Steady velocity

If we were to forget that the flow of current is due to a random motion which was already present before the field was applied—if we were to disregard the random motion entirely and assume that each and every electron, in the uniform field X, moves with the same steady velocity, the distance traveled by each electron in unit time would be the distance v used in the construction of Fig. 16 this is the value which would lead to a current density j under these assumptions, since all electrons initially within a distance v of the plane AB on one side would cross AB in unit time, and no others would cross. Further, in a field of unit intensity, the uniform velocity ascribed to every electron would be the u of (34) this quantity is known as the mobility of the charged particle. (If the mobility is given in centimeters per second, the value will depend on whether electrostatic units or volts per centimeter are used for expressing the field strength.)... 

An evaluation of the retardation effects of surfactants on the steady velocity of a single drop (or bubble) under the influence of gravity has been made by Levich (L3) and extended recently by Newman (Nl). A further generalization to the domain of flow around an ensemble of many drops or bubbles in the presence of surfactants has been completed most recently by Waslo and Gal-Or (Wl). The terminal velocity of the ensemble is expressed in terms of the dispersed-phase holdup fraction and reduces to Levich s solution for a single particle when approaches zero. The basic theoretical principles governing these retardation effects will be demonstrated here for the case of a single drop or bubble. Thermodynamically, this is a case where coupling effects between the diffusion of surfactants (first-order tensorial transfer) and viscous flow (second-order tensorial transfer) takes place. Subject to the Curie principle, it demonstrates that this retardation effect occurs on a nonisotropic interface. Therefore, it is necessary to express the concentration of surfactants T, as it varies from point to point on the interface, in terms of the coordinates of the interface, i.e.,... 

In the present discussion only the problem of steady flow will be considered in which the time average velocity in the main stream direction X is constant and equal to ux. in laminar flow, the instantaneous velocity at any point then has a steady value of ux and does not fluctuate. In turbulent flow the instantaneous velocity at a point will vary about the mean value of ux. It is convenient to consider the components of the eddy velocities in two directions—one along the main stream direction X and the other at right angles to the stream flow Y. Since the net flow in the X-direction is steady, the instantaneous velocity w, may be imagined as being made up of a steady velocity ux and a fluctuating velocity ut, . so that ... 

As a fluid is deformed because of flow and applied external forces, frictional effects are exhibited by the motion of molecules relative to each other. The effects are encountered in all fluids and are due to their viscosities. Considering a thin layer of fluid between two parallel planes, distance y apart as shown in Figure 3.4 with the lower plane fixed and a shearing force F applied to the other, since fluids deform continuously under shear, the upper plane moves at a steady velocity ux relative to the fixed lower plane. When conditions are steady, the force F is balanced by an internal force in the fluid due to its viscosity and the shear force per unit area is proportional to the velocity gradient in the fluid, or ... 

Fig. 6. (a) Schematic view of an extruder channel with an undulating baffle, (b) B, C, D Steady flow streamlines with and without baffles for initial condition A. E, F Mixing in a cavity flow with an oscillating baffle. The upper plate moves with a steady velocity while the lower plate with the baffle undergoes linear oscillatory motion (Jana, Tjahjadi, and Ottino, 1994). 

There are, in principle, three ways in which material may be transported to the electrode surface diffusion, convection and migration. Of these, perhaps the most straightforward is migration, which simply consists of the movement of a charged particle under the influence of an electric field. Experimentally, it is well established that after an extremely short time an ion in solution in an electric field will behave as if it had acquired a steady velocity in the direction of the field. The reason why a steady velocity is established rather... 

Figure 8.9 Control volume energy conservation for a thermally thick solid with flame spread steady velocity, Up...

What is the mass of a sphere of material of density 7500 kg/m3 which falls with a steady velocity of 0.6 m/s in a large deep tank of water ... 

When an explosive gas mixture is placed in a tube having one or both ends open, a combustion wave can propagate when the tube is ignited at an open end. This wave attains a steady velocity and does not accelerate to a detonation wave. 

