Viscous depth

Viscous losses of acoustic energy occur due to the oscillation of particles in an acoustic pressure field. The oscillations of particles are caused by the difference in density of the particles and the dispersion medium. As a result of these oscillations the liquid layers in the vicinity of the particles are involved in non-stationary sliding motion, referred to as a shear wave. The shear waves generated by the oscillating particles dampen exponentially in the vicinity of the particles. The extent of penetration of the shear wave into the dispersion medium is characterized by the viscous depth, 8visc, which is the distance at which the amplitude of the shear wave decreases by the factor of exp. The viscous depth is given by... 

The above expression exhibits the strong dependence of avisc on the parameter Y, i.e., on whether the viscous depth is small or large compared to the particle radius. In these two cases substantially different types of dependence of avisc on ultrasound frequency, a>, and on the particle radius are obtained. Indeed, for the case when 5visc rp, i.e., when Y 0, which corresponds to the low-frequency limit, eq. (V-46) becomes... 

Like viscous depth, thermal depth is a function of the ultrasound frequency, oi. The... 

The difference between the viscous depth and the thermal depth provides an answer to the observed differences between emulsions and solid particle dispersions. These parameters characterize the penetra tion of the shear wave and thermal wave, respectively, into the liquid. Particles oscillating in the sound wave generate these waves which damp in the particle vicinity. The characteristic distance for the shear wave amplitude to decay is the viscous depth 5y. The corresponding distance for the thermal wave is the thermal depth 5. The following expressions give these parameter values in dilute systems ... 

The relationship between <5y and 5 has been considered before. For instance, McClements plots thermal depth and viscous depth versus frequency (4). It is easy to show that viscous depth is 2.6 times more than thermal depth in aqueous dispersions (15). As a result, the particle viscous... 

This parameter is 2.6 for water, as was mentioned before. Figure 3 shows values of this parameter for all hquids from our database relative to flie viscous depth of water. It is seen that this parameter is even larger for many hquids. 

Salt acts as a completely mobile plastic below 7600 m of overburden and at temperatures above 200°C (2). Under lesser conditions, salt domes can grow by viscous flow. Salt stmctures originate in horizontal salt beds at depths of 4000—6000 m or more beneath the earth s surface. The resulting salt dome or diapir is typically composed of relatively pure sodium chloride in a vertically elongated, roughly cylindrical, or inverted teardrop-shaped mass. 

In the equation shown above, the first term—including p for density and the square of the linear velocity of u—is the inertial term that will dominate at high flows. The second term, including p. for viscosity and the linear velocity, is the viscous term that is important at low velocities or at high viscosities, such as in liquids. Both terms include an expression that depends on void fraction of the bed, and both change rapidly with small changes in e. Both terms are linearly dependent on a dimensionless bed depth of L/dp. 

In this introduction, the viscoelastic properties of polymers are represented as the summation of mechanical analog responses to applied stress. This discussion is thus only intended to be very introductory. Any in-depth discussion of polymer viscoelasticity involves the use of tensors, and this high-level mathematics topic is beyond the scope of what will be presented in this book. Earlier in the chapter the concept of elastic and viscous properties of polymers was briefly introduced. A purely viscous response can be represented by a mechanical dash pot, as shown in Fig. 3.10(a). This purely viscous response is normally the response of interest in routine extruder calculations. For those familiar with the suspension of an automobile, this would represent the shock absorber in the front suspension. If a stress is applied to this element it will continue to elongate as long as the stress is applied. When the stress is removed there will be no recovery in the strain that has occurred. The next mechanical element is the spring (Fig. 3.10[b]), and it represents a purely elastic response of the polymer. If a stress is applied to this element, the element will elongate until the strain and the force are in equilibrium with the stress, and then the element will remain at that strain until the stress is removed. The strain is inversely proportional to the spring modulus. The initial strain and the total strain recovery upon removal of the stress are considered to be instantaneous. 

The entrance and exit regions in the spiral dam were also modified to eliminate the stagnant sections of the channel. The modification Is shown In Fig. 11.40. This modification allowed a relatively small amount of resin to flow Into the smaller channel at the entry such that stagnation of the resin cannot occur. A similar modification was made at the exit to allow a small amount of resin to flow out of the smaller channel into the main flow channel. To eliminate the unmelted particles or the particles that appeared to be more viscous because they were at a lower temperature, the clearance to the spiral dam was decreased from 0.76 to 0.25 mm. Since the meter channel depth was unchanged, the specific rotational flow rate for the modified screw was unchanged at 0.94 kg/(h-rpm). 

An extrusion symposium (El) contains papers which deal extensively with the mathematics of viscous flow in screw extruders but which are limited to Newtonian materials. An extension of this work to materials which may be assumed to be Bingham plastic in behavior has been reported in Japan (M18, M19). The first of these papers deals with a screw extruder with a uniform channel the second with an extruder for which the depth of the channel decreases linearly with channel length. The mathematical results are shown graphically in terms of four dimensionless groups ... 

The velocity distribution equation (27) indicates that in the absence of surface tension effects the maximum velocity in a film flowing in a flat channel of finite width should occur at the free surface of the film at the center of the channel. The surface velocity should then fall off to zero at the side walls. However, experimental observations have shown (BIO, H18, H19, F7) that the surface velocity does not follow this pattern but shows a marked increase as the wall is approached, falling to zero only within a very narrow zone immediately adjacent to the walls. The explanation of this behavior is simple because of surface tension forces, the liquid forms a meniscus near the side walls. Equation (12) shows that the surface velocity increases with the square of the local liquid depth, so the surface velocity increases sharply in the meniscus region until the side wall is approached so closely that the opposing viscous edge effect becomes dominant. 

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