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.     

Heat transfer by magnetohydrodynamic laminar flow in circular pipe

This paper presents the influence of magnetic field on heat due to viscous and electrical dissipations for an incompressible, viscous, electrically conducting fluid through a circular pipe in the presence of an applied (transverse) uniform magnetic field. The walls of the pipe are assumed to be non-conducting and kept at uniform temperature gradient in one case and at a constant temperature gradient in another case. The heat equation governing the present problem is solved exactly in hypergeometric series. The temperature at the centre of the pipe T_e, unweighted mean temperature T_m and weight mean temperature T_M are calculated. The temperature profiles are shown graphically for different values of Hartmann number M, Brinkman number B_r and a non-dimensional number S. Numerical calculations are made for the Nusselt number and are entered in the table.     

Convergence of some finite element iterative methods related to different Reynolds numbers for the 2D/3D stationary incompressible magnetohydrodynamics

Based on finite element method (FEM), some iterative methods related to different Reynolds numbers are designed and analyzed for solving the 2D/3D stationary incompressible magnetohydrodynamics (MHD) numerically. Two-level finite element iterative methods, consisting of the classical m-iteration methods on a coarse grid and corrections on a fine grid, are designed to solve the system at low Reynolds numbers under the strong uniqueness condition. One-level Oseen-type iterative method is investigated on a fine mesh at high Reynolds numbers under the weak uniqueness condition. Furthermore, the uniform stability and convergence of these methods with respect to equation parameters R_e,R_m, S_c, mesh sizes h,H and iterative step m are provided. Finally, the efficiency of the proposed methods is confirmed by numerical investigations.     

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.

     

Duality in hypercomplex function theory

Let $\text{A}$ be the Clifford algebra constructed over a quadratic n-dimensional real vector space with orthogonal basis {e₁,…, e_n}, and e₀ be the identity of $\text{A}$. Furthermore, let M_k(Ω;$\text{A}$) be the set of $\text{A}$-valued functions defined in an open subset Ω of R^m+1 (1 ? m ? n) which satisfy D^kf = 0 in Ω, where D is the generalized Cauchy-Riemann operator $\text{D = ∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{m}}\text{e}{}_{\mathrm{i}}\text{(}\text{?}\text{?x}{}_{\mathrm{i}}\text{)}$ and k? N. The aim of this paper is to characterize the dual and bidual of M_k(Ω;$\text{A}$). It is proved that, if M_k(Ω;$\text{A}$) is provided with the topology of uniform compact convergence, then its strong dual is topologically isomorphic to an inductive limit space of Fréchet modules, which in its turn admits M_k(Ω;$\text{A}$) as its dual. In this way, classical results about the spaces of holomorphic functions and analytic functionals are generalized.     

Reynolds operator on functors

Let be an affine R-monoid scheme. We prove that the category of dual functors (over the category of commutative R-algebras) of G-modules is equivalent to the category of dual functors of A^∗-modules. We prove that G is invariant exact if and only if A^∗=R×B^∗ as R-algebras and the first projection A^∗→R is the unit of A. If M is a dual functor of G-modules and w_G?(1,0)∈R×B^∗=A^∗, we prove that M^G=w_G⋅M and M=w_G⋅M⊕(1−w_G)⋅M; hence, the Reynolds operator can be defined on M.     

Hartmann duct flow at moderate magnetic Reynolds numbers

The application of a uniform external magnetic field on the turbulent duct flow of an electrically conducting fluid leads to several interesting changes in the structure and the mean charateristics of the flow. This is fairly well understood from the existing studies of duct flows in the low magnetic Reynolds number (R_m) limit. In this paper, we present the results for magnetohydrodynamic duct flow at moderate R_m obtained from direct numerical simulations (DNS). Several differences are observed to occur in this case as compared to low R_m flows, such as increased Hartmann layer thickness and enhanced large scale turbulence in the core region of the duct cross-section due to partial expulsion of magnetic flux. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary

