Velocity arbitrary

Initial velocity (arbitrary units) [S] (mM) Without inhibitor With inhibitor... 

The Champ-Sons model has been developed to quantitatively predict the field radiated by water- or solid wedge- eoupled transdueers into solids. It is required to deal with interfaces of complex geometry, arbitrary transducers and arbitrary excitation pulses. It aims at computing the time-dependent waveform of various acoustical quantities (displacement, velocity, traction, velocity potential) radiated at a (possibly large) number of field-points inside a solid medium. 

Flow which fluctuates with time, such as pulsating flow in arteries, is more difficult to experimentally quantify than steady-state motion because phase encoding of spatial coordinate(s) and/or velocity requires the acquisition of a series of transients. Then a different velocity is detected in each transient. Hence the phase-twist caused by the motion in the presence of magnetic field gradients varies from transient to transient. However if the motion is periodic, e.g., v(r,t)=VQsin (n t +( )q] with a spatially varying amplitude Vq=Vq(/-), a pulsation frequency co =co (r) and an arbitrary phase ( )q, the phase modulation of the acquired data set is described as follows ... 

Circulating fluidized beds (CFBs) are high velocity fluidized beds operating well above the terminal velocity of all the particles or clusters of particles. A very large cyclone and seal leg return system are needed to recycle sohds in order to maintain a bed inventory. There is a gradual transition from turbulent fluidization to a truly circulating, or fast-fluidized bed, as the gas velocity is increased (Fig. 6), and the exact transition point is rather arbitrary. The sohds are returned to the bed through a conduit called a standpipe. The return of the sohds can be controUed by either a mechanical or a nonmechanical valve. 

Consider a body undergoing a smooth homogeneous admissible motion. In the closed time interval [fj, fj] with < fj, let the motion be such that the material particle velocity v(t) and deformation gradient /"(t), and hence (r), and p(r), have the same values at times tj and tj. Such a finite smooth closed cycle of homogeneous deformation will be denoted by tj). Consider an arbitrary region in the body of volume which has a smooth closed boundary of surface area with outward unit normal vector n. The work W done by the stress s on and by the body force A in during... 

The velocity and density have the same values at the end of the closed cycle as at the beginning. Thus, the kinetic energy is the same at the beginning and the end, and the first term in (5.36) vanishes. The work assumption (5.34) is expected to hold for any arbitrary region of the body and must therefore hold locally at every material particle, so that (5.36) reduces to... 

Note that the spatial velocity (u) is arbitrary and may be the material velocity (u). If the spatial velocity is the material velocity (u = v), then the region of space moves with the material and the Lagrangian forms of the equations are generated. If the spatial velocity is zero, then the region of space is fixed and the equations take the Eulerian form. 

Arbitrary-Lagrangian-Eulerian (ALE) codes dynamically position the mesh to optimize some feature of the solution. An ALE code has tremendous flexibility. It can treat part of the mesh in a Lagrangian fashion (mesh velocity equation to particle velocity), part of the mesh in an Eulerian fashion (mesh velocity equal to zero), and part in an intermediate fashion (arbitrary mesh velocity). All these techniques can be applied to different parts of the mesh at the same time as shown in Fig. 9.18. In particular, an element can be Lagrangian until the element distortion exceeds some criteria when the nodes are repositioned to minimize the distortion. 

Semi-Theoretical and Empirical Velocity Fields Since the use of formulas to calculate the velocities outside an arbitrary opening could be very tedious, only some examples of these formulae are given. These calculations are best done on computers and there are some dedicated programs to calculate and visualize the flow fields outside exhaust openings. There could sometimes be problems when calculating the velocity field outside an opening close to... 

Here v is the space- and time-dependent velocity field, p is the density of the fluid, p is the local pressure, v is the kinematic viscosity, and / is some arbitrary body-force acting on each small element of the fluid (gravitation, for example). 

The volume-source method is not only useful in a spherical approach, but can also be used in more arbitrary geometries, where it is possible to express the volume source strength in a product of burning velocity and flame surface area ... 

The double integral in Equation (8.4) is a fairly general definition of the mixing-cup average. It is applicable to arbitrary velocity profiles and noncircular cross sections but does assume straight streamlines of equal length. Treatment of curved streamlines requires a precise and possibly artificial definition of the system boundaries. See Nauman and Buffham. ... 

Consider Equations (6-10) that represent the CVD reactor problem. This is a boundary value problem in which the dependent variables are velocities (u,V,W), temperature T, and mass fractions Y. The mathematical software is a stand-alone boundary value solver whose first application was to compute the structure of premixed flames.Subsequently, we have applied it to the simulation of well stirred reactors,and now chemical vapor deposition reactors. The user interface to the mathematical software requires that, given an estimate of the dependent variable vector, the user can return the residuals of the governing equations. That is, for arbitrary values of velocity, temperature, and mass fraction, by how much do the left hand sides of Equations (6-10) differ from zero ... 

Here V is the barycentrie velocity. For a system in mechanical equilibrium, V can be replaced by any arbitrary velocity and therefore it is replaced here by the velocity of the volume fixed reference w (=0). Then Eq. (37) becomes... 

Figure 1.24 Outlet concentration and specific molar flux of product (PI) in arbitrary units as a function of the space velocity of a feed component.

Am(q) and Am(p) are masses which may have arbitrary values, and they are measured in kilograms. As follows from Newton s second law, mass is a quantitative measure of inertia, since with an increase of mass the rate of a change of the particle velocity for a fixed force becomes smaller. Also is the vector ... 

Assume that origins of two Cartesian systems of coordinates are located at the same point and the frame of reference P rotates about a point 0 of the frame P with constant angular velocity co. Let us imagine two planes, one above another, so that the upper plane P rotates and, correspondingly, unit vectors iiand ji change their direction, Fig. 2.2b. Consider an arbitrary point p, which has coordinates x, y on the plane P and xi, yi on P, and establish relationships between these pairs of coordinates. For the radius vector of the point p in both frames we have... 

In electrochemical cells we often find convective transport of reaction components toward (or away from) the electrode surface. In this case the balance equation describing the supply and escape of the components should be written in the general form (1.38). However, this equation needs further explanation. At any current density during current flow, the migration and diffusion fluxes (or field strength and concentration gradients) will spontaneously settle at values such that condition (4.14) is satisfied. The convective flux, on the other hand, depends on the arbitrary values selected for the flow velocity v and for the component concentrations (i.e., is determined by factors independent of the values selected for the current density). Hence, in the balance equation (1.38), it is not the total convective flux that should appear, only the part that corresponds to the true consumption of reactants from the flux or true product release into the flux. This fraction is defined as tfie difference between the fluxes away from and to the electrode ... 

Thus, both the angular frequency u> k) and the phase velocity Uph are dependent on the choice of the zero-level of the potential energy and are therefore arbitrary neither has a physical meaning for a wave packet representing a particle. 

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