Viscous fluctuations

J.T. Jenkins and D.F.. McTigue, Viscous Fluctuations and the Rheology of Concentrated Suspensions. Submitted to Journal of Fluid Mechanics (1995)... 

Drop breakage occurs when surrounding fluid stresses exceed the surface resistance of drops. Drops are first elongated as a result of pressure fluctuations and then spHt into small drops with a possibiUty of additional smaller fragments (Fig. 19). Two types of fluid stresses cause dispersions, viscous shear and turbulence. In considering viscous shear effects, it is assumed that the drop size is smaller than the Kohnogoroff microscale, Tj. 

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... 

With turbulent flow, shear stress also results from the behavior of transient random eddies, including large-scale eddies which decay to small eddies or fluctuations. The scale of the large eddies depends on equipment size. On the other hand, the scale of small eddies, which dissipate energy primarily through viscous shear, is almost independent of agitator and tank size. 

Inertial forces are developed when the velocity of a fluid changes direction or magnitude. In turbulent flow, inertia forces are larger than viscous forces. Fluid in motion tends to continue in motion until it meets a sohd surface or other fluid moving in a different direction. Forces are developed during the momentum transfer that takes place. The forces ac ting on the impeller blades fluctuate in a random manner related to the scale and intensity of turbulence at the impeller. 

Under certain conditions the energy dissipation may lead to an oscillatory regime of laminar flow in micro-channels. The relation of hydraulic diameter to channel length and the Reynolds number are important factors that determine the effect of viscous energy dissipation on flow parameters. The oscillatory flow regime occurs in micro-channels at Reynolds numbers less than Recr- In this case the existence of velocity fluctuations does not indicate change from laminar to turbulent flow. 

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. 

Although the transport properties, conductivity, and viscosity can be obtained quantitatively from fluctuations in a system at equilibrium in the absence of any driving forces, it is most common to determine the values from experiments in which a flux is induced by an external stress. In the case of viscous flow, the shear viscosity r is the proportionality constant connecting the magnitude of shear stress S to the flux of matter relative to a stationary surface. If the flux is measured as a velocity gradient, then... 

To measure the strength of the forces exerted on particles, various analytical techniques have been developed [6, 7]. Unfortunately, since most of these techniques are based on hydrodynamics, assumption of the potential profiles is required and the viscosities of the fiuid and the particle sizes must be precisely determined in separate experiments, for example, using the viscous flow technique [8,9] and power spectrum analysis of position fluctuation [10]. Furthermore, these methods provide information on ensemble averages for a mass of many particles. The sizes, shapes, and physical and chemical properties of individual particles may be different from each other, which will result in a variety of force strengths. Thus, single-particle... 

When electrically insulated strip or spot electrodes are embedded in a large electrode, and turbulent flow is fully developed, the steady mass-transfer rate gives information about the eddy diffusivity in the viscous sublayer very close to the electrode (see Section VI,C below). The fluctuating rate does not give information about velocity variations, and is markedly affected by the size of the electrode. The longitudinal, circumferential, and time scales of the mass-transfer fluctuations led Hanratty (H2) to postulate a surface renewal model with fixed time intervals based on the median energy frequency. 

Roy and Davidson (1989) considered the validity of the Ml and viscous limit scaling laws at elevated pressures and temperatures. The nondimensional dominant frequency and amplitude of the pressure drop fluctuations were used as the basis of the comparison. They concluded that when the full set of scaling parameters is matched, similarity is achieved. They also suggested that it is not necessary to match the density ratio (ps/pf) and dp/D, the simplification for viscous limit scaling, for particle Reynolds numbers (Re ) less than 30. Although the only run with Redp near 30 which was similar to the low Reynolds number test had the same density ratio as the lowRedp runs. These conclusions may be open to different interpretations. As shown in Table 6, the scaling parameters neither matched closely nor varied in a systematic manner. 

In this equation, the superscript ( ) indicates that a term is computed based upon the most recent information, which complies with the ( + l)th time level when all iterative loops have converged. Further, the convective transport and viscous generation of fluctuating kinetic energy have been collected in the explicit term D. The iterative solution procedure for the granular energy equations continues until the convergence criteria... 

Chemical reactions will take place only when the reactant molecules are in intimate contact. In some cases, especially with very fast reactions or viscous liquids, segregation of the reactants can exist, which make the reaction rates and selectivities dependent on the mixing intensity. In chemical reactor engineering, the assumption is usually made that only mean concentrations need be considered. In reality, concentration values fluctuate about a mean, and in some cases these fluctuations must be considered in detail. This field is very complex and is still the subject of much research. This example serves only to introduce these concepts and to show how simulations can be made for certain simple situations. 

In equation 1.94, (Tyx)v is the viscous shear stress due to the mean velocity gradient dvjdy and pv yv x is the extra shear stress due to the velocity fluctuations v x and v y. These extra stress components arising from the velocity fluctuations are known as Reynolds stresses. (Note that if the positive sign convention for stresses were used, the sign of the Reynolds stress would be negative in equation 1.94.)... 

As the fluid s velocity must be zero at the solid surface, the velocity fluctuations must be zero there. In the region very close to the solid boundary, ie the viscous sublayer, the velocity fluctuations are very small and the shear stress is almost entirely the viscous stress. Similarly, transport of heat and mass is due to molecular processes, the turbulent contribution being negligible. In contrast, in the outer part of the turbulent boundary layer turbulent fluctuations are dominant, as they are in the free stream outside the boundary layer. In the buffer or generation zone, turbulent and molecular processes are of comparable importance. 

From equation 1.41, the total shear stress varies linearly from a maximum fw at the wall to zero at the centre of the pipe. As the wall is approached, the turbulent component of the shear stress tends to zero, that is the whole of the shear stress is due to the viscous component at the wall. The turbulent contribution increases rapidly with distance from the wall and is the dominant component at all locations except in the wall region. Both components of the mean shear stress necessarily decline to zero at the centre-line. (The mean velocity gradient is zero at the centre so the mean viscous shear stress must be zero, but in addition the velocity fluctuations are uncorrelated so the turbulent component must be zero.)... 

All non-linear terms involving spatial gradients require transported PDF closures. Examples of such terms are viscous dissipation, pressure fluctuations, and scalar dissipation. 

The first term on the right-hand side of (6.20) is unclosed, and corresponds to the effects of viscous dissipation and pressure fluctuations. The closure of this term is the principal... 

The two terms in A[ V, r/t) represent the effects of the fluctuating viscous forces and the fluctuating pressure field, respectively. The molecular viscous-dissipation term in (6.43) is negligible at high Reynolds numbers. However, it becomes important in boundary layers where the Reynolds number is low, and must be included in the boundary conditions as described below. 

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