Let M be an m-dimensional differentiable manifold with a nontrivial circle action S={St_}t∈R, St+1=St, preserving a smooth volume μ. For any Liouville number α we construct a sequence of area-preserving diffeomorphisms Hn such that the sequence converges to a smooth weak mixing diffeomorphism of M. The method is a quantitative version of the approximation by conjugations construction introduced in [Trans. Moscow Math. Soc. 23 (1970) 1].For m=2 and M equal to the unit disc D²={x²+y²?1} or the closed annulus A=T×[0,1] this result proves the following dichotomy: α∈R?Q is Diophantine if and only if there is no ergodic diffeomorphism of M whose rotation number on the boundary equals α (on at least one of the boundaries in the case of A). One part of the dichotomy follows from our constructions, the other is an unpublished result of Michael Herman asserting that if α is Diophantine, then any area preserving diffeomorphism with rotation number α on the boundary (on at least one of the boundaries in the case of A) displays smooth invariant curves arbitrarily close to the boundary which clearly precludes ergodicity or even topological transitivity.     

Injectivity and operator spaces

Let B(H) be the bounded operators on a Hilbert space H. A linear subspace R ? B(H) is said to be an operator system if 1 ?R and R is self-adjoint. Consider the category $\text{b}$ of operator systems and completely positive linear maps. R ∈ $\text{C}$ is said to be injective if given A ? B, A, B ∈ $\text{C}$, each map A → R extends to B. Then each injective operator system is isomorphic to a conditionally complete C¹-algebra. Injective von Neumann algebras R are characterized by any one of the following: (1) a relative interpolation property, (2) a finite “projectivity” property, (3) letting M_m = B(C^m), each map R → N ? M_m has approximate factorizations R → M_n → N, (4) letting K be the orthogonal complement of an operator system N ? M_m, each map $\text{M}{}_{\mathrm{m}}\text{K}\text{→ R}$ has approximate factorizations $\text{M}{}_{\mathrm{m}}\text{K}\text{→ M}{}_{\mathrm{n}}\text{→ R}$. Analogous characterizations are found for certain classes of C¹-algebras.     

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.     

Linear preservers of row-dense matrices

Let M_m,n be the set of all m × n real matrices. A matrix A ∈ M_m,n is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions T: M_m,n → M_m,n that preserve or strongly preserve row-dense matrices, i.e., T(A) is row-dense whenever A is row-dense or T(A) is row-dense if and only if A is row-dense, respectively. Similarly, a matrix A ∈ M_n,m is called a column-dense matrix if every column of A is a column-dense vector. At the end, the structure of linear preservers (strong linear preservers) of column-dense matrices is found.     

Spectral radius and infinity norm of matrices

Let Mn(R) be the linear space of all n×n matrices over the real field R. For any A∈Mn(R), let ρ(A) and ‖A_‖∞ denote the spectral radius and the infinity norm of A, respectively. By introducing a class of transformations φa on Mn(R), we show that, for any A∈Mn(R), ρ(A)<‖A_‖∞ if . If A∈Mn(R) is nonnegative, we prove that ρ(A)<‖A_‖∞ if and only if , and ρ(A)=‖A_‖∞ if and only if the transformation φ‖A_‖∞ preserves the spectral radius and the infinity norm of A. As an application, we investigate a class of linear discrete dynamic systems in the form of X(k+1)=AX(k). The asymptotical stability of the zero solution of the system is established by a simple algebraic method.     

Approximations in the l∞ norm and the generalized inverse

We introduce the concept of a strict l_∞-metric projector, based in the definition of strict approximation, to prove that for each matrix A of order m×n with coefficients in the field R of real numbers there exists a set of operators G: R^m → Rⁿ homogeneous and continuous, but not necessarily linear (strict generalized inverse) such that AGA = A and 6AGy?y6 is minimized for all y, when the norm is the l_∞ norm. We investigate the properties of these operators and prove that there are two distinguished operators A^-1_{∞, β} and A^-1_∞ which are extensions of the generalized inverse introduced by Newman and Odell in the case of a strictly convex norm.     

Complementary bases of a matroid

Let e₁, e′₁, e₂, e′₂, …, e_n, e′_n be the elements of matroid M. Suppose that {e₁, e₂, …;, e_n} is a base of M and that every circuit of M contains at least m + 1 elements. We prove that there exist at least 2^m bases, called complementary bases, of M with the property that only one of each complementary pair e_j, e′_j is contained in any base.We also prove an analogous result for the case where E is partitioned into E₁, E₂, …, E_n and the initial base contains |E_j| ? 1 elements from partition E_j.     

