Friction velocity-independent

Figure 9. Average kinetic friction F (independent of a) in the athermal Prandtl Tomlinson model at low velocities v for two different spring strengths k and various damping coefficients 7. The symbols at r o = 0 indicate the static friction force for k = 0.1k. All units are reduced units.

We are no more able to calculate the pressure drop in steady, turbulent flow in a noncircular conduit than we are in a circular one. However, it seems reasonable to expect that we could use the friction-loss results for circular pipes to estimate the results for other shapes. Let us assume that the shear stress at the wall of any conduit is the same for a given average fluid flow velocity independent of the shape of the conduit. Then, from a force balance on a horizontal section like that leading to Eq. 6.3, we conclude that in steady flow... 

Coulomb (1785) The coefficient of friction is independent of the velocity between the two surfaces, provided that velocity is not zero,... 

Consequently, Prandtl [124] postulated that, at high Reynolds numbers, close to the wall iy/S 1) there is an inner layer in which the mean velocity profile is determined by the viscous scales, independent of 6 and the free stream velocity, Vxiy = S). Since ample experimental analyzes support the Prandtl hypothesis, it is evident that close to the wall the viscosity, o, and the wall shear stress, uw, are important parameters. From these quantities one can define viscous scales that are appropriate velocity scales and length scales in the near-wall region. Dimensional analysis confirmed by experiments indicates that the relevant velocity scale for the inner region is the friction velocity,given by ... 

Bottom water currents in sluggish streams (i.e., bayous), lakes, estuaries, and other near-shore marine waters are moved by the wind at the surface. Both thermal and salinity stratification in these waters is a factor influencing the magnitudes of the bottom-water transport coefficients. Although this subject of MTCs has received limited study, some estimation methods are proposed. For unstratified water bodies. Equation 12.10 is useful wind speed is a key independent variable. For stratified lakes surface winds cause seiches that generate bottom water currents. Equations 12.11 through 12.13 can be used with seiche water displacement heights. To estimate bottom currents, these values are converted to bottom friction velocities with Equation 12.8, Equation 12.1 is then used for the MTC estimate. Bed characteristics can be used as proxies for bottom currents see Table 12.5. 

In this chapter, the mode coupling instability in the lead screw drives is studied. As mentioned in Sect. 4.2, mode coupling is exclusive to multi-DOF systems and can destabilize a system even when the coefficient of friction is independent of sliding velocity. 

The often-cited Amontons law [101. 102] describes friction in tenns of a friction coefiBcient, which is, a priori, a material constant, independent of contact area or dynamic parameters, such as sliding velocity, temperature or load. We know today that all of these parameters can have a significant influence on the magnitude of the measured friction force, especially in thin-film and boundary-lubricated systems. 

Equation (Cl.4.35) yields two remarkable predictions first, tliat tire sub-Doppler friction coefficient can be a big number compared to since at far detuning Aj /T is a big number and second, tliat a p is independent of tire applied field intensity. This last result contrasts sharjDly witli tire Doppler friction coefficient which is proportional to field intensity up to saturation (see equation (C1.4.24). However, even tliough a p looks impressive, tire range of atomic velocities over which is can operate are restricted by tire condition tliat T lcv. The ratio of tire capture velocities for Doppler versus sub-Doppler cooling is tlierefore only uipi/uj 2 Figure Cl. 4.6 illustrates... 

A number of simulation methods based on Equation (7.115) have been described. Thess differ in the assumptions that are made about the nature of frictional and random forces A common simplifying assumption is that the collision frequency 7 is independent o time and position. The random force R(f) is often assumed to be uncorrelated with th particle velocities, positions and the forces acting on them, and to obey a Gaussiar distribution with zero mean. The force F, is assumed to be constant over the time step o the integration. 

If we scale time as t = xr, then the frst term in (5.52) decreases as l/>/, while the other two are independent of friction. Therefore, at large rj the second derivative term in (5.52), as well as the kinetic energy term in the action, can be neglected, and the entire effect of friction is to change the timescale. That is, the solution to (5.52) is Q x) = Q x/ri) where Q is a function independent of rj. The instanton velocity is scaled as Q cc and the action (5.38) grows linearly with r, ... 

In the nonadiabatic limit ( < 1) B = nVa/Vi sF, and at 1 the adiabatic result k = k a holds. As shown in section 5.2, the instanton velocity decreases as t] increases, and the transition tends to be more adiabatic, as in the classical case. This conclusion is far from obvious, because one might expect that, when the particle loses energy, it should increase its upside-down barrier velocity. Instead, the energy losses are saturated to a finite //-independent value, and friction slows the tunneling motion down. 

The friction loss through cyclones may range from 1 to 20 inlet-velocity heads, depending on its geometric proportions. For a cyclone of specific geometric proportions, Fev and Apev. are essentially constant and independent of the actual cyclone size. 

Unlike traditional textbooks of tribology, in this book we regard boundary lubrication as a limit state of hydrodynamic lubrication when film thickness is down to molecular dimension and independent of the velocity of relative motion. The discussions are based on the existing results, some from literatures but mostly from the authors own work. The topics are mainly focused on the mechanical properties of boundary films, including rheology transitions, molecular ordering, and shear responses. Ordered molecule films, such as L-B films and SAM, are discussed, with emphasis on the frictional performance, energy dissipation and the effects from structural features. Boundary films can be modeled either as a confined substance, or an adsorbed/reacted layer on the... 

Suppose you want to design a hydrocarbon piping system in a plant between two points with no change in elevation and want to select the optimum pipe diameter that minimizes the combination of pipe capital costs and pump operating costs. Prepare a model that can be used to carry out the optimization. Identify the independent and dependent variables that affect the optimum operating conditions. Assume the fluid properties (/i, p) are known and constant, and the value of the pipe length (L) and mass flowrate (m) are specified. In your analysis use the following process variables pipe diameter (D), fluid velocity (v), pressure drop (A/ ), friction factor (/). 

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