Tube coordinate

Suppose we want to postulate a closed evolution equation for the tube coordinates only, without having to worry about the chain coordinates. Let us specify the tube by a set of Z + 1 points Vo. Vz- Alternatively, we may want to use Z connector vectors u,=V,-.,i - Vj. The first question we should ask is about the static properties of these coordinates, that is, what is the equilibrium joined probability distribution Peq(Vo, Vz). Until the tube is properly defined from microscopic 3D chain coordinates, there is no definitive or verifiable answer to this question. However, one can reasonably assume that at length scales much larger than the step size of the tube, the statistics should become that of the random walk following the statistics of the chain. In other words, for a pair of tube vertices V,- and Vj with i-j 1, we expect (see eqn [17])... 

Now we are ready to create the simplest tube model for the tube coordinates only. We assume that at each time interval r a random number f is generated, which can be either 1 or -1 with equal probability. The tube can either reptate one segment toward its head if f = 1 or toward its tail Vq if f = -1. When... 

The next important question now is how to calculate all observables of the previous section from the tube coordinates Vj(t). In the slithering snake model, we will not distinguish the chain from the tube and assume that N = Z and Ri = Vj. This means that R can be thought of as the centers of the blobs of size Ne = N/Z, and one understands that the results of such a model are not valid at timescales treptation motion is associated with the center-of-mass motion of the chain inside the tube, it is reasonable to assume that the jump time t scales linearly with the molecular weight, that is, t=Ztss. Thus, the parameters of the model areZ, a, and z s- Since the last two parameters are just imits of space and time (similar to b and zq in the Rouse model), Z is the only nontrivial parameter (analogous to N in the Rouse model). 

An additional difficulty arises when one tries to define the stress in tube models. Indeed, since the forces were not defined in the equation of motion, one carmot use the miaoscopic stress definition (eqn [ 1 ]). Thus, one has to make an additional assumption about the stress tensor expression as a function of the tube coordinates. The tube theory routinely uses an expression... 

One disadvantage of the slithering snake model is that the time step cannot be controlled in one step the chain moves exactly one tube segment. This in particular leads to the unphysical oscillations in i,mid(t) at early time in Figure 17. In order to resolve the motion on smaller timescales (and more importantly to account for fluctuations of the chain inside the tube, see below), one has to distinguish between the tube and chain coordinates. Now we introduce the main set of variables of the tube model the 3D tube coordinates V),(t), fe = 0...Z as in the previous section plus the one-dimensional (ID) chain coordinates inside the tube Xj( ), i = 0...N. In total, we have 3(Z-r 1) -r (N-f 1) variables, and their equations of motion are coupled. The main idea of the tube theory is that the chain inside the tube moves independent of the tube coordinates, whereas the tube segments are deleted at the ends when the chain does not occupy them any more, and are created when the chain sticks out of the tube. In the pure reptation model, only the center-of-mass of X coordinates moves according to... 

The stress can now be defined in at least four different ways. The first definition is eqn [58], which uses only the tube coordinates Vi,. This however ignores the fluctuations of number of monomers in each tube segment. To include these fluctuations, one should use the second definition, which uses both the tube coordinates Vi, and the ID chain coordinates x,-. To do so, one has to find the fractional monomer number /, associated with each tube vertex k. This can be done in several ways since the 1D chain can cross a point Lfe several times. The most symmetric way to deal with this is according to... 

Figure 6.11 Definition of the tube coordinate first used in Eq. 6.31.

T- and mapped to the image plane considering scaling (Si,Sy) of the coordinate axes and a shift Ci,Cy) of the center of the coordinate system. The distance between X-ray source and image intensifier tube is called /. 

The effect known either as electroosmosis or electroendosmosis is a complement to that of electrophoresis. In the latter case, when a field F is applied, the surface or particle is mobile and moves relative to the solvent, which is fixed (in laboratory coordinates). If, however, the surface is fixed, it is the mobile diffuse layer that moves under an applied field, carrying solution with it. If one has a tube of radius r whose walls possess a certain potential and charge density, then Eqs. V-35 and V-36 again apply, with v now being the velocity of the diffuse layer. For water at 25°C, a field of about 1500 V/cm is needed to produce a velocity of 1 cm/sec if f is 100 mV (see Problem V-14). 

