Initial velocity, definition

A more precise definition would include conditioning on the random initial velocity and compositions /li, , x Uo,. o.Y Vb XIY), V o, y 0- However, only the conditioning on initial location is needed in order to relate the Lagrangian and Eulerian PDFs. Nevertheless, the initial conditions (Uo, o) for a notional particle must have the same one-point statistics as the random variables U(Y, to) and (V. to). 

In Fig. 3.6 the dashed line OL is so drawn that the velocity changes are confined between this line and the trace of the wall. Because the velocity lines are asymptotic with respect to distance from the plate, it is assumed, in order to locate the dashed line definitely, that the line passes through all points where the velocity is 99 percent of the bulk fluid velocity Line OL represents an imaginary surface that separates the fluid stream into two parts one in which the fluid velocity is constant and the other in which the velocity varies from zero at the wall to a velocity substantially equal to that of the undisturbed fluid. This imaginary surface separates the fluid that is directly affected by the plate from that in which the local velocity is constant and equal to the initial velocity of the approach fluid. The zone, or layer, between the dashed line and the plate constitutes the boundary layer. 

SEPARATION FROM VELOCITY DECREASE Boundary-layer separation can occur even where there is no sudden change in cross section if the cross section is continuously enlarged. For example, consider the flow of a fluid stream through the trumpet-shaped expander shown in Fig. 5.16. Because of the increase of cross section in the direction of flow, the velocity of the fluid decreases, and by the Bernoulli equation, the pressure must increase. Consider two stream filaments, one, aa, very near the wall, and the other, bb, a short distance from the wall. The pressure increase over a definite length of conduit is the same for both filaments, because the pressure throughout any single cross section is uniform. The loss in velocity head is, then, the same for both filaments. The initial velocity head of filament aa is less than that of filament bb, however, because filament aa is nearer... 

Galileo s Acceleration Formula The acceleration (a) represents the velocity increase in unit time. The total increase over a period t is therefore given by at. Hence the final velocity is w = w -f- at, where u is the initial velocity. A graph of the acceleration is given by the straight line AB, on axes of velocity vs time. The distance traversed at uniform velocity u over a period t is given, by definition, as s = ut, which corresponds to the area of the shaded rectangle. The total area OABC represents the distance traversed in the accelerated motion, i.e. 

We are about to provide a microscopic definition of the collision cross-section. Two ingredients come in. One is the definition of a collision. On the basis of Newton s laws of motion we took it that a collision occurs whenever two molecules exercise a force on one another. The outcome of the collision can be that a chemical reaction took place, or only that the two molecules deflected from flieir unperturbed straight-line motion, or anything in between. Whatever the outcome, when a force due to the potential acted, a collision is said to have taken place. Nor need this force be repulsive, and indeed the long-range part of the force is in general weakly attractive. Section 2.1.8. The other point is that the cross-section is that area, drawn in a plane perpendicular to the initial velocity, that the relative motion of the molecules needs to cross if a coUision is to take place. 

Worth noting that there is no monotonically form between 0 and 1 other that that of equation (1.35) to reproduce basic Michaelis-Menten term (1.34) when approximated for small x = [S t) K For instance, if one decides to use exp(-x then the unreactive probability will give 1/(1+x ) as the approximation for small x, definitely different of what expected in basic Michaelis-Menten treatment (1.34). This way, the physico-chemical meaning of Eq. (1.36) is that the Michaelis-Menten term (1.34) and its associated kinetics apply to fast enzymatic reactions, i.e., for fast consumption of [ S](t), which also explains the earlier relative success in applying linearization and graphical analysis to the initial velocity equation (1.18). 

Momentum is, by definition, the product of mass and velocity. For the molecule in Figure 6-18, the initial velocity is +u therefore, the momentum before the collision is +mU . When the molecule hits the wall, it is reflected without losing any energy and travels in the opposite direction. The velocity and momentum of the particle after the collision are, respectively, —u and —mu. The momentum change of the molecule is the final momentum minus the initial momentum —mu — +mu ) = —2mu. The momentum transfer to the wall is +2mu. This represents the momentum transfer per collision. We must multiply this quantity by the number of collisions per unit time the molecule makes with the end wall. The time between collisions is equal to the time it takes for the molecule to travel twice the length of the box, 2L. For a particle moving with speed u, the time it takes to travel a distance 2L is t = 2LjUy.. Because IL/Uy. is the time between collisions, the reciprocal of this quantity is the number of collisions per unit time or the collision frequency. Notice that the collision frequency, Uy.j2h, is directly proportional to the molecular speed. 

