Asymptotic Stability of a Stationary Flowing Regime of an Ideal Incompressible Fluid

We give sufficient conditions for asymptotic stability of a stationary solution to a flowing problem of a homogeneous incompressible fluid through a given planar domain. We consider a planar problem for the Euler equation and boundary conditions for the curl and the normal component of the velocity; moreover, the latter is given on the whole boundary of the flow domain and the curl is given only on the inlet part of the boundary. We establish asymptotic stability of a stationary flow (in linear approximation), assuming it to have no rest points and to satisfy some smallness condition which means that the perturbations leave the flow domain before they become to affect the main flow. In particular, we prove asymptotic stability for an arbitrary stationary flow in a rectangular canal close to the Couette flow without rest points. Moreover, we show that stability of the main flow in the L ₂-norm under curl perturbations implies its stability in higher-order norms depending, for example, on the derivatives of the curl.     

Uniformly convergent H(div)-conforming rectangular elements for Darcy-Stokes problem

We consider a singular perturbation problem which describes 2D Darcy-Stokes flow.An H(div)-conforming rectangular element,DS-R14,is proposed and analyzed frst.This element has 14 degrees of freedom for velocity and is proved to be uniformly convergent with respect to perturbation constant.We then simplify this element to get another H(div)-conforming rectangular element,DS-R12,which has 12 degrees of freedom for velocity.The uniform convergence is also obtained for this element.Finally,we construct a de Rham complex corresponding to DS-R12 element.     

The steady two-dimensional flow over a rectangular obstacle lying on the bottom

We study a plane problem with mixed boundary conditions for a harmonic function in an unbounded Lipschitz domain contained in a strip. The problem is obtained by linearizing the hydrodynamic equations which describe the steady flow of a heavy ideal fluid over an obstacle lying on the flat bottom of a channel. In the case of obstacles of rectangular shape we prove unique solvability for all velocities of the (unperturbed) flow above a critical value depending on the obstacle depth. We also discuss regularity and asymptotic properties of the solutions.     

A numerical procedure for viscous free surface flows

We present a numerical procedure for two-dimensional unsteady viscous free surface flow problems with surface tension. The procedure is based on a finite difference approach to a primitive variable formulation; a coordinate transformation is used to transform the irregularly shaped flow domain onto a fixed rectangular domain. The procedure is tested on a standing wave problem and a cavity flow problem with a free surface. Satisfactory numerical solutions are obtained for both problems for Reynolds numbers up to 200.     

Longitudinal flow through an array of rectangular cylinders

The flow parallel to an array of rectangular cylinders is solvedby an efficient domain decomposition and matching method. Thepermeability constant is determined for a variety of geometries,including strips of zero thickness. Formulae for the drag ofa single rectangular cylinder and a single strip are derived.     

Modeling of the base of a hydraulic structure with constant flow velocity sections and a curvilinear confining layer

We construct the underground contour of an embedded rectangular dam, whose corners are rounded by curves of constant flow velocity. We consider the case of a water-permeable base underlain by a curvilinear confining layer with a horizontal part, whereas the remainder parts of the layer are characterized by a constant flow velocity. We obtain an analytical solution to the corresponding mixed problem of the theory of analytic functions, we present results of numerical computations and consider the limiting case studied earlier by P. Ya. Polubarinova-Kochina and I. N. Kochina.     

Modeling duct flow by the R-function method

The article deals with a flow in the tube of rectangular cross-section with an inner circular cylindrical element. We use the numerical method basing on R-functions. This method is meshfree and therefore more efficient than the finite element method that requires remeshing when the geometry of problem is changed. The dependence of the flow on the diameter of the central cylinder and its position in the duct is investigated under constant pressure gradient. It was found that the resistance decreases if the inner element is moved from the center of the duct.     

Maxwell流体在震荡的矩形输送管道中的流动

分析了不可压缩Maxwell流体在震荡矩形截面管道中的非稳定流动问题.利用Fourier变换和Laplace变换作为数学工具,提出了问题的解,该解可以看成稳态解和暂态解之和.大倍数时,暂态消失,解可以表示为稳态解.在极限情况的案例中给出了Newton流体的解.当震荡频率不存在时,得到了Maxwell流体在震荡矩形截面管道中流动问题的解.最后,以图形形式给出不同参数时,矩形管道正弦震荡达到稳态所需要的时间.同时,分别描绘了x和y变化时的速度曲线.     

A Proof of the Convexity of the Free Boundary for Porous Flow Through a Rectangular Dam Using the Maximum Principle

We consider the steady two-dimensional flow under gravity ofwater from one reservoir (on the left) to a lower reservoir(on the right) through a porous rectangular isotropic homogeneousdam with impervious bottom. Because of gravity the water doesnot flow through the entire dam and the dam is dry near itsupper right corner. The interface separating the dry and wetregions of the dam is a free boundary. Recently, Friedman &Jensen (1977) have proved that the free boundary is convex.We give a different proof which uses only the maximum principleand its generalizations.     

