Viscous resistance terms

In the viscous resistance term in Eq. (163), the radius r can be taken as 1.25 rfb without causing a great loss in accuracy. Equations (162) and (163) reduce to constant flow or constant pressure conditions, respectively, when Q is constant or when it is expressed in terms of orifice equation. 

The term (-ur]lk) in Eq. 3.23 is the Darcy resistance term, and the term (rjW u) is the viscous resistance term the driving force is still considered to be the pressure gradient. When the permeability k is low, the Darcy resistance dominates the Navier-Stokes resistance, andEq. 3.23 reduces to Darcy s law. Therefore, the Brinkman equation has the advantage of considering both viscous drag along the walls and Darcy effects within the porous medium itself. In addition, because Brinkman s equation has second-order derivatives of u, it can satisfy no-slip conditions at solid surfaces bounding the porous material (e.g. the walls of a packed bed reactor), whereas Darcy s law cannot. In that sense, Brinkman s equation is more exact than Darcy s law. 

The model for inviscid liquids is equally well applicable to viscous liquids also, provided that the resistance due to viscous drag is included in the analysis. As a first approximation, the viscous drag may be evaluated by a Stokes resistance term, since the bubble is not followed by a wake. Thus we proceed as before, first evaluating the force-balance bubble volume Kr and then the total bubble volume by reference to the detachment stage. 

The left-hand side of the latter equation is related to the liquid inertia, whereas both terms in the right-hand side are related to capillarity (the driving force), and viscous resistance, respectively. Under steady conditions, capillarity is balanced by the viscous drag of the liquid, and the famous Lucas-Washbum s equation can be derived (De Geimes et al., 2002) ... 

The two terms on the right-hand side represent the contributions due to viscous and turbulent flow resistances. It was assumed that the pressure drop due to viscous resistance is negligible ky = 0). However, the parametric sensitivity of K on the critical superficial gas velocity was also studied. The turbulent pressure drop was estimated using the orifice equation... 

For a viscous dispersed phase the derivation of the drop size equations are modified (Calabrese and Berkman, 1988 Streiff et al., 1997) with an extra term representing the viscous resistance to drop breakup. This adds a new term ... 

K. Tanaka (Kanazawa University, Japan) What is the relationship between the shearing strength of a lubricant film and the viscous resistance of the film I have the impression that physicists do not use the term "shearing strength" when referring to a viscous liquid. 

It is evident that the Giesekus model can quantitatively describe the shear thinning behavior of the entangled solutions of rod-shaped micelles. The decrease of the viscous resistance is caused by the alignment of the anisometric aggregates in the streaming solutions. Similar conclusions can be drawn from measurements of the first normal stress difference. This parameter is often represented in terms of the first normal stress coefficient ... 

Astrom and Pipes (28) showed that (he importance of the thermal expansion is small (less than 2% in terms of the matrix pressure). The viscous resistance and compactation are the major contribution to the pulling force and the friction resistance is negligible. This is a consequence of the small magnitude of (he pressure load reached in the die so that the matrix pressure does not contribute to die pulling force. 

The dielectric relaxation time, r, is approximately the mean time required for the molecular dipoles to turn 180° against the viscous resistance in a glassy polymer or the mean time for the dipoles in a crystalline structure to jump from one position over an energy barrier to another position rotated by 180°. The relationships shown in Figure 27 can be expressed (49) in terms of a single relaxation time (Debye eq. 15 and 16),... 

Viscoelasticity illustrates materials that exhibit both viscous and elastic characteristics. Viscous materials tike honey resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain instantaneously when stretched and just as quickly return to their original state once the stress is removed. Viscoelastic materials have elements of both of these properties and, as such, exhibit time-dependent strain. Viscoelasticity is the result of the diffusion of atoms or molecules inside an amorphous material. Rubber is highly elastic, but yet a viscous material. This property can be defined by the term viscoelasticity. Viscoelasticity is a combination of two separate mechanisms occurring at the same time in mbber. A spring represents the elastic portion, and a dashpot represents the viscous component (Figure 28.7). 

A major complication, especially for separated flows, arises from the effect of slip. Slip occurs because the less dense and less viscous phase exhibits a lower resistance to flow, as well as expansion and acceleration of the gas phase as the pressure drops. The result is an increase in the local holdup of the more dense phase within the pipe (phase density, pm), as given by Eq. (15-11). A large number of expressions and correlations for the holdup or (equivalent) slip ratio have appeared in the literature, and the one deduced by Lockhart and Martinelli is shown in Fig. 15-7. Many of these slip models can be summarized in terms of a general equation of the form... 

Majerus (61, 62) has approached the failure behavior of highly filled polymers by a thermodynamic treatment in which the ability to resist rupture is related to the propellant s ability to absorb and dissipate energy at a certain rate. An energy criterion which requires failure to be a function of both stress and strain was originally stated by Griffith (36) for brittle materials and later adapted to polymers by Rivlin and Thomas (80). Williams (115) has applied an energy criterion to viscoelastic materials such as solid propellants where appropriate terms are included for viscous energy dissipation. 

Equation 8.40 shows that the two terms in the right-hand side are equal at Re = 85.7(1 - eb). So, for typical bed void fractions of 0.4 the viscous and inertial resistance s are approximately equal at Re 50. 

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