Comparison of the simplified and full MHD models for laminar incompressible flow past a circular cylinder

In the majority of research on incompressible magnetohydrodynamic (MHD) flows, the simplified model with the low magnetic Reynolds number assumption has been adopted because it reduces the number of equations to be solved. However, because the effect of flow on magnetic field is also neglected, the solutions of the simplified model may be different from those of the full model. As an example, the flow of an electrically conducting fluid past a circular cylinder under a magnetic field is investigated numerically using the simplified and full models in this paper. To solve the problems, two second-order compact finite difference algorithms based on the streamfunction-velocity formulation of the simplified model and the quasi-streamfunction-velocity formulation of the full model are developed respectively.Numerical simulations are carried out over a wide range of Hartmann number for steady-state laminar problems with both models. For the full model, magnetic Reynolds number (Re_m) is chosen from 0.01 to 10. The computed results show that solutions of the simplified MHD model are not exactly the same as those of the full MHD model for this flow problem in most cases even if Re_m in the full model is very low. Only in the special case that a strong external magnetic field is exerted perpendicular to the dominant flow direction, can the simplified MHD model be regarded as an approximation of the full MHD model with low Re_m.     

A new approach to the nonlinear stability of viscous flow in a coplanar magnetic field

We present a new Lyapunov function for laminar flow, in the x‐direction, between two parallel planes in the presence of a coplanar magnetic field for three‐dimensional perturbations with stress‐free boundary planes that provides conditional nonlinear stability for all Reynolds numbers(R_e) and magnetic Reynolds numbers(R_m) below π²/2M. Compared with previous results on the nonlinear stability of this problem, the radius of stability ball and the energy decay rate obtained in this paper are independent of the magnetic field. Copyright © 2016 John Wiley & Sons, Ltd.     

The giant spiral

Cocycles ofZ ^m actions on compact metric spaces provide a means for constructingR ^m actions or flows, called suspension flows. It is known that allR ^m flows with a free dense orbit have an almost one-to-one extension which is a suspension flow. Whenm=1, examples of cocycles are easy to construct; there is a one-to-one correspondence between cocycles and real valued continuous functions. However, whenm>1 the construction of examples of cocycles becomes more problematic. The only existing class of examples, the close to linear cocycles, have strong linearity properties and are well understood. In fact, when theZ ^m action is uniquely ergodic, all cocycles are close to linear. We will show that in general this need not be the case. We present a method, suggested to us by Hillel Furstenberg, for constructing examples of cocycles whenm>1 and use this method to construct a non close to linear cocycle on a minimalZ ² action.     

Magnetohydrodynamic flow past a circular cylinder

This paper deals with the slow-flow problem of an incompressible, viscous, electrically conducting fluid past a circular cylinder in an aligned magnetic field. The solutions for the velocity and magnetic fields are sought by the method of matched asymptotic expansions under the assumptions:R, R _mM1, such thatR=O(R _m). The influence ofR andR _m on this solution is studied toO(R/(logM)²) andO(R/logM), respectively.

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Zusammenfassung Die Arbeit behandelt das Problem des langsamen Fliessens einer nicht zusammendrückbaren zähen elektrisch leitenden Flüssigkeit, die um einen Kreiszylinder mit parallel zur Strömung gerichtetem Magnetfeld strömt. Die Lösungen für die Geschwindigkeit und das Magnetfeld werden gefunden durch die Methode der angepassten asymptotischen Entwicklungen unter den VoraussetzungenR, R _mM1, so dassR=O(R _m). Der Einfluss vonR undR _m auf diese Lösungen wird fürO(R/(logM)²) bzw.O(R _m /logM) untersucht.

     

Stability study for the MHD problem in perforated and parallel walls

We have studied the stability of a conducting fluid when it is continuously injected or ejected through a pair of parallel porous walls and escapes in both directions along the channel. The flow forms a stagnation point at the center and the effluence is restricted by a magnetic field perpendicular to the walls. A theoretical analysis of the steady state solutions of the MHD equations in the incompressible case is given as a function of three parameters R_e, R_m and M_A (R_e: Reynolds number, R_m: magnetic Reynolds number, M_A: Alfvenic Mach number), for some asymptotic limits. In the case of suction, critical values of the parameters are found for which there are bifurcations in the system. Such bifurcations are pitchfork type. In the case of injection, the flux results always stable.     