Length problems for automorphisms of modules over local rings

Let R be a local ring and M a free module of a finite rank over R. An element τ ∈ Aut_RM is said to be simple if τ ≠ 1 fixes a hyperplane of M.We shall show that for any σ ∈ Aut_RM there exist a basis X for M and ρ ∈ Aut_RM such that ρ acts as a permutation on X and ρ⁻¹σ is a product of m or less than m simple elements in Aut_RM, where m is the order of the invariant factors of σ modulo the maximal ideal of R.Also we shall investigate the problem treated by E.W. Ellers and H. Ishibashi [Factorizations of transformations over a valuation ring, Linear Algebra Appl. 85 (1987) 17-27], in which they showed that σ is a product of simple elements and gave an upper bound of the smallest number of such factors of σ, whereas in the present paper we will give lower bounds of σ in case that R is a local domain. Moreover we will factorize θσ as a product of symmetries and transvections for some θ the matrix of which is diagonal.     

Products of reflections in the general linear group over a division ring

Let F be a division ring and A?GL_n(F). We determine the smallest integer k such that A admits a factorization A=R₁R₂?R_k?1B, where R₁,…,R_k?1 are reflections and B is such that rank(B?I_n)=1. We find that, apart from two very special exceptional cases, k=rank(A?I_n). In the exceptional cases k is one larger than this rank. The first exceptional case is the matrices A of the form I_m⊕αI_n?m where n?m?2, α≠?1, and α belongs to the center of F. The second exceptional case is the matrices A satisfying (A?I_n)²=0, rank(A?I_n)?2 in the case when char F≠2 only. This result is used to determine, in the case when F is commutative, the length of a matrix A?GL_n(F) with detA=±1 with respect to the set of all reflections in GL_n(F).     

Degree functions and projectively full ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal

Let (R,m) be a 2-dimensional rational singularity with algebraically closed residue field and for which the associated graded ring is an integrally closed domain. According to Göhner, (R,m) satisfies condition (N): given a prime divisor v, there exists a unique complete m-primary ideal A_v in R with T(A_v)={v} and such that any complete m-primary ideal with unique Rees valuation v, is a power of A_v. We use the theory of degree functions developed by Rees and Sharp as well as some results about regular local rings, to investigate the degree coefficients d(A_v,v). As an immediate corollary, we find that for a simple complete m₁-primary ideal I₁ in an immediate quadratic transform (R₁,m₁) of (R,m); the inverse transform of I₁ in R is projectively full.     

Bialgebroid actions on depth two extensions and duality

A general notion of depth two for ring homomorphism N→M is introduced. The step two centralizers A=End_NM_N and in the Jones tower above N→M are shown in a natural way via H-equivalence to be dual bimodules for Morita equivalent endomorphism rings, the step one and three centralizers, R=C_M(N) and C=End_N-M(M⊗_NM). We show A and B to possess dual left and right R-bialgebroid structures which generalize Lu's fundamental bialgebroids over an algebra. There are actions of A and B on M and with Galois properties. If M|N is depth two and Frobenius with R a separable algebra, we show that A and B are dual weak Hopf algebras fitting into a duality-for-actions tower extending previous results in this area for subfactors and Frobenius extensions.     

Subproper splitting for rectangular matrices

In this paper iterative schemes for approximating a solution to a rectangular but consistent linear system Ax = b are studied. Let A?C^{m × n}_r. The splitting A = M ? N is called subproper if R(A) ? R(M) and $\text{R(A}{}^1\text{) ?R(M}{}^1\text{)}$. Consider the iteration $\text{x}{}_{\mathrm{i}}\text{= M}{}^2\text{Nx}{}_{\mathrm{i}}{}_{\mathrm{?1}}\text{+ M}{}^2\text{b}$. We characterize the convergence of this scheme to a solution of the linear system. When A?R^m×ⁿ_r, monotonicity and the concept of subproper regular splitting are used to determine a necessary and a sufficient condition for the scheme to converge to a solution.