Topological classification of flows on closed two-dimensional manifolds

Let M be a smooth manifold with tangent bundle TM. A k-plane field (or k-distribution) on M is a k-dimensional subbundle a of TM. Equivalently let a denote the section of the Grassmann bundle Gk(M) of k-planes associated to TM whose value at e M is the k-plane ax c TM,. Two k-plane fields are homotopic if they are homotopic as sections of Gk(M). Homotopic k-plane fields are equivalent as k-plane bundles over M, but not conversely. If L is an injectively immersed, smooth submanifold of M such that TLX = ax c TMx for all e L, L is called an integral submanifold of a. A kplane field a is called completely integrable if the following three equivalent conditions are satisfied. A. M is covered by open sets U with local coordinates x1, * , x such that the submanifolds defined by xk+l = constant, * * *, xm = constant are integral submanifolds of a. B. a is smooth and through every point e M there is an integral submanifold L of a. C. a is smooth and if X and Y are vector fields on M with Xx, Yx C ax for all e M then the bracket [X, Y]x e ax. The equivalence of these conditions is the Frobenius theorem. An integrable k-plane field is also called a foliation (this is equivalent to other definitions) and the maximal connected integral submanifolds are called leaves. The leaves of a foliation partition the manifold. The existence theorem for ordinary differential equations says that smooth line-fields are always integrable. In general for k > 1 the set of kplane fields which are not integrable is open and dense in the space of sections of Gk(M). G. Reeb [16] has asked if the existence of a k-plane field on a manifold implies the existence of a foliation. He has given an example of a foliation of codimension one on S3. W. B. R. Lickorish [10] and, independently, S. P. Novikov and H. Zeischang have exhibited foliations of codimension one on any closed, orientable 3-manifold. In ? 4 we consider the unorientable

The following theorem is proved. For any natural numbers and , , on a two-dimensional orientable compact manifold without boundary of class and genus there exists a topologically transitive flow of class having fixed points and exactly invariant ergodic normalized measures such that the measure of each trajectory is equal to zero.Bibliography: 3 items.

Numerical simulations of compressible rocket flows are conducted in laminar, transitional, and turbulent regimes. The laminar simulation is carried out on a planar rocket flow without nozzle using the unsteady two-dimensional Navier-Stokes system. The transitional and turbulent flows are performed in three-dimensional on an extended rocket geometry with a divergent outlet using compressible large eddy simulation (LES) models. In both cases, the compressibility effect plays an important role. In the laminar case, pressure oscillation is forced at the outflow boundary. The time-averaged part of the solution is compared with the inviscid theory of compressible rocket flow of Balakrishnan et al. (Balakrishnan, G., Linan, A., and Williams F. A., Compressibility Effects in Thin Channels with Injection, AIAA Journal, Vol.29, No. 12, 1991, pp. 2149-2154) and the oscillatory part with the acoustic layer model of Majdalani and Van Moorhem (Majdalani, J., and Van Moorhem, W. K., Improved Time-Dependent Flowfield Solution for Solid Rocket Motors, AIAA Journal, Vol. 36, No. 2, 1998, pp. 241-248). The mean flow from the present numerical result is in better agreement with the compressible theory than the conventional Taylor's profiles (Taylor, G. I., Fluid Flow in Regions Bounded by Porous Surfaces, Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences, Vol. 234, 1956, pp. 456-475), as expected. The oscillatory part of the flow agrees well in the first quarter of the axial extent, near the head end. Farther downstream, the discrepancies develop rapidly between the numerical result and the acoustic-layer model. Possible causes of the difference are the effect of compressibility, which alters the local speed of sound, hence, acoustic properties, and the interference of hydrodynamic instabilities. In the transition and turbulent regimes, the dynamic LES model is applied on different resolutions. The measurements data of Traineau et al. (Traineau, J. C., Hervat, P., and Kuentzmann, P., Cold Flow Simulation of a Two Dimensional Nozzleless Solid Rocket Motor, AIAA Paper 86-1447, June 1986) are employed for comparison purposes. The refinement study by comparison with the measurement data suggests the importance of resolving the laminar and transition region for a reliable application of LES in transitional flows. With the consideration of this aspect, LES with efficient grid size can produce resonable accuracy. Forcing hydrodynamic instabilities and a more realistic injection fluctuations model are recommended.