On stable range conditions

In this paper, we show that a ring R satisfies unit 1-stable range if and only if a₁R + ? + a_mR = dR with m ≥ 2,a ₁, ?a_m ?R implies that there exist u₁ , ?u_m ? U(R) such that a₁u₁ +?+a_mu_m = d and an exchange ring R has stable range one if and only if a₁R+?+a_mR = dR with m ≥ 2,a ₁,?,a_m ? R implies that there exist unit-regular w ₁,?,w_m ? R such that a₁w₁ +?+ a_mw_m = d. Also we show that an exchange ring R satisfies the n-stable range condition if and only if a( ⁿR)+bR = dR with a ? Rⁿ,b,d ? R implies that there exist unimodular regular w ? ⁿ R and: y ? R such that aw+by = d.     

Periodic fluid transients in rectangular ducts with transverse magnetic fields III

Summary This paper treats periodic fluid transients in a liquid metal contained in a constant-area, perfectly conducting rectangular duct with a uniform magnetic field applied perpendicular to one pair of the duct's walls. The Mach numberM is assumed to be small and viscous effects are assumed to be negligible. A previous paper treated this problem with the additional assumption that the magnetic Reynolds numberR _m is much smaller thanM, which excluded the Alfvén wave mechanism. On the other hand, ifR _m is much larger thanM, then the transients are the well-known Alfvén waves in an incompressible fluid. The present paper treats the case between these two extremes withR _m=M, where is an arbitrary constant. For this case there are three classes of wave modes: one class of Alfvén waves which involve no disturbances to the fluid pressure and two classes of waves which involve a coupling of the acoustic and Alfvén wave mechanisms. Dispersion relations are presented for all three classes.

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Résumé Cet article traite d'ondes de compression périodiques d'un métal liquide contenu dans une conduite rectangulaire à section droite constante et avec des parois parfaitement conductrices. On applique un champ magnétique homogène et qui est perpendiculaire à deux parois. On admet que le nombre de Mach,M, est petit et que la viscosité est negligeable. Dans un article précédent ce probléme a été traité avec l'hypothèse que le nombre magnétique de Reynolds,R _m, est plus petit queM. Dans ce cas, il n'y a pas d'ondes d'Alfvén et on trouve les deux types d'ondes acoustiques. Si l'on admet queR _m est plus grande queM, on ne trouve que des ondes d'Alfvén. Le présent article traite le cas intermédiaire. On admet queR _m=M et que est une constante arbitraire. On trouve les trois types d'ondes. Pour le premier type d'ondes, il n'y a pas de perturbation de la pression. Pour les deux autres, les mécanismes des ondes acoustiques et des ondes d'Alfvén sont couplés. On détermine les équations de dispersion pour les trois types d'ondes.

     

An algorithm for the divisors of monic polynomials over a commutative ring

Gilmer and Heinzer proved that given a reduced ring R, a polynomial f divides a monic polynomial in R[X] if and only if there exists a direct sum decomposition of R = R₀ ⊕ … ⊕ R_m (m ≤ deg f), associated to a fundamental system of idempotents e₀, … , e_m, such that the component of f in each R_i[X] has degree coefficient which is a unit of R_i. We propose to give an algorithm to explicitly find such a decomposition. Moreover, we extend this result to divisors of doubly monic Laurent polynomials.     

The Blow-up Locus of Heat Flows for Harmonic Maps

Abstract Let M and N be two compact Riemannian manifolds. Let u _k (x, t) be a sequence of strong stationary weak heat flows from M×R ⁺ to N with bounded energies. Assume that u _k→u weakly in H ^{1, 2}(M×R ⁺, N) and that Σ^t is the blow-up set for a fixed t > 0. In this paper we first prove Σ^t is an H ^m−2-rectifiable set for almost all t∈R ⁺. And then we prove two blow-up formulas for the blow-up set and the limiting map. From the formulas, we can see that if the limiting map u is also a strong stationary weak heat flow, Σ^t is a distance solution of the (m− 2)-dimensional mean curvature flow [1]. If a smooth heat flow blows-up at a finite time, we derive a tangent map or a weakly quasi-harmonic sphere and a blow-up set ∪_t<0Σ^t× {t}. We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion. This work is supported by NSF grant     