What this shows is that, from the definition of off-bottom motion to complete uniformity, the effect of mixer power is much less than from going to on-bottom motion to off-bottom suspension. The initial increase in power causes more and more solids to be in active communication with the liquid and has a much greater mass-transfer rate than that occurring above the power level for off-bottom suspension, in which slip velocity between the particles of fluid is the major contributor (Fig. 18-23). 

That the terminal acceleration should most likely vanish is true almost by definition of the steady state the system returns to equilibrium with a constant velocity that is proportional to the initial displacement, and hence the acceleration must be zero. It is stressed that this result only holds in the intermediate regime, for x not too large. Hence and in particular, this constant velocity (linear decrease in displacement with time) is not inconsistent with the exponential return to equilibrium that is conventionally predicted by the Langevin equation, since the present analysis cannot be extrapolated directly beyond the small time regime where the exponential can be approximated by a linear function. 

As shown above in (6.162), the Lagrangian fluid-particle PDF can be related to the Eulerian velocity, composition PDF by integrating over all initial conditions. As shown below in (6.168), for the Lagrangian notional-particle PDF, the same transformation introduces a weighting factor which involves the PDF of the initial positions y) and the PDF of the current position /x.(x t). If we let V denote a closed volume containing a fixed mass of fluid, then, by definition, x, y e V. The first condition needed to reproduce the Eulerian PDF is that the initial locations be uniform ... 

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... 

Unless 7 1, all terms in Eq. (11-33) must be retained. Since Eq. (11-30) has no formal justification, the individual terms cannot definitely be ascribed to added mass or history effects. Even so, the relative magnitudes of the terms are of interest. Figure 11.7 shows the three terms for specific values of 7 and Rejs, expressed as fractions of the immersed particle weight. Added mass dominates initially history passes through a maximum and decays slowly steady drag increases monotonically to become the sole component at the terminal velocity. Both A and Ah depart from unity early in the motion. For smaller Rexs, history may be the dominant drag component for a brief period (02). 

In eq. (13-1) the first equality gives the definition of the oscillator strength in the dipole-length representation, while the second equality gives the definition in the momentum (or velocity) representation. As usual, and T, are the total molecular wavefunctions for the final and initial states, and Ef and Et are the energies of the final and initial states, respectively. 

The Michaelis constant, KM, for an enzyme-substrate interaction has two meanings (1) Ku is the substrate concentration that leads to an initial reaction velocity of V" /2 or, in other words, the substrate concentration that results in the filling of one-half of the enzyme active sites, and (2) KM = (k2 + ki)/kv The second definition of Ku has special significance in certain... 

In the case of an orifice, Pr = 1. For other types of pipes and ducts Pr < 1. A significant part of assessing molecular flow in ducts involves the estimation of Pr. An initial assumption is that molecules arrive at the entrance plane of a duct with an isotropic velocity distribution. Conductance under molecular flow conditions is independent of the pressure but obviously the throughput is proportional to the Ap as stated by the definition of C (e.g Equation (2.6)). 

Using the experimental values for the width of the traveling wave front (portion be, Fig. 8), let us estimate the propagation velocity for the case of a thermal mechanism based on the Arrhenius law of heat evolution from the known relationship U = a/d, where a 10"2 cm2/s is the thermal conductivity determined by the conventional technique. We obtain 5 x 10"2 and 3 x 10-2cm/s for 77 and 4.2 K, respectively, which are below the experimental values by about 1.5-2 orders of magnitude. This result is further definite evidence for the nonthermal nature of the propagation mechanism of a low-temperature reaction initiated by brittle fracture of the irradiated reactant sample. 

The subject of molecular beam kinetics is very extensive and in this section, therefore, we will deal only briefly with the relevant aspects of the topic. Molecular beam sources are often thermal, operating as a flow system with a gas or a vapour from a heated oven. The velocity distribution of species in such beams is Maxwell—Boltzmann in form. For many experiments, this does not provide sufficient definition of initial translational energy and some form of velocity selection may be used [30], usually at the expense of beam intensity. 

In the Mint model, we have to take into account the following considerations (i) the initial filtration coefficient Xq, which is a parameter, presents a constant value after time and position (ii) the detachment coefficient, which is another constant parameter (iii) the quantity of the suspension treated by deep filtration depends on the quantity of the deposited solid in the bed this dependency is the result of the definition of the filtration coefficient (iv) the start of the deep bed filtration is not accompanied by an increase in the filtration efficiency. These considerations stress the inconsistencies of the Mint model 1. valid especially when the saturation with retained microparticles of the fixed bed is slow 2. unfeasible to explain the situations where the detachment depends on the retained solid concentration and /or on the flowing velocity 3. unfeasible when the velocity of the mobile phase inside the filtration bed, varies with time this occurrence is due to the solid deposition in the bed or to an increasing pressure when the filtration occurs with constant flow rate. Here below we come back to the development of the stochastic model for the deep filtration process. 

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