This paper treats geodesic triangles on two-dimensional orientable Riemannian manifolds $M$. Fixing two vertices $A$ and $B$, we can consider the area and the interior angles of the geodesic triangle $\Delta PAB$ as smooth functions of $P$. Applying the Laplace operator to these functions, we obtain formulas for the area and interior angles of $\Delta APAB$. It is shown that if $M$ is of constant curvature, the area and interior angles of geodesic triangles are harmonic.

The structure of flows exhibiting nontrivial recurrence on two-dimensional manifolds

In this paper some aspects on chaotic behavior and minimality in planar piecewise smooth vector fields theory are treated. The occurrence of non-deterministic chaos is observed and the concept of orientable minimality is introduced. Some relations between minimality and orientable minimality are also investigated and the existence of new kinds of non-trivial minimal sets in chaotic systems is observed. The approach is geometrical and involves the ordinary techniques of non-smooth systems.

For any gas lift system installation it is important to know the dynamic opening and closing pressures and valve throughput to yield an efficient and optimized gas lift system installation. A comprehensive theoretical and experimental study was performed to develop an improved mechanistic model of a pressure operated nitrogen charged gas-lift valve. Experiments are conducted on a Teledyne Merla N-15R, 1.5 inch, nitrogen charged, bellows operated gas-lift valve. A static probe tester is used to determine the load rates as a function of the initial dome pressure. Empirical equations quantifying bellows behavior are developed. Quasi-one dimensional flow theory for a convergent-divergent nozzle is used to model and describe the complex high velocity gas flow through the ball-seat configuration. The reduction of flow rate in the critical flow region of the throttling and transition flow regimes is achieved by the occurrence of a normal shock wave in the ball-seat configuration. A model incorporating compressible flow theory and the behavior of the bellows assembly is developed. Experimental and theoretical sensitivity studies indicate that a nitrogen charged pressure operated valve is very sensitive to performance parameters. Changes in pressures and temperatures by as little as 1% can change the maximum flow rate by as much as 30%.

We investigated the speed time series of the vehicles at a section of a highway in the city of Isfahan, Iran. Using k-means clustering algorithm, we find that the natural number of clustering for this set of data is 3. This is in agreement with the three-phase theory of uninterrupted traffic flows. According to this theory, the three traffic phases are categorized as free flow (F), synchronized (S) and wide moving jam (J). We obtain the transition speeds and densities at F–S and also S–J transitions. We also apply the Shannon entropy analysis on the speed time series over finite windows, which equips us to monitor in areal time the instant state of a traffic flow.

The topological equivalence of low-dimensional Morse-Smale flows without fixed point (NMS-flows) under assumptions of various generality is the subject of a number of publications. Starting from dimension 4, there are only few results on classification. However, it is known that there exists nonsingular flows with wildly embedded invariant saddle manifolds. In this paper the class of nonsingular Morse-Smale flows on closed orientable 4-manifolds with a unique saddle orbit which is, moreover, nontwisted, is considered. It is shown that the equivalence class of a certain knot embedded in $\mathbb S^2\times\mathbb S^1$ is a complete invariant of such a flow. Given a knot in $\mathbb S^2\times\mathbb S^1$, a standard representative in the class of flows under consideration is constructed. The supporting manifold of all such flows is shown to be the manifold $\mathbb S^3\times\mathbb S^1$. Bibliography: 24 titles.