Concentration sets of the Landau–Lifshitz system and quasi-mean curvature flows

The aim of this work is to analyze the concentration set of the stationary weak solutions to the Landau-Lifshitz system of the ferromagnetic spin chain from R ^m into the unit sphere S ² of R ³ . Suppose that u _k → u weakly in W ^1,2(R ^m × R ₊, S ²) and that Σ ^t is the concentration set for fixed t. In the present paper we first prove that Σ ^t is a -rectifiable set for almost all t ∈ R ₊. And then we verify that Σ ^t moves by the quasi-mean curvature under some assumptions, which is a new codimension 2 curvature flow. Finally we analyze the behavior of the solution at the singular point and get the blow up formulas. The main barrier to Landau–Lifshtiz system is that there is no energy monotonicity inequality. After the seminal work to on the study of the concentration set of minimizing energy harmonic maps by Leon Simon, there are several papers dealing with the stationary harmonic maps and its heat flows, and so on. This investigation is inspired by the study on the heat flow of harmonic maps and it largely depends on our result of the partial regularity.     

On the linear stability study of zonal incompressible flows on a sphere

The normal mode (linear) stability of zonal flows of a nondivergent fluid on a rotating sphere is considered. The spherical harmonics are used as the basic functions on the sphere. The stability matrix representing in this basis the vorticity equation operator linearized about a zonal flow is analyzed in detail using the recurrent formula derived for the nonlinear triad interaction coefficients. It is shown that the zonal flow having the form of a Legendre polynomial P_n(μ) of degree n is stable to infinitesimal perturbations of every invariant set I_m with |m| ≥ n. For each zonal number m, I_m is here the span of all the spherical harmonics $Y^{m}_{k}(x)$, whose degree k is greater than or equal to m. It is also shown that such small-scale perturbations are stable not only exponentially, but also algebraically. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 649–665, 1998     

Orthogonal basis for spherical monogenics by step two branching

Spherical monogenics can be regarded as a basic tool for the study of harmonic analysis of the Dirac operator in Euclidean space \mathbb R^m{{\mathbb R}^m}. They play a similar role as spherical harmonics do in case of harmonic analysis of the Laplace operator on \mathbb R^m{{\mathbb R}^m}. Fix the direct sum \mathbb R^m=\mathbb R^p ?\mathbb R^q{{\mathbb R}^m={\mathbb R}^p \oplus {\mathbb R}^q}. In this article, we will study the decomposition of the space M_n(\mathbb R^m, \mathbb C_m){{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)} of spherical monogenics of order n under the action of Spin(p) × Spin(q). As a result, we obtain a Spin(p) × Spin(q)-invariant orthonormal basis for M_n(\mathbb R^m, \mathbb C_m){{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}. In particular, using the construction with p = 2 inductively, this yields a new orthonormal basis for the space M_n(\mathbb R^m, \mathbb C_m){{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}.     

Length Four Polynomial Automorphisms

We study the structure of length four polynomial automorphisms of R[X, Y] when R is a unique factorization domain. The results from this study are used to prove that, if SL _m (R[X ₁, X ₂,…, X _n ]) = E _m (R[X ₁, X ₂,…, X _n ]) for all n, m ≥ 0, then all length four polynomial automorphisms of R[X, Y] that are commutators are stably tame.     

极限跟踪的不变性

It is showed that the LmSP (limit shadowing property) for flows is invariant under topological equivalence, and the suspension flow ϕ _f of a homeomorphism f under a continuous function ϑ: X → R>0 has LmSP₊ if and only if f has LmSP₊. As examples of application, the LmSP₊ of the ergodic flow on torus which has irrational number is discussed, and the characterization of a class of flows on torus with LmSP₊ is given. Supported by the National Natural Science Foundation of China (10371030) and the doctor’s foundation of Hebei Normal Univ. (L2003B05).     

