Friction definition

In the Smoluchowski limit the reaction is by definition the slow coordinate, such that y(kr) y(0) = dL yW Lrand Agii fO). Though the time-dependent friction... 

Thus, under conditions of plastic defonnation the real area of contact is proportional to the nonnal force. If the shear force during sliding is proportional to that area, one has the condition that the shear force is proportional to the nonnal force, thus leading to the definition of a coefficient of friction. 

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. 

Noncircular Channels Calciilation of fric tional pressure drop in noncircular channels depends on whether the flow is laminar or tumu-lent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter shoiild be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraiilic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraiilic diameter for a circiilar pipe is = D, for an annulus of inner diameter d and outer diameter D, = D — d, for a rectangiilar duct of sides 7, h, Dij = ah/[2(a + h)].T ie hydraulic radius Rii is defined as one-fourth of the hydraiilic diameter. 

The angle of internal friction, a, is defined as the equilibrium angle between flowing particles and bulk or stationary solids in a bin. Figure 4 illustrates the definition. The angle of internal friction is greater than the angle of repose. 

See nomenclature for definition of symbols and units. The units presented are English engineering units, unless a conversion is required. The friction factor is the only experimental variable that must be determined by reference to the above equations and it is represented by Figure 2-3. Note that this may sometimes be referred to as the Fanning formula, and may be modified to )held a fric-... 

The ideal soil is defined as a loose, granular medium that is devoid of cohesion but possesses internal friction. In contrast, an ideal cohesive medium is one that is devoid of internal friction. Real soils generally fall between the foregoing two limiting definitions. 

Dissipative systems whether described as continuous flows or Poincare maps are characterized by the presence of some sort of internal friction that tends to contract phase space volume elements. They are roughly analogous to irreversible CA systems. Contraction in phase space allows such systems to approach a subset of the phase space, C P, called an attractor, as t — oo. Although there is no universally accepted definition of an attractor, it is intuitively reasonable to demand that it satisfy the following three properties ([ruelle71], [eckmanSl]) ... 

Equation 3.11 is due to Blasius(6) and the others are derived from considerations of velocity profile. In addition to the Moody friction factor / = 8R/pu2, the Fanning or Darcy friction factor / = 2R/pu2 is often used. It is extremely important therefore to be clear about the exact definition of the friction factor when using this term in calculating head losses due to friction. 

Temperature, shock, shockwaves, friction and light may be the physical agency of instability. Unsaturated organic substances can sometimes undergo violent chemical transformations under the influence of some of these but do not come within the above definition. In these specific cases, dangerous chemical reactions, which often involve catalytic impurities, are the cause and are treated in chapter 4 as dangerous reactions . 

In Eq. (13), the vector q denotes a set of mass-weighted coordinates in a configuration space of arbitrary dimension N, U(q) is the potential of mean force governing the reaction, T is a symmetric positive-definite friction matrix, and , (/) is a stochastic force that is assumed to represent white noise that is Gaussian distributed with zero mean. The subscript a in Eq. (13) is used to label a particular noise sequence For any given a, there are infinitely many... 

Group N6 (or some multiple thereof) is also known as a friction factor (/), because the driving force (AP) is required to overcome friction (i.e., the energy dissipated) in the pipeline (assuming it to be horizontal), and N3 is known as the Reynolds number (N e). There are various definitions of the pipe friction factor, each of which is some multiple of N6 e.g., the Fanning friction factor is N6/2, and the Darcy friction factor is 2N6. The group N4 is also known as the Euler number. 

Although pV2/2 represents kinetic energy per unit volume, pV2 is also the flux of momentum carried by the fluid along the conduit. The latter interpretation is more logical in Eq. (5-50), because rw is also a flux of momentum from the fluid to the tube wall. However, the conventional definition includes the (arbitrary) factor i. Other definitions of the pipe friction factor are in use that are some multiple of the Fanning friction factor. For example, the Darcy friction factor, which is equal to 4/, is used frequently by mechanical and civil engineers. Thus, it is important to know which definition is implied when data for friction factors are used. 

Here, p is the density of the fluid, V is the relative velocity between the fluid and the solid body, and A is the cross sectional area of the body normal to the velocity vector V, e.g., nd1/4 for a sphere. Note that the definition of the drag coefficient from Eq. (11-1) is analogous to that of the friction factor for flow in a conduit, i.e.,... 

The expressions for the hydraulic diameter and the superficial velocity can be incorporated into the definition of the friction factor to give an equivalent expression for the porous medium friction factor ... 

Most references use Eq. (13-10) without the numerical factor of 3 as the definition of the porous medium friction factor, i.e.,... 

If the definitions of /PM and ARcRM are inserted into the Ergun equation, the resulting expression for the frictional energy loss (dissipation) per unit mass of fluid in the medium is... 

Rewriting the definition of the friction factor f from equation 21.3-5, and the Ergun correlation for/given by equation 21.3-7, both at mf, we obtain... 

Equation (e) is merely a definition of the mass flow rate. Equation (/) is a standard correlation for the friction factor for turbulent flow. (Note that the correlation between /and the Reynold s number (Re) is also available as a graph, but use of data from a graph requires trial-and-error calculations and rules out an analytical solution.)... 

Dividing equation 3.57 by equation 3.56 and using the definition of the Fanning friction factor (equation 2.10) gives... 

Obviously, it is better to use the smaller multiplier. The curves for each combination of flow regimes cross at X = 1, as they must from the definition of X, so the only gas reference is preferable for X < 1 and the only liquid reference for X>1. The reader is reminded that this correlation should be used only when the frictional component of the pressure gradient is dominant. 

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