A concept of the laminar flowmeter is based on the well-known relationship between laminar flow rate of fluid Q through a measuring element of length L and pressure drop ΔP. Usually the measuring element is a capillary tube, capillary tube bundle, parallel plate bundle, or concentric annular duct (Fig. 1).The pressure drop in the measuring element ("long duct") consists of two components:1ΔP=CρνL2ADh2Q+K∞ρ2A2Q2where: C is a constant (depends on a duct cross-section shape), A is the duct cross-sectional area, Dh is the hydraulic diameter of the duct, K∞ is the incremental pressure drop number in a fully developed flow region (depends both on the duct cross-section shape and on the inflow conditions), L is the distance between the flowmeter pressure taps, ρ is fluid density, and ν is fluid kinematic viscosity.The first component on the right-hand side of Eq. 1 is the pressure drop in the fully developed flow. The second one describes some additional pressure drop due to momentum change and accumulated increment in wall shear between developing flow and developed flow (1). It is seen that the relationship between produced differential pressure ΔP and flow rate Q is not linear. As a result, in the majority of commercial laminar flowmeters, to obtain precise Q values software corrections are required. In some of the commercial "laminar flow elements" the entrance effect is ignored as a relatively low one. In order to eliminate the necessity of reading corrections and to achieve inherent linearity the second term of Eq. 1 must be removed. It can be done by locating the first pressure tap P1 at the beginning of the fully developed flow region. As the entrance length Le increases with the Reynolds number [Re=DhQ∕(Aν)], the distance between the duct entrance and the pressure tap P1 should not be shorter than the entrance length for the maximum flow rate, i.e., for the laminar-turbulent transition Reynolds number.The main aim of this work was to determine the design recommendation for the inherently linear laminar annular-duct flowmeter. The pressure distribution along the annular duct was measured. The entrance lengths for various inlet conditions and Reynolds numbers were investigated.A solution of the continuity and momentum equations for the steady-state laminar fully developed flow of the Newtonian fluid in a concentric annulus provides us with the formula for the C constant in Eq. 1:2C=64(D−d)2(D2−d2)D4−d4−(D2−d2)2∕ln(D∕d)where (see Fig. 2) d is the cylinder diameter and D is the tube internal diameter.For a narrow annular duct we have limd→DC=96, which is the well-known value for parallel plates. One may examine that for D∕d<1.2 the ratio CFLAT∕CANNULAR<1.00055. It means that the flow rate, even in a relatively "wide" annulus, can be calculated from an equation valid for the developed flow between parallel plates. If the first pressure tap of our flowmeter is located behind the entrance length Le, the second component of the right-hand side of Eq. 1 will not exist. In this case the flow rate Q can be calculated using the simple linear relationship between Q and ΔP:3Q=2ADh2CρνLΔPin which C=96 (for D∕d<1.2), Dh=D−d, and A=π(D2−d2)∕4. As Eq. 3 is valid for a laminar flow and for a flowmeter having correctly located pressure tap P1 it is very important to check the Reynolds number and the hydrodynamic entrance length Le limit values.The experimental flowmeter test section is shown in Fig. 2. Air of precisely controlled temperature (23±0.2°C) and of known barometric pressure was used as a working fluid. An annular duct was formed between 215mm long circular cylinders (stainless steel) of diameter d=28.000±0.005mm and inner surface of calibrated thick-walled tube (polymethylmatacrylate) of diameter D=30.23±0.01mm. Both the cylinder and tube surfaces were polished and their diameters were precisely measured at 23°C. Two stainless steel turning covers and a special holding system were used to obtain rigid and concentric cylinder-tube connection. As a result, the obtained eccentricity was less than 0.1mm. The inlet edge of the cylinder was rounded with the radius of about R=0.3mm. Two test sections were investigated. The first one was equipped with a