. The object here was to try to discover which formof the inequality one should regard as being fundamental. Oneimportant conclusion is that entropy S must be defined simultaneouslywith the identification of the inequality, and that in generalthis cannot be done until the constitutive equations are known.The empirical element enters with the notion of irreversibility,which is given a precise meaning with the aid of the motionreversed parity (x), a variable x having = +1 or = –1if, when time and motions are reversed, x x or xx.The macroscopic parity of x, *(x), is obtained by first replacingx by the constitutive equation for x. The entropy production rate has both irreversible (f) and reversible(r) parts. It is shown that the reciprocal relations followfrom the requirement that the macroscopic parity of (i) mustbe +1. Continuum thermodynamics is based on various principles extractedfrom theory developed for uniform systems, the example chosento illustrate the ideas being the simple monatomic gas. Second-orderconstitutive relations are introduced, and the expressions forentropy and its production rate per unit volume, , obtained.It is shown that the stability condition 0 cannot, in general,be satisfied merely by imposing constraints on the constitutiverelations. To second-order = 1 + 2, where 1 is the usual bilinearform, and the terms in 2 have an additional derivative. Thesecond-order term 2 can have both signs, and is not dissipative.The relation between this fact and the frame-dependence of constitutiverelations is explained. The final section illustrates the errors frequently found inthe thermodynamic arguments appearing in books and papers onrational continuum mechanics. The principle of these is that 0 is interpreted as being a constraint on the constitutiverelations alone. Another is the idea that the balance equationscan be set aside as constraints by regarding them as mere definitionsof a heat source and a body force, an error based partly onthe misconception that constitutive relations should be frame-indifferent.Finally, an inequality due to Glansdorff & Prigogine isexamined and found to be in error.  相似文献   

3.
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   

4.
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.  相似文献   

5.
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   

6.
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   

7.
A mathematical model for change of phase is presented, accountingfor the presence of regions in which liquid and solid coexist.The basic variables are temperature and solid fraction v. Westart from a relationship of the type =(v), supposed valid inthermodynamical equilibrium. Then for dynamical processes weintroduce a perturbation causing v to be less than its equilibriumvalue in any solidification process. This solid fraction deficiencyis governed by an ordinary differential equation containingt, in the forcing term. The heat-balance equation is in turncoupled to the ordinary differential equation through the termvt, ( is latent heat). Some existence and uniqueness resultsare proved and some monotonicity properties are described forpure melting or pure solidification processes.  相似文献   

