Viscous effects

It remains only to take into account viscous effects described by... 

We commented above that the elastic and viscous effects are out of phase with each other by some angle 5 in a viscoelastic material. Since both vary periodically with the same frequency, stress and strain oscillate with t, as shown in Fig. 3.14a. The phase angle 5 measures the lag between the two waves. Another representation of this situation is shown in Fig. 3.14b, where stress and strain are represented by arrows of different lengths separated by an angle 5. Projections of either one onto the other can be expressed in terms of the sine and cosine of the phase angle. The bold arrows in Fig. 3.14b are the components of 7 parallel and perpendicular to a. Thus we can say that 7 cos 5 is the strain component in phase with the stress and 7 sin 6 is the component out of phase with the stress. We have previously observed that the elastic response is in phase with the stress and the viscous response is out of phase. Hence the ratio of... 

This intricate mode of crystallization requires more time to accomplish than, say, the entry of small ions into growing salt crystals. This, coupled with low chain mobility due to viscous effects, makes the rate of crystallization slow and accounts in part for the fact that with rapid cooling-called quenching-the temperature drops below T without crystallization. 

Here (0 is the magnitude of the vorticity vector, which is directed along the z axis. An irrotational flow is one with zero vorticity. Irro-tational flows have been widely studied because of their useful mathematical properties and applicability to flow regions where viscous effects m be neglected. Such flows without viscous effec ts are called in viscid flows. 

Turbulent velocity fluctuations ultimately dissipate their kinetic energy through viscous effects. MacroscopicaUy, this energy dissipation requires pressure drop, or velocity decrease. The ener dissipation rate per unit mass is usually denoted . For steady ffow in a pipe, the average energy dissipation rate per unit mass is given by... 

Typically, Ro is small to the order of 1 with the high end of the range showing possible effect due to inertia, whereas the Ek number is usually very small, 10" or smaller. Therefore, the viscous effect is confined to thin boundaiy layers with thickness Ek L. 

If deb > 1 flien the process is predominantly elastic whereas if A deb < 1 then viscous effects dominate the flow. 

In shear layers, large-scale eddies extract mechanical energy from the mean flow. This energy is continuously transferred to smaller and smaller eddies. Such energy transfer continues until energy is dissipated into heat by viscous effects in the smallest eddies of the spectrum. 

As indicated earlier, non-Newtonian characteristics have a much stronger influence on flow in the streamline flow region where viscous effects dominate than in turbulent flow where inertial forces are of prime importance. Furthermore, there is substantial evidence to the effect that for shear-thinning fluids, the standard friction chart tends to over-predict pressure drop if the Metzner and Reed Reynolds number Re R is used. Furthermore, laminar flow can persist for slightly higher Reynolds numbers than for Newtonian fluids. Overall, therefore, there is a factor of safety involved in treating the fluid as Newtonian when flow is expected to be turbulent. 

As noted before, thin film lubrication (TFL) is a transition lubrication state between the elastohydrodynamic lubrication (EHL) and the boundary lubrication (BL). It is widely accepted that in addition to piezo-viscous effect and solid elastic deformation, EHL is featured with viscous fluid films and it is based upon a continuum mechanism. Boundary lubrication, however, featured with adsorption films, is either due to physisorption or chemisorption, and it is based on surface physical/chemical properties [14]. It will be of great importance to bridge the gap between EHL and BL regarding the work mechanism and study methods, by considering TFL as a specihc lubrication state. In TFL modeling, the microstructure of the fluids and the surface effects are two major factors to be taken into consideration. 

Hydrodynamic effects on suspended particles in an STR may be broadly categorized as time-averaged, time-dependent and collision-related. Time-averaged shear rates are most commonly considered. Maximum shear rates, and accordingly maximum stresses, are assumed to occur in the impeller region. Time-dependent effects, on the other hand, are attributable to turbulent velocity fluctuations. The relevant turbulent Reynolds stresses are frequently evaluated in terms of the characteristic size and velocity of the turbulent eddies and are generally found to predominate over viscous effects. 

The predicted effects of temperature on (/m/ wcrc experimentally observed by Botterill and Teoman (1980) as shown in Fig. 3. For sand particles 462 microns in diameter, they observed that Umf- decreased with temperature. For larger material (2320 microns) in the transition region in Fig. 2 (where (/ changes from increasing with temperature to decreasing with temperature), they observed an initial increase in f/ / w ith temperature at low temperatures, which was followed by a decrease in Umjras increasing temperature caused viscous effects to become dominant. 

The first term on the right-hand side is known as the buoyant force, the second is known as thrust. If this were just a puff of hot air without the balloon exhaust, we would only have the buoyant force acting. In this case we could not ignore (d/dt) Jjf pvxdV since the puff would rise (vx +) solely due to its buoyancy, with viscous effects retarding it. Buoyancy generated flow is an important controlling mechanism in many fire problems. 

It should be noted that the dynamic conditions of droplet impact processes discussed above cover a large range of the actual conditions in many industrial processes, such as spray forming, thermal spray, spray combustion, spray cooling, and aircraft flight. Under these conditions, the spreading behavior of droplets on a flat surface is essentially governed by inertia and viscous effects (Fig. 

The spreading behavior of droplets on a non-flat surface is not only dependent on inertia and viscous effects, but also significantly influenced by an additional normal stress introduced by the curved surface. This stress leads to the acceleration-deceleration effect, or the hindering effect depending on the dimensionless roughness spacing, and causes the breakup and ejection of liquid. Increasing impact velocity, droplet diameter, liquid density, and/or... 

The procedure is the same as for constant flow conditions. The only forces to be considered are the inertia force, the surface-tension force, and the buoyancy force. The viscous effects are neglected. Making a force balance, we obtain ... 

In this region, the slope of the curve changes progressively from —1 to 0 as Re increases. Several workers have suggested approximate equations for flow in this intermediate region. Dallavelle proposed that R /pu2 may be regarded as being composed of two component parts, one due to Stokes law and the other, a constant, due to additional non-viscous effects. 

In practice all real fluids have nonzero viscosity so that the concept of an inviscid fluid is an idealization. However, the development of hydrodynamics proceeded for centuries neglecting the effects of viscosity. Moreover, many features (but by no means all) of certain high Reynolds number flows can be treated in a satisfactory manner ignoring viscous effects. 

Equation (7-30) gives the natural frequency of the fundamental mode for stationary fluid particles undergoing small oscillations with viscous forces neglected. It has been modified to account for viscous effects (L4, MIO, SIO), surface impurities (MIO), finite amplitudes (S5, Yl), and translation (SIO). Observed oscillation frequencies are generally less than those given by Eq. (7-30), typically by 10-20% for drops in free motion in impure systems (S4) and by 20-40% for pure systems (El, E3, W8, Yl). The amplitude tends to be larger for pure systems (E3) and this explains the reduction in frequency. 

For the special case (a) above where surface tension and viscous effects are negligible, the terminal velocity of a slug in a column of rectangular cross section (D x D2) is given by... 

For the more general case when surface tension and viscous effects are appreciable, there are few data available. Grigorev and Krokhin (GIO) presented some results for the rise of bubbles in thin rectangular slits and wedge-shaped channels, while Schad and Bishop (S2) investigated bubble rise in thin annular and planar gaps. 

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