The system considered in this chapter is a rigid or fluid spherical particle of radius a moving relative to a fluid of infinite extent with a steady velocity U. The Reynolds number is sufficiently low that there is no wake at the rear of the particle. Since the flow is axisymmetric, it is convenient to work in terms of the Stokes stream function ij/ (see Chapter 1). The starting point for the discussion is the creeping flow approximation, which leads to Eq. (1-36). It was noted in Chapter 1 that Eq. (1-36) implies that the flow field is reversible, so that the flow field around a particle with fore-and-aft symmetry is also symmetric. Extensions to the creeping flow solutions which lack fore-and-aft symmetry are considered in Sections II, E and F. 

Prediction of fluid motion, drag, and transfer rates becomes much more complex when the motion is unsteady. Dimensional analysis gives an indication of the problems. A rigid sphere moving with steady velocity in a gravitational field is governed by an equation of the general form... 

Zel dovich Kompaneets (Ref 21) stated that the values of steady velocities of spherically propagating detonation waves in gaseous explosions are the same as the values measured in tubes at corresponding mixture ratios... 

When an expl of deton velocity ca 8000m/se c was detonated in contact with an expl of 5000 m/sec, the velocity of the first expl was "carried over into the second expl for a distance of less than a few mm from the boundary. The same effect was shown when the transition was from the low component to the high. This result was, however, not obtd if, for example, the second component was loosely packed and therefore of low density. In this case the transition in velocity was gradual and took several centimeters to reach a characteristic steady velocity... 

This layer of diffuse charge qd will experience an electric force q C. The charged layer on the solution side will begin to move. But the motion of the charged fluid is opposed by a viscous force that is once again given [see Eq. 6.309)] by T v/K-1) or (47iT A /0e)9dv. When the electrolyte attains a steady velocity, the electric and viscous forces are exactly equal. Hence,... 

Travelling wavefronts, are familiar in non-isothermal gas-phase systems, as flames. Once established, these generally propagate along a tube into a stationary reactant mixture at a steady velocity or can be stabilized on-a burner,... 

In dimensionless terms, the wave travels with a steady velocity... 

Again the scenario we envisage is similar to that shown qualitatively in Fig. 11.7 we expect our best chance of such behaviour if the decay rate is small, i.e. k2 1. The reaction wave has a leading front moving with a steady velocity c1, through which most of the conversion of A to B occurs. After this front, the dimensionless concentration of A is almost zero and that of B is almost unity. At some distance, the first front is followed by a recovery wave, possibly more diffuse, in which A is completely removed and the autocatalyst also decays. The velocity of the recovery wave is c2. If ct exceeds c2, the first front will move away from the second, so the pulse will increase in width if c, = c2, the pulse will move with a constant shape if, however, c2 exceeds c, we can expect the second wave to catch the first, in which case propagation may fail. 

Detonation velocity (also called detonation rate) is the speed at which detonation progresses thru an expl- Commonly it is designated by the symbol D, and it is usually understood to be a steady velocity, and not a transient velocity such as that observed in build-up to detonation or in overdriven detonations. The contents of this article are limited to steady high-velocity detonations. Detonation velocity is the most easily and most accurately measurable detonation parameter (other detonation parameters are detonation pressure, particle velocity, temp, etc). Consequently there exists a large body of exptl and theoretical literature on this subject. Much of this literature, up to about 1965, was summarized in Vol 4, pp D223-224,232-235, 242-245,280,352,356,362,384-389,460-461, 463-464, 629-637, 641-652, 657-660, 663—675 and 718—721. The present article is a selected review of published information on detonation velocity from 1967 to date... 

At sufficiently large values of X the saturation curves approach a constant pattern form, and thereafter the concentration front progress through the columii at a steady velocity, governed by the capacity of the adsorbent and the feed concentration, with no further change in the shape of the curve. Such behavior is characteristic of systems with a favorable equilibrium isotherm (12). The constant pattern limit is reached when the dimensionless concentration profile in fluid phase and adsorbed phase become practically coincident, and the asymptotic form of the break-... 