8.
Bull London Math. Soc, 4 (1972), 370–372. The proof of the theorem contains an error. Before giving acorrect proof, we state two lemmas. LEMMA 1. Let K/k be a cyclic Galois extension of degree m, let generate Gal (K/k), and let (A, I, ) be defined over K. Supposethat there exists an isomorphism :(A,I,) (A, I, ) over K suchthat vm–1 ... = 1, where v is the canonical isomorphism(Am, Im, m) (A, I, ). Then (A, I, ) has a model over k, whichbecomes isomorphic to (A, I, ) over K. Proof. This follows easily from [7], as is essentially explainedon p. 371. LEMMA 2. Let G be an abelian pro-finite group and let : G Q/Z be a continuous character of G whose image has order p.Then either: (a) there exist subgroups G' and H of G such that H is cyclicof order pm for some m, (G') = 0, and G = G' x H, or (b) for any m > 0 there exists a continuous character m ofG such that pm m = . Proof. If (b) is false for a given m, then there exists an element G, of order pr for some r m, such that () ¦ 0. (Considerthe sequence dual to 0 Ker (pm) G pm G). There exists an opensubgroup Go of G such that (G0) = 0 and has order pr in G/G0.Choose H to be the subgroup of G generated by , and then aneasy application to G/G0 of the theory of finite abelian groupsshows the existence of G' (note that () ¦ 0 implies that is not a p-th. power in G). We now prove the theorem. The proof is correct up to the statement(iv) (except that (i) should read: F' k1 F'ab). To removea minor ambiguity in the proof of (iv), choose to be an elementof Gal (F'ab/k2) whose image $$\stackrel{\¯}{\sigma}$$ in Gal (k1/k2) generates this last group. The error occursin the statement that the canonical map v : AP A acts on pointsby sending ap a; it, of course, sends a a. The proof is correct, however, in the case that it is possibleto choose so that p = 1 (in Gal (F'/k2)). By applying Lemma 2 to G = Gal (F'ab/k2) and the map G Gal(k1/k2) one sees that only the following two cases have to beconsidered. (a) It is possible to choose so that pm = 1, for some m, andG = G' x H where G' acts trivially on k1 and H is generatedby . (b) For any m > 0 there exists a field K, F'ab K k1 k2is a cyclic Galois extension of degree pm. In the first case, we let K F'ab be the fixed field of G'.Then (A, I, ), regarded as being defined over K, has a modelover k2. Indeed, if m = 1, then this was observed above, butwhen m > 1 the same argument applies. In the second case, let : (A, I, ) (A$$\stackrel{\¯}{\sigma}$$, I$$\stackrel{\¯}{\sigma }$$, $$\stackrel{\¯}{\sigma}$$) be an isomorphism defined over k1 and let v ... p–1 = µ(R). If is replaced by for some Autk1((A, I, )) then is replacedby P. Thus, as µ(R) is finite, we may assume that pm–1= 1 for some m. Choose K, as in (b), to be of degree pm overk2. Let m be a generator of Gal (K/k2) whose restriction tok1 is $$\stackrel{\¯}{\sigma }$$. Then : (A, I, ) (A$$\stackrel{\¯}{\sigma }$$, I$$\stackrel{\¯}{\sigma}$$, $$\stackrel{\¯}{\sigma }$$ = (A$$\stackrel{\¯}{\sigma}$$m, I$$\stackrel{\¯}{\sigma }$$m, $$\stackrel{\¯}{\sigma}$$m is an isomorphism defined over K and v mpm–1, ... m =pm–1 = 1, and so, by) Lemma 1, (A, I, ) has a model overk2 which becomes isomorphic to (A, I, over K. The proof may now be completed as before. Addendum: Professor Shimura has pointed out to me that the claimon lines 25 and 26 of p. 371, viz that µ(R) is a puresubgroup of R*t, does not hold for all rings R. Thus this condition,which appears to be essential for the validity of the theorem,should be included in the hypotheses. It holds, for example,if µ(R) is a direct summand of µ(F).  相似文献   

9.
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   

10.
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   

11.
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   

12.
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   

13.
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.  相似文献   

14.
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).  相似文献   

15.
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   

16.
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   

17.
Let be an irrational number in [0, 1] and A the correspondingirrational rotation C*-algebra. Let Aut (A) be the group ofall automorphisms of A and Int (A) the normal subgroup of Aut(A) of all inner automorphisms of A. Let Pic (A) be the Picardgroup of A. In the present note we shall show that if is notquadratic, then Pic (A)Aut (A)/Int (A) and that if is quadratic,then Pic (A) is isomorphic to a semidirect product of Aut (A)/Int(A) with Z. Furthermore, in the last section we shall discussPicard groups of certain Cuntz algebras.  相似文献   

18.
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   

19.
Based on the construction of infinite-dimensional balanced realizations,an alternative solution to the following inverse spectral problemis presented. Given a decreasing sequence of positive numbers(n)n>1 (i.e. 123...0), does there exist a Hankel operatorwhose sequance of singular values is (n)n>1? This paper isan extension of a previously published paper in which the sameapproach was taken in the case of a monotonically decreasingsequence(n)n>1.  相似文献   

20.
This paper determines the electrostatic potential and field taking place both inside and outside a slender dielectric body embedded in a given potential 0. This task is actually reducedto the determination of the occurring polarization surface-chargedensity qwhich depends on 0, on the body shape but also onthe ratio = 2/1of the dielectic constants (2outside and 1inside the body). The adopted procedure consists in asymptoticallyexpanding and inverting (with respect to the small slendernessratio of the body) the well-known Fredholm boundary integralequation of the second kind governing the function q. The technicaldifficulties such an approach encounters are bypassed by employinga systematic formula in getting the asymptotic estimate ofcertain integrals depending upon a small parameter. Contraryto other works in the field, this method authorizes us to handlethe case of non-axisymmetric slender bodies. As an illustrationthe theory is applied to a body of elliptical cross-sectionand comparisons are presented for a slender dielectric ellipsoidembedded in a special potential 0for which the exact densityqis obtained in a closed form.  相似文献   

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号