Local Interaction Theory for Laminar Transonic Flows in Slender Channels

Transonic flows through channels so narrow that the classical boundary layer approach fails are considered. As a consequence the properties of the inviscid core and the viscosity dominated boundary layer region can no longer be determined in subsequent steps but have to be calculated simultaneously. The resulting viscous inviscid interaction problem for weakly three dimensional laminar flows is formulated for perfect gases under the requirement that the channel is sufficiently narrow so that the flow outside the viscous wall layers becomes planar in the leading order approximation. Representative solutions for subsonic as well as for supersonic flows disturbed by three dimensional surface mounted obstacles will be presented. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weakly Nonhydrostatic Effects in Compositionally-Driven Gravity Flows

In the study of compositionally-driven gravity currents it is customary to adopt the hydrostatic assumption for the pressure field which, in turn, leads to a depth-independent horizontal velocity field and significant simpilifications to the governing equations. The hydrostatic assumption is reasonable in, say, the case of a two-layer flow when the depth variations of the lower layer are small when considered as a function of space and time. However, for larger deflections of the interface (such as those caused by bottom topography) the flow will deviate in its behavior from the low aspect ratio, slowly varying purely hydrostatic flow because of the presence of vertical accelerations. In this paper we present an approach to capture the contribution of interface curvature to nonhydrostatic effects in fully time-dependent flows in two-fluid systems. Our approach involves expanding the relevant dependent variables in the form of an asymptotic expansion   f = f ⁽⁰⁾+δ² f ⁽¹⁾+ o (δ²)  , where  0 < δ≪ 1  is the aspect ratio of the flow, and obtaining the first-order correction to hydrostatic theory. Numerical results and comparisions with the purely hydrostatic theory are included.     

Periodic Solution of Weakly Damped 3D Schrodinger-Boussinesq Equations

In this paper the authors consider a model of the interaction of a nonlinear complex Schrodinger field and a real Boussinesq field in a 3D domain with the weakly damping which arises in the laser and plasma physics and prove the existence of the periodic solution.     

弱P-调和映射流的紧性性质

分别考虑了映入球面及紧致的齐性Ｒｉｅｍａｎｎｉａｎ空间的弱Ｐ－调和映射流；通过球面及齐性Ｒｉｅｍａｎｎｉａｎ空间的对称性质，证明了弱Ｐ－调和映射流的紧性性质．     

三维具弱阻尼Schrǒdinger-Boussinesq方程组的周期解

In this paper the authors consider a model of the interaction of a nonlinear complex Schrdinger field and a real Boussinesq field in a 3D domain with the weakly damping which arises in the laser and plasma physics and prove the existence of the periodic solution.     

3‐D Inviscid Transonic Condensing Flow around a Swept Wing

Transonic condensing flow is an interesting phenomena because of the large change in temperature over a small area. This drop in temperature allows the moist air to condense. It is the purpose of this paper to examine the effect of sweep on condensing flow. The geometry of the wing model starts with NACA‐0014 at the wall and reduces to a NACA‐0010 at the tip. The span of the wing is 2.5 times the maximum chord length. The effect of sweep is examined by comparing a model wing with a sweep angle of 11.3 with a straight trailing edge that has no thickness and then a straight leading edge with a 11.3 trailing edge sweep. The free stream Mach number is 0.8 and angle of attack is 0. A 2‐D calculation shows that the NACA‐0014 and NACA‐0010 have a region of supersonic flow but due to the effect of sweep the sonic line does not extend to the tip. This change of the supersonic region influences the area of condensation on the wing. The swept wing has a lower total drag coefficient for the adiabatic and all condensation cases compared to the straight leading edge wing and second for the each wing the trend of increasing drag with humidity is shown.     

Weakly Nonlinear Lower-Branch Stability of Fully Developed and Developing Free-Surface Flows

We examine the weakly nonlinear stability of both fully developedand of developing liquid layers. The study of these free-surfaceflows is more complicated than that of many other flows owingto the fully nonlinear boundary conditions present. The scalingsused for the two problems follow from the work of J. S. B. Gajjar,who described their linear stability properties. We use thetechnique given by F. T. Smith to derive amplitude equationsof the type presented by J. T. Stuart and J. Watson. Both flowsare found to be supercritically stable in general and a varietyof asymptotic cases are considered.     