The problem of a kinematic dynamo in a steady velocity field can be treated mathematically as a problem of the effect of a small diffusion or round-off error on the Kolmogorov-Sinai entropy (or Lyapunov exponent) of a dynamical system which is specified by the velocity field v. This problem, on which Ya.B. worked actively, therefore has a general mathematical nature as well, and each step toward its solution is simultaneously a step forward in several seemingly distant areas of modern mathematics. 

The solution of the problem (8)-(10) is some approximation to the solution of the original problem (1)—(3) for sufficiently large times when the influence of the initial conditions disappears. To find the degree of correspondence between the solutions we performed numerical calculations on the determination of the non-steady velocity and temperature distribution in the combustion wave using direct and approximate methods. 

The physical processes which occur here axe as follows the externally heated c-phase begins to burn, but it burns at a velocity exceeding the steady velocity and with a temperature of the combustion products which exceeds the calculated theoretical temperature. Such intensive combustion occurs... 

The condition of the limit of steady combustion is related to the temperature dependence of the combustion velocity. The last formula states that steady combustion is possible only at an initial temperature at which the combustion velocity is not less than 1/e, i.e., is not less than 37% of the combustion velocity of the c-phase heated completely to the temperature Tb. Above the limit the dependence of the steady velocity on the initial temperature does not undergo any changes. 

Practical application of the conditions (11) is complicated by the fact that when it arises, the detonation wave very often has a velocity and pressure which are larger than in the steady regime (see, e.g., data of Bone, Fraser, and Wheeler [25], according to which, in a mixture of CO and 02, the detonation velocity at the moment when it appears reaches 3000 m/sec, whereas the steady velocity is equal to 1760 m/sec. References to superpressures and breakage of the tube at the point where detonation begins are numerous). 

Figure 1 schematically depicts the buildup of a steady velocity profile for a fluid contained between two plates, where one plate is held stationery and the other set in motion. The lack of slip condition requires that the fluid velocity be zero at the stationery wall and the same as the solid at the moving wall. The lack of slip at the walls is the basic premise of rheometry and the interpretation of rheometrical data is usually based on the assumption of no slip at the boundary. 

When a charged particle q moves with a steady velocity uE through a fluid under an electric field Ef, the electrostatic force on the particle is counter-balanced by the fluid drag on the particle. For globular proteins, the drag force can be approximated from Stokes law. Therefore, the balance is... 

We consider the motion of a single spherical particle immersed in a stationary fluid and falling at a constant velocity and we examine the forces exerted on the particle to estimate the steady velocity of the particle relative to the fluid, the particle terminal fall velocity, ut. It is important to emphasise at this point that the equilibrium conditions experienced by the particle falling at ut are equivalent to that of a motionless particle suspended in an upwardly flowing fluid with velocity ut, as represented in Figure 5. 

This is termed the boundary layer momentum integral equation. As previously mentioned, it is equally applicable to laminar and turbulent flow. In laminar flow, u is the actual steady velocity while in turbulent flow it is the time averaged value. 

For a nonlinear system the behavior depends on the shape of the isotherm. If the isotherm is unfavorable (Fig. 11),/(c) increases with concentration so that w decreases with concentration. This leads to a spreading profile, as illustrated in Figure 12b. However, if the isotherm is favorable (in the direction of the concentration change), an entirely different situation arises. ThenJ(c) decreases with concentration so that w increases with concentration. This would lead to the physically unreasonable overhanging profiles shown in Figure 12a. In fact, what happens is that the continuous solution is replaced by the equivalent shock transition so that response becomes a shock wave which propagates at a steady velocity vv given by... 

When an ion of valence z is moving with a steady velocity through a solution, under the influence of an electrical force esF, where F is the... 

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