Flow of a nonlinear viscoelastic liquid in cylindrical channels

The problem of nonrectilinear steady-state flow of a nonlinear viscoelastic liquid in an arbitrary cylindrical channel is examined. On the assumption that the cross flows are insignificant as compared with the longitudinal flows an equation of state is derived for the flow regime in question. A variational principle established for steady-state flows of the investigated media is proposed as the basis of a method of solving problems of the flow of polymer materials in arbitrary cylindrical channels. The flow of a polymer solution in rectangular channels is investigated.Institute of Mechanics, AS UkrSSR, Kiev. Translated from Mekhanika Polimerov, Vol. 4, No. 6, pp. 1103–1111, November–December, 1968.     

Wilson Loops in Exact Holographic RG Flows at Zero and Finite Temperatures

Theoretical and Mathematical Physics - We study rectangular timelike Wilson loops at long distances in the exact renormalization group flow in the context of the holographic duality. We consider...     

Control of the jet of a pulsed flywheel water cannon

We describe a pulsed water cannon in which the compression of the water occurs as a result of a cam located on a flywheel. We pose and solve the variational problem of optimal control of the spray of the pulsed water cannon. The control is the shape of the cam. We compute a way of finding a pressure pulse that is nearly rectangular, providing a uniform fluid flow. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 107–111.     

Stokes flow through a staggered array of rectangular cylinders and the junction resistance

The micro-fluid mechanical problem of viscous flow across an array of staggered rectangular cylinders is studied by eigenfunction expansions and matching. Streamlines and permeabilities are determined for (mean) flows normal and parallel to the stagger. A variety of approximate formulas are deduced. The resistance of a T junction is determined.     

Rectangular group congruences on a semigroup

We study rectangular group congruences on an arbitrary semigroup. Some of our results are an extension of the results obtained by Masat (Proc. Am. Math. Soc. 50:107–114, 1975). We show that each rectangular group congruence on a semigroup S is the intersection of a group congruence and a matrix congruence and vice versa, and this expression is unique, when S is E-inversive. Finally, we prove that every rectangular group congruence on an E-inversive semigroup is uniquely determined by its kernel and trace.     

Spline Representation of Incompressible Flow

We describe a scheme for reconstructing a distribution of fluidvelocities from given values of the net flow across the facesof a rectangular mesh in two or three dimensions. Assuming thedata to be consistent, the reconstructed velocities will havethe properties that the continuity condition is everywhere satisfied,that the track of any fluid particle is a smooth curve and thatthere is no flow across any part of the boundary that has beenspecified as representing a solid wall.     

Modelling avalanche flows

We present a lattice model that illustrates avalanche flow ina driven, disordered system. Structural disorder is associatedwith rectangular grains that have orientational degrees of freedom.Grain orientation transitions couple the evolving disorder withsurface instabilities. The simulated event size distributionhas features in common with observed surface granular flow and,in contrast to previous sandpile models, does not have a simplescaling behaviour. We have identified properties of the sandpilesurface that correlate with the internal structural disorder.The simulation results support a model of granular dynamicsin which surface flow and granular relaxation processes arestrongly coupled.     

Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices

Let be a {0, 1}-homomorphism of a finite distributive lattice D into the congruence lattice Con L of a rectangular (whence finite, planar, and semimodular) lattice L. We prove that L is a filter of an appropriate rectangular lattice K such that ConK is isomorphic with D and is represented by the restriction map from Con K to Con L. The particular case where is an embedding was proved by E.T. Schmidt. Our result implies that each {0, 1}-lattice homomorphism between two finite distributive lattices can be represented by the restriction of congruences of an appropriate rectangular lattice to a rectangular filter.     

Singular values of nonnegative rectangular tensors

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. Some properties concerning the singular values of a real rectangular tensor were discussed by K. C. Chang et al. [J. Math. Anal. Appl., 2010, 370: 284–294]. In this paper, we give some new results on the Perron-Frobenius Theorem for nonnegative rectangular tensors. We show that the weak Perron-Frobenius keeps valid and the largest singular value is really geometrically simple under some conditions. In addition, we establish the convergence of an algorithm proposed by K. C. Chang et al. for finding the largest singular value of nonnegative primitive rectangular tensors.     

Rectangular random matrices, related convolution

We characterize asymptotic collective behavior of rectangular random matrices, the sizes of which tend to infinity at different rates. It appears that one can compute the limits of all noncommutative moments (thus all spectral properties) of the random matrices we consider because, when embedded in a space of larger square matrices, independent rectangular random matrices are asymptotically free with amalgamation over a subalgebra. Therefore, we can define a “rectangular-free convolution”, which allows to deduce the singular values of the sum of two large independent rectangular random matrices from the individual singular values. This convolution is linearized by cumulants and by an analytic integral transform, that we called the “rectangular R-transform”.