梯形复式断面均匀流水力计算

对由主槽、斜边坡以及边滩三部分组成的梯形复式断面渠道均匀流的水力计算进行了研究,采用表观切应力表示各子断面间的横向动量交换,结合各部分力的平衡关系式,推导出了主槽、斜边坡和边滩平均流速的计算公式．采用UK-FCF的实验数据对公式进行了验证,滩槽相对水深为0.057~0.4的7组工况计算值与实测值的比较表明,不考虑表观切应力的计算值与实测值的一致性都较差．比较而言,斜边坡部分平均流速在水深较小时计算值大于实测值,在水深较大时计算值小于或接近实测值,表观切应力对边滩部分平均流速的影响不明显,而主槽部分若不考虑表观切应力的影响,则计算值与实测值差别较大．运用三段式模型与刘沛清等提出的两段式模型分别进行了有边坡梯形断面复式渠道的水力计算,结果表明对于有边坡段的梯形断面渠道（河道）,三段式模型更加有效．最后,将计算的表观切应力与实测表观切应力进行了比较,两者的一致性表明采用的动量输运系数是合适的．     

Weakly coupled systems of semilinear elastic waves with different damping mechanisms in 3D

We consider the following Cauchy problem for weakly coupled systems of semilinear damped elastic waves with a power source nonlinearity in three dimensions: where with b² > a² > 0 and θ ∈ [0,1]. Our interests are some qualitative properties of solutions to the corresponding linear model with vanishing right‐hand side and the influence of the value of θ on the exponents p₁,p₂,p₃ in to get results for the global (in time) existence of small data solutions.     

Transonic shock in a nozzle I: 2D case

In this paper we establish the existence and uniqueness of a transonic shock for the steady flow through a general two‐dimensional nozzle with variable sections. The flow is governed by the inviscid potential equation, and is supersonic upstream, has no‐flow boundary conditions on the nozzle walls, and a given pressure at the exit of the exhaust section. The transonic shock is a free boundary dividing two regions of C flow in the nozzle. The potential equation is hyperbolic upstream where the flow is supersonic, and elliptic in the downstream subsonic region. In particular, our results show that there exists a solution to the corresponding free boundary problem such that the equation is always subsonic in the downstream region of the nozzle when the pressure in the exit of the exhaustion section is appropriately larger than that in the entry. This confirms exactly the conjecture of Courant and Friedrichs on the transonic phenomena in a nozzle [10]. Furthermore, the stability of the transonic shock is also proved when the upstream supersonic flow is a small steady perturbation for the uniform supersonic flow or the pressure at the exit has a small perturbation. The main ingredients of our analysis are a generalized hodograph transformation and multiplier methods for elliptic equation with mixed boundary conditions and corner singularities. © 2004 Wiley Periodicals, Inc.     

Nonlinear Evolution Equations and the Braiding of Weakly Transporting Flows Over Gravel Beds

The flow of a river that transports sediment in the form of gravel as bedload is investigated for the case when the transport is small. The linear stability of such flows is discussed and used to formulate some strongly nonlinear investigations describing the interaction of bar instabilities that are known to occur. The key spatial scales in the asymptotic limit of small transport are identified, and highly nonlinear evolution equations derived for each case. A generalized KDV equation is found to govern the nonlinear evolution at small wavenumbers, while at O(1) wavenumbers an infinite set of "triad-like" amplitude equations describes the flow. The interactions demonstrate the natural tendency of rivers of width significantly higher than the critical width at which instability first occurs to form complex patterns that may be associated with braided rivers. The weak transport limit used in our anaysis makes our work directly relevant to rivers experiencing flood conditions where the onset of a flood causes transport to begin. The results shown suggest that in the highly nonlinear stages, bars take the form of slabs tilted in the flow direction with steep edges. In addition, it is found that there is no equilibrium state. These findings are consistent with observations.     

Logarithmically Improved Criteria for the 3D Nematic Liquid Crystal Flows in the Multiplier Spaces

This paper proves some logarithmically improved regularity criteria in terms of the multiplier space for the 3D nematic liquid crystal flows.     

Flows in 3-edge-connected bidirected graphs

It was conjectured by A. Bouchet that every bidirected graph which admits a nowhere-zero k-flow admits a nowhere-zero 6-flow. He proved that the conjecture is true when 6 is replaced by 216. O. Zyka improved the result with 6 replaced by 30. R. Xu and C. Q. Zhang showed that the conjecture is true for 6-edge-connected graph, which is further improved by A. Raspaud and X. Zhu for 4-edge-connected graphs. The main result of this paper improves Zyka’s theorem by showing the existence of a nowhere-zero 25-flow for all 3-edge-connected graphs.     