Many experimental methods may be distinguished by whether and how they achieve time resolution—directly or indirectly. Indirect methods avoid the requirement for fast detection methods, either by detemiining relative rates from product yields or by transfonuing from the time axis to another coordinate, for example the distance or flow rate in flow tubes. Direct methods include (laser-) flash photolysis [27], pulse radiolysis [28]... 

Consider now the problem of steady motion in an infinitely long cylindrical tube of circular cross-section and radius a, and let (r,2) denote cylindrical coordinates about the tube axis. Since satisfies... 

Chromium (II) also forms sulfides and oxides. Chromium (II) oxide [12018-00-7], CrO, has two forms a black pyrophoric powder produced from the action of nitric acid on chromium amalgam, and a hexagonal brown-red crystal made from reduction of Cr202 by hydrogen ia molten sodium fluoride (32). Chromium (II) sulfide [12018-06-3], CrS, can be prepared upon heating equimolar quantities of pure Cr metal and pure S ia a small, evacuated, sealed quartz tube at 1000°C for at least 24 hours. The reaction is not quantitative (33). The sulfide has a coordination number of six and displays a distorted octahedral geometry (34). 

V Radius cylindrical and spherical coordinate distance from midplane to a point in a body i i for inner wall of annulus Vo for outer wall of annulus for inside radius of tube for distance from midplane or center of a body to the exterior surface of the body m ft... 

Figure 2.12. A flow tube used to derive one-dimensional flow equations in Lagrangian coordinates. Internal surfaces are massless, impermeable partitions to aid in visualizing elements of fluid in Lagrangian coordinates.

It is not common practice, but quite viable, to inject the flow marker at a predetermined volume offset from the polymer injection. This can be accomplished either with a second injection at a predetermined time into the run or by having two coordinated injection valves separated by a fixed volume of tubing. This approach can avoid many of the pitfalls described earlier. However, the mathematics of this correction is slightly different from that for a coinjected marker. The proper correction for the delayed volume injection is shown in Eq. (3) ... 

Plain tubes (either as solid wall or duplex) are available in carbon steel, carbon alloy steels, stainless steels, copper, brass and alloys, cupro-nickel, nickel, monel, tantalum, carbon, glass, and other special materials. Usually there is no great problem in selecting an available tube material. However, when its assembly into the tubesheet along with the resulting fabrication problems are considered, the selection of the tube alone is only part of a coordinated design. Plain-tube mechanical data and dimensions are given in Tables 10-3 and 10-4. 

Coordinated phosphate control charts assume either that all contribution to pH level is derived from phosphate or that the buffering action of phosphate is sufficient to overcome the presence of other alkaline species, such as amines. Neither assumption is true. This may lead operators to conclude perhaps that a higher than anticipated bulk water pH level (caused by the presence of amine) should be rectified by the addition of MSP. This action may lower localized Na P04 ratios below 2.2 1.0, producing acid phosphate corrosion (phosphate wastage) and resulting in tube thinning and ultimately tube failure. 

In mammals, ciliated cells line the respiratory air passages, the fallopian tubes, and the ventricles of the brain. The cilia beat in a coordinated manner in waves that propel fluids, suspended cells, and small particles along a surface. The motility of the sperm cell is provided by a single flagellum. 

Flow velocity field determined by PIV. Lean limit flames propagating upward in a standard cylindrical tube in methane/air and propane/ air mixtures, (a) Methane/air—laboratory coordinates, (b) propane/air—laboratory coordinates, (c) methane/air—flame coordinates, and (d) propane/air—flame coordinates. 

We demonstrate the procedure with an experiment conducted on a Bentheimer sandstone sample. For simplicity, we use a relatively thin sample and resolve only the two in-plane spatial coordinates. The sample is a rectangular parallelepiped shape having a length of 50 mm extending in the z direction, width 25 mm along the z2 direction and thickness 5 mm in the z3 direction. The sample was sealed laterally with epoxy and mounted in Plexiglass end-plates with O-rings and tube... 

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