Forced velocity, definition

Since the common engineering units for both mass and force are 1 lb, it is essential to retain gc in all force-mass relations. The interconversions may be illustrated with the example of viscosity whose basic definition is force/(velocity)(distance). Accordingly the viscosity in various units relative to that in SI units is... 

The mobility can also be viewed as the drift velocity that would be attained by the particles under unit external force. Recall (9.50), which is the mobility in the special case of an electrical force. By definition, the electrical mobility is related to the particle mobility by Be = qB, where q is the particle charge. A particle with zero charge, has a mobility given by (9.78) and zero electrical mobility. 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

Figure 2.1 The relationship between the applied force per unit area and the velocity used in the definition of viscosity.

Substituting this into Eq. (14.19) and then combining this with the definition of Re, Eq. (14.20), and the force equilibrium condition, Eq. (14.22), we obtain the following equation for the free-falling velocity ... 

The first pseudo force, Fi, is called the Coriolis force, and its magnitude is directly proportional to the angular velocity of the rotating frame of reference and the linear velocity of the particle in this frame. By definition, this force is perpendicular to the plane where vectors Vi and o are located, Fig. 2.3a, and depends on the mutual position of these vectors. The second fictitious force, F2, is called the centrifugal force. Its magnitude is directly proportional to the square of the angular velocity and the distance from the particle to the center of rotation. It is directed outward from the center and this explains the name of the force. It is obvious that with an increase of the angular velocity the relative contribution of this force... 

By definition, the anion-free water is free of salt. When pressure is applied to a clay-brine slurry to force out water (as that described in the experimental section), the solution that flows out of the cell should maintain the same chloride concentration as the brine s if the anion-free water is immobile. Otherwise, the concentration of the chloride decreases. Pressure forces water to flow through the pores with a certain velocity meanwhile, the pore size... 

By definition, the terminal velocity of a particle (ut) is the superficial gas velocity which suspends an isolated particle without translational motion—i.e., the terminal free fall velocity for that particle. From force balance on the particle, the terminal velocity for an approximately spherical particle can be shown to be... 

None of the methods currently used to study molecular dynamics can span the whole time range of motions of interest, from picoseconds to seconds and minutes. However, the structural resolution of a method is of equal importance. A method has to not only provide information about the existence of motions with definite velocities but also to identify what structural element is moving and what is the mechanism of motion. Computer simulation of molecular dynamics has proved to be a very important tool for the development of theories concerning times and mechanisms of motions in proteins. In this approach, the initial coordinates and forces on each atom are input into the calculations, and classical equations of motions are solved by numerical means. The lengthy duration of the calculation procedure, even with powerful modem computers, does not permit the time interval investigated to be extended beyond hundreds of picoseconds. In addition, there are strong... 

An interesting recent development is the application of an electron-nuclear-dynamics code [68] to penetration phenomena [69]. The scheme is capable of treating multi-electron systems and may he particularly useful for low-velocity stopping in insulating media, where alternative treatments are essentially unavailable. However, conceptional problems in the data analysis need attention, such as separation of nuclear from electronic stopping and, in particular, the very definition of stopping force as discussed in Section 5.2. 

Substituting all the possible combinations of characteristics, i.e. values of p and q, info equation 1.10 gives rise to a number of differenf definitions of the mean size of a distribution. At minimum fluidization the drag force acting on a particle due to the flow of fluidizing gas over the particle is balanced by the net weight of fhe particle. The former is a function of surface area and the latter is proportional to particle volume. Consequently the surface-volume mean diameter, with p = 2 and = 3, is the most appropriate particle size to use in expressions for minimum fluidizing velocity. It is defined by equafion 1.11... 

A particle drag coefficient Cd can now be defined as the drag force divided by the product of the dynamic pressure acting on the particle (i.e. the velocity head expressed as an absolute pressure) and the cross-sectional area of the particle. This definition is analogous to that of a friction factor in conventional fluid flow. Hence... 

It will be assumed throughout that H and are symmetric positive-definite tensors. We write the mobility as a contravariant tensor, with raised bead indices, to reflect its function the mobility H may be contracted with a covariant force vector Fv (e.g., the derivative of a potential energy) to produce a resulting contravariant velocity Fy. 

Just as process translation or scaling-up is facilitated by defining similarity in terms of dimensionless ratios of measurements, forces, or velocities, the technique of dimensional analysis per se permits the definition of appropriate composite dimensionless numbers whose numeric values are process-specific. Dimensionless quantities can be pure numbers, ratios, or multiplicative combinations of variables with no net units. 

In Chapter 2 considerable effort is devoted to establishing the relationship between the stress tensor and the strain-rate tensor. The normal and shear stresses that act on the surfaces of a fluid particle are found to depend on the velocity field in a definite, but relatively complex, manner (Eqs. 2.140 and 2.180). Therefore, when these expressions for the forces are substituted into the momentum equation, Eq. 3.53, an equation emerges that has velocities (and pressure) as the dependent variables. This is a very important result. If the forces were not explicit functions of the velocity field, then more dependent variables would likely be needed and a larger, more complex system of equations would emerge. In terms of the velocity field, the Navier-Stokes equations are stated as... 

These work contributions have a negative sign because, by definition, positive velocities are flowing in the coordinate direction and positive stresses are opposite to the coordinate directions. The work is a scalar quantity the subscripts on W[ simply indicate the face and do not represent vector components as would be the case for the force vector. 

As a layer of gas at one velocity is pulled across an adjacent layer of gas at a slightly different velocity, gas in the faster layer tends to be slowed down by the interaction, and gas in the faster layer tends to speed up. There is velocity or, more precisely, momentum transfer between the layers. Thus it takes a force to maintain the velocity gradient across the fluid. (Recall the definition that force is the time rate of change of momentum.) Fundamentally, the viscosity is a transport property associated with momentum transfer. 

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