short (14mm) upstream chamber receiving air from the stub pipe 15mm in diameter through four orifices each 8mm in diameter. The air entering the gap was strongly disturbed (Fig. 2). The second one (not shown here) had a 3.3D long upstream chamber (tube of diameter D without the stub pipe) producing moderately disturbed inflow conditions. The tube of the test section was equipped with a set of 23 pressure taps having 0.7mm diameter. Distances between adjoining taps were 5mm. The first pressure tap was located precisely at the gap inlet.Pressure drop ΔP was measured with the electric micromanometer, and flow rate Q was measured with the calibrated laboratory rotary gasmeter. Uncertainty evaluation of the final results was performed in accordance with a 95% confidence interval to satisfy ISO document (2). Uncertainty of the pressure difference measurement ΔP was estimated to be ±2% for low and ±1% for moderate and high flow rates. Uncertainty of the volume flow rate measurement Q was about ±1.5% for low, ±1% for moderate, and ±0.5% for high flow rates. In the region of Re<1000 the uncertainties of Re and Le were estimated to be ±3.5% and ±5.5%, respectively. For Re>1000 they were ±2.5% and ±4.5%.As the diameter ratio is D∕d=1.0796<1.2 we may neglect the curvature effect and treat the annular duct as the parallel plates [C=96 in Eq. 3].In Fig. 3 the experimental data of the friction factor in the fully developed region for a strongly disturbed inflow condition versus Reynolds number are shown. It is seen that below the Reynolds number of Re≈1900 our experimental data agree very well with the laminar theory for the parallel plate flow. This Reynolds number is considerably lower (probably due to strong inflow disturbances) than the results of Beavers et al. (3) who determined the laminar-turbulent transition Reynolds number within the range from 2200 to 3400.A definition of the hydrodynamic entrance length Le proposed by Sparrow and Lin (4) was used, i.e., the length Le required to achieve the local incremental pressure drop number as about 98% of the incremental pressure drop number in the fully developed region K∞.In Fig. 4 the dimensionless pressure drop distribution ΔP*=(Px=0−Px)∕Pd along the duct for various Reynolds numbers and a graphic representation of the dimensionless entrance length Le*=Le∕Dh as well as the value of K∞ are shown [where Pd=0.5ρ(Q∕A)2, x is distance from the duct entrance]. The available data concerning the hydrodynamic entrance length for the flow between parallel plates (Table 1) and the measured entrance length for strongly and moderately disturbed inlet flows are presented in Fig. 5. The experimental relationship Le*=f(Re) seems to be linear and typical for laminar duct flow. Sudden change in the slope of the experimentally obtained entrance length at Re=2200 is attributed to laminar-turbulent transition. Higher values of Le* are observed for the disturbed inflow conditions. In this case the flow is likely not only in the course of developing flow but also in the process of damping of the large inlet disturbances. Moreover, some elongation of the entrance length is plausibly attributed to the flow separation from the cylinder edge and to the "vena contracta" phenomenon. As a result the measured pressure drop entrance length is larger than the theoretical values obtained for undisturbed inflow conditions (Table 1). As it is seen in Fig. 5 the dimensionless entrance length Le* at the Reynolds number Re=1900 is about 32 for the moderately disturbed inlet flow and about 35 for strongly disturbed inlet flow. The last value is the recommended distance from the duct inlet to the place in which the first flowmeter pressure tap should be located.The inherently linear annular-duct-type laminar flowmeter [Fig. 1d] was investigated with air. To achieve the inherent linearity a proper location of the upstream pressure tap (Le*⩾35) and the Reynolds number below Re⩽1900 are required. The flow characteristics of the flowmeter can be obtained theoretically using Eq. 3 for D∕d<1.2 and neither calibration nor software reading correction are necessary. This device can be relatively easily designed and produced in any laboratory workshop.