Weakly Cycle Complementary 3-Partite Tournaments

The vertex set of a digraph D is denoted by V(D). A c-partite tournament is an orientation of a complete c-partite graph. Let V ₁, V ₂, . . . ,V _c be the partite sets of D. If there exist two vertex disjoint cycles C and C′ in D such that V_i?(V(C)èV(C￠)) 1 ?{V_{\mathrm{i}}\cap(V(C)\cup V(C'))\neq\emptyset} for all i = 1, 2, . . . , c, then D is weakly cycle complementary. In 2008, Volkmann and Winzen gave the above definition of weakly complementary cycles and proved that all 3-connected c-partite tournaments with c ≥ 3 are weakly cycle complementary. In this paper, we characterize multipartite tournaments are weakly cycle complementary. Especially, we show that all 2-connected 3-partite tournaments that are weakly cycle complementary, unless D is isomorphic to D _3,2, D _3,2,2 or D _3,3,1.     

Dynamics of Poisson-Nernst-Planck Systems and Ionic Flows Through Ion Channels: A Review

Poisson-Nernst-Planck systems are basic models for electrodiffusion process, particularly, for ionic flows through ion channels embedded in cell membranes. In this article, we present a brief review on a geometric singular perturbation framework for analyzing the steady-state of a quasi-one-dimensional Poisson-Nernst-Planck model. The framework is based on the general geometric singular perturbed theory from nonlinear dynamical system theory and, most crucially, on the reveal of two specific structures of Poisson-Nernst-Planck systems. As a result of the geometric framework, one obtains a governing system–an algebraic system of equations that involves all physical quantities such as protein structures of membrane channels as well as boundary conditions, and hence, provides a complete platform for studying the interplay between protein structure and boundary conditions and effects on ionic flow properties. As an illustration, we will present concrete applications of the theory to several topics of biologically significant based on collaboration works with many excellent researchers.     

弯曲扩压管道中三维不可压粘性流的数值计算及湍流模型的研究

为了便于复杂形状管道中粘性流动的数值计算,本文在任意曲线坐标系下导出了控制方程的Favre质量加权平均形式,导出了一种考虑曲率影响的湍流模型.采用抛物化方法对弯曲扩压管道内的层流和湍流流动进行了数值计算.根据计算结果对压气机静叶流道内的气动性能进行了初步的分析.     

Generalization of the Crocco Invariant for 3D Gas Flows Behind Detached Bow Shock Wave

Russian Mathematics - We study a steady 3D flow of the ideal gas. In the flow between the bow shock wave and the nose part of the body streamlined by the uniform supersonic flow, we consider...     

Mathematical and Numerical Study of the 3D Atmospheric Boundary Layer Flows Including Pollution Dispersion

The paper presents a mathematical and numerical investigation of the atmospheric boundary layer (ABL) flow over 3D complex terrain part of which is represented by the real topography of the Krkonose mountains located in the Czech Republic. The flow is supposed to be turbulent, non‐stratified, viscous, incompressible and stationary. Two mathematical models have been formulated. The first model is based upon the RANS equations in the conservative form and the second one uses the Boussinesq approximation of RANS equations and takes the non‐conservative form. Also pollution dispersion over the complex 3D terrain has been considered in both models. The problem closure is achieved by an algebraic turbulence model and given boundary conditions.     

On v-Distal Flows on 3-Manifolds

In [2], H. Furstenberg studied a distal action of a locallycompact group G on a compact metric space X, and establisheda structure theorem. As a consequence, he showed that if G isabelian, then a simply connected space X does not admit a minimaldistal G-action. In this paper we concern ourselves with a nonsingular flow = {^t} on a closed 3-manifold M. Recall that is called distalif for any distinct two points x, y M, the distance d(^tx, ^ty)is bounded away from 0. The distality depends strongly uponthe time parametrization. For example, there exists a time parametrizationof a linear irrational flow on T² which yields a nondistal flow[4, 6]. 1991 Mathematics Subject Classification 58F25, 57R30.     

Every 4k-edge-connected graph is Weakly 3k-linked

We prove:

1. Fork ≥ 2 andα = 0, 1, every (4k + 2α)-edge-connected graph is weakly (3k + 2α)-linked.
2. IfG is ak-edge-connected graph (k ≥ 2),s, t are vertices andf is an edge, then there exists a pathP betweens andt such thatf ? E(P) andG ? E(P) ? f is (k ? 2)-edge-connected, whereE(P) denotes the edge set ofP.