Relation Between Force and Velocity

FIGURE 6.4. Vertical extraction of a plate from a pool of liquid. 

FIGURE 6.5. Vertical extraction Dependence of the velocity on the dynamical angle 9d.  

The dynamical properties of the triple line involve local phenomena—on a molecular scale—in the immediate vicinity of the line, as well as longer-range phenomena in the form of viscous flows in the overall liquid body. We shall see that both aspects are generally at play. Only in one particular case (when 0 is small) do macroscopic effects tend to dominate. 

FIGURE 6.6. Flow of a spreading liquid 6d Ob)- The velocity V of the line is the average of the velocity profile sketched in the figure. 

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Wilkie, D.R. The relation between force and velocity in human muscle. J. Physiol. (Lond) 110 249-280 (1950)... 

Under conditions of constant load the relationship between force and velocity is nearly hyperbolic [Hill, 1938], The shortening force-velocity relation was can be described by ... 

The units for viscosity are Pa-s. A Newtonian fluid is a fluid that follows Newton s law of viscosity, which states that the viscosity is independent of shear rate. This implies a linear relation between force and plate speed doubling the force on the plate will double its velocity. 

Our description of the fundamental principles is now nearly complete. We write a relation (at point A) between force and velocity, analogous to equation (10.37) ... 

The linear relations between forces and fluxes which were discussed in the previous section are empirical. To provide a justification based upon kinetic theory and simultaneously to obtain expressions for the transport coefficients x, rj, and D requires developing a molecular model for transport in dilute gases and extending the concept of a velocity distribution to nonequilibrium systems. 

The tower pressure losses are (1) tower packing or fill (70-80% of loss) (2) air inlet if induced draft (3) mist eliminators at top (4) air direction change losses and entrance to packing on forced draft units. These losses are a function of air velocity, number and spacing of packing decks, liquid rate and the relation between L and Ga. 

Discussion of the Equation.—The Boltzmann equation describes the manner in which the distribution function for a system of particles, /x = /(r,vx,f), varies in terms of its independent variables r, the position of observation vx, the velocity of the particles considered and the time, t. The variation of the distribution function due to the external forces acting on the particles and the action of collisions are both considered. In the integral expression on the right of Eq. (1-39), the Eqs. (1-21) are used to express the velocities after collision in terms of the velocities before collision the dynamics of the collision process are taken into account in the expression for x(6,e), from Eqs. (1-11) and (1-12), which enters into the k of Eqs. (1-21). Alternatively, as will be shown to be useful later, the velocities before and after collision may be expressed, by Eq. (1-20), in terms of G,g, and g the dynamics of the collision comes into the relation between g and g of Eq. (1-19). 

It may be assumed that the drag force F of the fluid on the particles under the free falling conditions is given by Stokes law and that the relation between the fluidisation velocity uc and voidage, e, for particles of terminal velocity, u0, is given by ... 

The most satisfactory way of representing the relation between drag force and velocity involves the use of two dimensionless groups, similar to those used for correlating information on the pressure drop for flow of fluids in pipes. 

Equation (12.33) is a measure of the distance of a nonequilibrium chemical reaction from equilibrium. At equilibrium, the affinity vanishes. Equation (12.34) shows a nonlinear relation between the reaction velocity Jr (flow) and the affinity (thermodynamic force). If the chemical system is close to equilibrium, that is, A/ RT) 1, then the contents of the square parentheses of Eq. (12.34) are approximated asAKRT), and we have the following linear flow-force relation between the reaction velocity and the affinity... 

A high phosphatase rate was taken as the basis of a new form of the latch model by Driska (1987) and Hai and Murphy (1988). In this model, LC20 phosphorylation is considered to be the only switch for crossbridge turnover, and the nonlinear relation between force (Jatp) and maximal shortening velocity (Vj ax) arises because of a significant rate of LC20 dephos-... 

It is easy to generalize Eq. (3.15) for a system which has many degrees of freedom. Let x = (xi, X2,...,xat) be the set of dynamical variables describing the state of Brownian particles. We need to know first the relation between the average velocity and the force Fm = —dVjdxm- Such a relation is generally expressed as... 

When the two forces are equilibrated, the so-called electrophoretic velocity Ve is reached and the relation between Ve, and E is... 

Zvonkov extended the concept of the four critical flow velocities to the process of wind erosion. Figure XI.6 shows the way in which the forces Ff and F f vary with the velocity of an air flow for soil particles 0.058 cm in diameter, with Kg = 1.0, and zero slope relative to the horizontal. The general character of the relationships obtained and the relation between the critical velocities... 

When dewetting occurs at velocities so high that inertial effects cannot be neglected, viscous effects are negligible and a simple analysis on the balance of capillary and inertial forces gives the relation between the dewetting velocity and the film thickness ... 

Fluid statics treats fluid in the equilibrium state (no motion), while fluid dynamics treats fluids when portion of the fluid are in motion (concerned with the relation between the fluid velocity and the forces acting on it). 

When a Newtonian fluid is placed between the two plates as shown in Figure 2.3 in which the top plate is moved to the right with constant velocity, V, the relation between force, F, divided by the area of the plates. A, and the velocity divided by the separation distance, //, is given as follows ... 

A related algorithm can be written also for the Brownian trajectory [10]. However, the essential difference between an algorithm for a Brownian trajectory and equation (4) is that the Brownian algorithm is not deterministic. Due to the existence of the random force, we cannot be satisfied with a single trajectory, even with pre-specified coordinates (and velocities, if relevant). It is necessary to generate an ensemble of trajectories (sampled with different values of the random force) to obtain a complete picture. Instead of working with an ensemble of trajectories we prefer to work with the conditional probability. I.e., we ask what is the probability that a trajectory being at... 

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