In 1911 M. Dehn posed the word and conjugacy problems for groups in general and provided algorithms which solved these problems for the fundamental groups of closed orientable two-dimensional manifolds. A crucial feature of these groups is that (with trivial exceptions) they are defined by a single relator r with the property that if s is a cyclic conjugate of r or r-1, with s ≠ r-1, there is very little cancellation in forming the product rs. Dehn’s algorithms have been extended to large classes of groups possessing presentations in which the defining relations have a similar small cancellation property. At first, investigations were concerned with the solution of the word problem for groups G presented as small cancellation quotients of a free group F. The theory was subsequently extended to the case where F is a free product, a free product with amalgamation, or an HNN extension. Moreover, strong results were obtained about algebraic properties; for example, one can classify torsion elements and commuting elements in small cancellation quotients.

We investigate the topological structure of flows on the Girl's surface which is one of two possible immersions of the projective plane in three-dimensional space with one triple point of self-intersection. First, we describe the cellular structure of the Boy's and Girl's surfaces and prove that there are unique images of the project plane in the form of a $2$-disk, in which the opposite points of the boundary are identified and this boundary belongs to the preimage of the $1$-skeleton of the surface. Second, we describe three structures of flows with one fixed point and no separatrices on the Girl's surface and prove that there are no other such flows. Third, we prove that Morse-Smale flows and they alone are structurally stable on the Boy's and Girl's surfaces. Fourth, we find all possible structures of optimal Morse-Smale flows on the Girl's surface. Fifth, we obtain a classification of Morse-Smale flows on the projective plane immersed on the Girl's surface. And finally, we describe the isotopic classes of these flows.

Papers of many mathematicians are devoted to the study of semisymmetric connections or metric connections with vector torsion on Riemannian manifolds. This type of connectivity is one of the three main types discovered by E. Cartan and finds its application in modern physics, geometry, and topology of manifolds. Geodesic lines and the curvature tensor of a given connection were studied by I. Agricola, K. Yano, and other mathematicians. In particular, K. Yano proved an important theorem on the connection of conformal deformations and metric connections with vector torsion. Namely: a Riemannian manifold admits a metric connection with vector torsion and the curvature tensor being equal to zero if and only if it is conformally flat. Although the curvature tensor of a hemisymmetric connection has a smaller number of symmetries compared to the Levi-Civita connection, it is still possible to define the concept of sectional curvature in this case. The question naturally arises about the difference between the sectional curvature of a semisymmetric connection and the sectional curvature of a Levi-Civita connection.This paper is devoted to the study of this issue, and the authors find the necessary and sufficient conditions for the sectional curvature of the semisymmetric connection to coincide with the sectional curvature of the Levi-Civita connection. Non-trivial examples of hemisymmetric connections are constructed when possible.

This paper gives a brief introduction and review of linear stability theory with emphasis on nonmodal effects. The temporal and spatial framework is discussed and the response to periodic and stochastic forcing is addressed. An extension to nonautonomous evolution problems is presented. Bypass transition due to the secondary instability of streaks is used as an example to illustrate the role of a nonmodal linear mechanism in the breakdown to turbulence.

Abstract. It is well known that a fractal set is not a submanifold of the ambient space. However, fractals arise as invariant subsets even in infinitely smooth conditions and the Minkowski dimension serves in this case as a characteristic of complexity of this scale. For example, when the equilibrium state during the Andronov-Hopf bifurcation losses its stability, the closure of the non-singular trajectory is a parametrically defined curve of the fractal type. In this work the fractal dimension of such curves is calculated. In addition, special two-parameter family of functions is studied such that Minkowski dimension of their graphs varies from 1 to 2. The obtained result allows us to implement a regular dynamic system with an isolated hyperbolic point such that the closure of two-dimensional stable manifold of this point may have Minkowski dimension greater than 2. To calculate the graph dimension, the segment of the argument defining the graph is split into two parts. The dimension of the first part of the graph can be estimated from above by direct calculation of the corresponding curve’s length. The upper estimation of the other part’s dimension is provided by means of the area of rectangle containing this curve. The lower estimation of the Minkowski dimension is based on calculating the cardinality of ε -distinguishable set of graph points.