Limiting velocity

Under the action of forces due to other particles j, or to external fields, a, given particle i is accelerated as  

Due to the solvent, a friction force appears. For a particle submitted to 

External forces on particles can be of origins other than electrostatic. Let us consider, as another example, the case of diffusion forces. 

Diffusion forces derive from macroscopic chemical potentials. In the simplest case, of an ideal solution we have 

The velocity taken under the action of the diffusion force by particle i is then  

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The equation proposed by Moore (1967) tends to follow the upper-limit velocity. This is not surprising, because the equation was based upon high levels of energy. Despite its simplicity, its results compare fairly well with other models for both low and high energy levels. 

A new tower diameter can be assumed, or the limiting velocities can be calculated, and the diameter set from these. For illustration purposes, use the latter approach ... 

In fact, ut is the limiting velocity of the ion in an electric field of unit strength, with the dimensions cms-1 per Vcm-1 or cm2V-1s-1. By inserting eqn. 2.14 into eqn. 2.16 we find... 

The assumption of a steady state corresponds to laminar flow of the liquid. This is fulfilled only under certain conditions (limited velocity of the liquid, smooth phase boundary between the flowing liquid and the other phase, etc.). Otherwise, turbulent flow occurs, where the local velocity depends on time, pulsation of the system, etc. Mathematically the turbulent flow problem is a very difficult task and it is often doubtful whether, in specific cases, it is possible to obtain any solution at all. 

Williams (1964) derived the relation T = kBTrQV3De2, where T is the recombination time for a geminate e-ion pair at an initial separation of rg, is the dielectric constant of the medium, and the other symbols have their usual meanings. This r-cubed rule is based on the use of the Nernst-Einstein relation in a coulom-bic field with the assumption of instantaneous limiting velocity. Mozumder (1968) criticized the rule, as it connects initial distance and recombination time uniquely without allowance for diffusional broadening and without allowing for an escape probability. Nevertheless, the r-cubed rule was used extensively in earlier studies of geminate ion recombination kinetics. 

This limiting velocity ( obsd caic) = (3 1) X 10 s is only 0.06% of the value of fesoiv = 0.049 s. This provides evidence that the ion pair reaction intermediate is trapped by solvent less than once for every 1000 times the... 

The effect of suction velocity on the flame stabilization can be observed in Fig. 18.5, where the suction mass flow rate is kept constant while varying T. For a given there is a limiting velocity ratio —U2/U1) below which the flame will blow out. For example, at = 1, the limiting velocity ratio is about 0.1. At higher values of F, this ratio decreases as shown in the figure. 

Also, the data for different gap widths are included in the figure. The gap width seems to have an insignificant effect on the limiting velocity ratio. 

According to Stokes the limiting velocity imparted to a sphere moving through a liquid of viscosity rj under a constant applied force F is given by... 

According to Stokes law the limiting velocity of a spherical particle of radius r moving through a liquid of viscosity vf under the influence of a force F is... 

The stability of many colloidal solutions depends critically on the magnitude of the electrostatic potential ( /q) at the surface of the colloidal particles. One of the most important tasks in colloid science is therefore to obtain an estimate of V /o under a wide range of electrolyte conditions. In practice, one of the most convenient methods for obtaining /o uses the fact that a charged particle will move at some constant, limiting velocity under the influence of an applied electric field. Even quite small particles (i.e. <1 pm) can be observed using a dark-field micro-... 

Critical Detonation Velocity and Limiting Velocity. See Detonation Velocity, Critical and Limiting... 

The limiting density, P, may be defined as the lowest p at which. deton velocity will reach its maximum value, called limiting velocity, De. For HE s like TNT, PETN, RDX,... 

Although erosion is not serious in some cases at velocities as high as 10-15 ft/sec, conservative practice in the absence of specific knowledge limits velocities to 5-6 ft/sec. 

These results indicate that in the present linear elastic model, the limiting velocity for the screw dislocation will be the speed of sound as propagated by a shear wave. Even though the linear model will break down as the speed of sound is approached, it is customary to consider c as the limiting velocity and to take the relativistic behavior as a useful indication of the behavior of the dislocation as v — c. It is noted that according to Eq. 11.20, relativistic effects become important only when v approaches c rather closely. 

The behavior of an edge dislocation is more complicated since its displacement field produces both shear and normal stresses. The solution consists of the superposition of two terms, each of which behave relativistically with limiting velocities corresponding to the speed of transverse shear waves and longitudinal waves, respectively [2, 4, 5]. The relative magnitudes of these terms depend upon v. 

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