Comparison of Some Purities,Flatnesses and Injectivities

In this article, we compare (n, m)-purities for different pairs of positive integers (n, m). When R is a commutative ring, these purities are not equivalent if R does not satisfy the following property: there exists a positive integer p such that, for each maximal ideal P, every finitely generated ideal of R _P is p-generated. When this property holds, then the (n, m)-purity and the (n, m′)-purity are equivalent if m and m′ are integers ≥np. These results are obtained by a generalization of Warfield's methods. There are also some interesting results when R is a semiperfect strongly π-regular ring. We also compare (n, m)-flatnesses and (n, m)-injectivities for different pairs of positive integers (n, m). In particular, if R is right perfect and right self (?₀, 1)-injective, then each (1, 1)-flat right R-module is projective. In several cases, for each positive integer p, all (n, p)-flatnesses are equivalent. But there are some examples where the (1, p)-flatness is not equivalent to the (1, p + 1)-flatness.     

A note about two properties of matrix rings

We show that there exists for eachm≧2 a (non-commutative) integral domainR with a nilpotent matrixC ∈R _m whose order of nilpotency is greater thanm, and anyA ∈R _m with a right (or a left) inverse is invertible.     

On indecomposable definite unimodular hermitian forms

For any natural numbersm andn≥17 we can construct explicitly indecomposable definite unimodular normal Hermitian lattices of rankn over the ring of algebraic integersR _m in an imaginary quadratic field . It is proved that for anyn (in casem=11, there is one exceptionn=3) there exist indecomposable definite unimodular normal HermitianR ₁₅(R ₁₁)-lattices of rankn, and we exhibit representatives for each class. In the exceptional case there are no lattices with the desired properties. The method given in this paper can solve completely the problem of constructing indecomposable definite unimodular normal HermitianR _m -lattices of any rankn for eachm. Dedicated to the memory of Prof. Lee Hwa-Chung.     

Convergence of the Dirichlet solutions of the very fast diffusion equation

For any −1<m<0, positive functions f, g and u₀≥0, we prove that under some mild conditions on f, g and u₀ as R→∞ the solution u^R of the Dirichlet problem u_t=(u^m/m)_xx in (−R,R)×(0,∞), u(R,t)=(f(t)|m|R)^1/m, u(−R,t)=(g(t)|m|R)^1/m for all t>0, u(x,0)=u₀(x) in (−R,R), converges uniformly on every compact subset of R×(0,T) to the solution of the equation u_t=(u^m/m)_xx in R×(0,T), u(x,0)=u₀(x) in R, which satisfies some mass loss formula on (0,T) where T is the maximal time such that the solution u is positive. We also prove that the solution constructed is equal to the solution constructed in Hui (2007) [15] using approximation by solutions of the corresponding Neumann problem in bounded cylindrical domains.     

Unified characterizations for modifications ofR 0 andR 1 topological spaces

Cammaroto and Noiri [14] introduced a separation axiom calledm-R ₀ in anm-space (X, m). In this paper, we introduce the notion ofm-R ₁ spaces and offer many characterizations ofm-R ₀ (resp.m-R ₁) spaces which enable us to obtain unified characterizations of separation axiomsR ₀, semi-R ₀, pre-R ₀,α-R ₀,δ-semiR ₀, (δ, p)-R ₀ (resp.R ₁, semi-R ₁, pre-R ₁,α-R ₁,δ-semiR ₁, (δ, p)-R ₁).     

IBN rings and orderings on grothendieck groups

LetR be a ring with an identity element.R∈IBN means thatR ^m⋟Rⁿ impliesm=n, R∈IBN ₁ means thatR ^m⋟Rⁿ⊕K impliesm≥n, andR∈IBN ₂ means thatR ^m⋟R^m⊕K impliesK=0. In this paper we give some characteristic properties ofIBN ₁ andIBN ₂, with orderings on the Grothendieck groups. In addition, we obtain the following results: (1) IfR∈IBN ₁ and all finitely generated projective leftR-modules are stably free, then the Grothendieck groupK ₀(R) is a totally ordered abelian group. (2) If the pre-ordering of the Grothendieck groupK ₀(R) of a ringR is a partial ordering, thenR∈IBN ₁ orK ₀(R)=0. Supported by National Nature Science Foundation of China.