Viscous shear effects

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. 

This postulate imposes an idealization, and is the basis for all subsequent property relations for PVT systems. The PVT system sei ves as a satisfactoiy model in an enormous number of practical applications. In accepting this model one assumes that the effects of fields (e.g., elec tric, magnetic, or gravitational) are negligible and that surface and viscous-shear effects are unimportant. 

The theory of hydrodynamics similarly describes an ideal liquid behavior making use of the viscosity (see Sect 5.6). The viscosity is the property of a fluid (liquid or gas) by which it resists a change in shape. The word viscous derives from the Latin viscum, the term for the birdlime, the sticky substance made from mistletoe and used to catch birds. One calls the viscosity Newtonian, if the stress is directly proportional to the rate of strain and independent of the strain itself. The proportionality constant is the viscosity, q, as indicated in the center of Fig. 4.157. The definitions and units are listed, and a sketch for the viscous shear-effect between a stationary, lower and an upper, mobile plate is also reproduced in the figure. Schematically, the Newtonian viscosity is represented by the dashpot drawn in the upper left comer, to contrast the Hookean elastic spring in the upper right. 

The viscous shear properties at any given shear rate are primarily determined by two factors, the free volume within the molten polymer mass and the amount of entanglement between the molecules. An increase in the former decreases the viscosity whilst an increase in the latter, i.e. the entanglement, increases viscosity. The effects of temperature, pressure, average molecular weight, branching and so on can largely be explained in the these terms. 

Thus, the extra terms arising from the fluctuating velocity components are such that their effects are the same as an increase in the viscous shear stress and it is for this reason that they are termed the turbulent or Reynolds stress terms. 

Expressing the effective viscosity of the gas-liquid emulsion with Einstein s equation, and replacing the viscous shear stress T as the product of effective emulsion viscosity and the maximum liquid velocity gradient, and after combining Eq. (2) into Eq. (1), one arrives at the following relationship between a and a° ... 

The region of the flow above the plate bounded by 5 in which the effects of the viscous shearing forces caused by fluid viscosity are fell is called the velocity boundary layer. The boundary layer iliickiiess, 8, is typically defined as the distance) from the. surface at which u = 0.99F. 

When the viscous shear stresses are not negligible, their effect is accounted for by expressing the energy equation as... 

Many important coating processes are of liquids that are not Newtonian, and so the effects of non-Newtonian rheology on flow between rolls is of great interest. The code used here has been applied to the simplest non-Newtonian model, namely the purely viscous, shear-thinning fluid. Viscoelasticity, though also important, is more difficult to treat and is not considered here. 

Next we turn to the stability of Couette flow for parallel rotating cylinders. This is an important flow for various applications, and, though it is a shear flow, the stability is dominated by the centrifugal forces that arise because of centripetal acceleration. This problem is also an important contrast with the first two examples because it is a case in which the flow can actually be stabilized by viscous effects. We first consider the classic case of an inviscid fluid, which leads to the well-known criteria of Rayleigh for the stability of an inviscid fluid. We then analyze the role of viscosity for the case of a narrow gap in which analytic results can be obtained. We show that the flow is stabilized by viscous diffusion effects up to a critical value of the Reynolds number for the problem (here known as the Taylor number). 

This is a semiheuristic volume-averaged treatment of the flow field. The experimental observations of Dybbs and Edwards [27] show that the macroscopic viscous shear stress diffusion and the flow development (convection) are significant only over a length scale of i from the vorticity generating boundary and the entrance boundary, respectively. However, Eq. 9.22 predicts these effects to be confined to distances of the order oi Km and KuDN, respectively. We note that Km is smaller than d. Then Eq. 9.22 predicts a macroscopic boundary-layer thickness, which is not only smaller than the representative elementary volume i when i d, but even smaller than the particle size. However, Eq. 9.22 allows estimation of these macroscopic length scales and shows that for most practical cases, the Darcy law (or the Ergun extension) is sufficient. 

Composite wicking structures accomplish the same type of effect in that the capillary pumping and axial fluid transport are handled independently. In addition to fulfilling this dual purpose, several wick structures physically separate the liquid and vapor flow. This results from an attempt to eliminate the viscous shear force that occurs during countercurrent liquid-vapor flow. 

The wall shear term in Eq. (4.10) increases significantly with increasing superficial liquid (I/l) and gas (11 ) velocities and can amount to 20% of the total gas holdup (Hills, 1976 Merchuk and Stein, 1981). This is because the wall shear stress increases significantly with [/l and f/g (Liu, 1997 Magaud etal., 2001 Wallis, 1969). When the liquid phase is highly viscous, the wall shear term can be significant even at superficial liquid velocities on the order of 2-lOcm/s (Al-Masry, 2001). Hence, it is necessary to include the wall shear effect in the total gas holdup value for most cocurrent or viscous flow bioreactors. 

As mentioned previously, at the Y-shaped junction droplet formation takes place in a one-step mechanism that is determined by the viscous shear force and the interfacial tension force [11]. Because of this special feature, it was possible to directly measure the effect of interfacial tension on the droplet size using various systems with different static interfacial tensions. Water/ ethanol mixtures were used as continuous phase, and hexadecane and silicon oils as to-be-dispersed phase. The size of the droplets was recorded and a calibration curve constructed, and based on that curve, the dynamic interfacial tension could be estimated in systems that contain surfactants. 

The shear-driven fluidic approach is based on a radical modification of the fluidic channel concept. This recently developed technique for the transport of fluids in ultrathin channels based on the SDF [2] relies on a very basic hydrodynamic effect the viscous drag. This effect is present in every fluid flow, be it a liquid or a gas flow. In pressure-driven flows, the viscous drag manifests itself in an undesirable manner, as the stationary column and particle surfaces tend to slow down the fluid flow. In SDF, the viscous drag effect is... 

Boundary layer. For all flow regimes, whether laminar or turbulent, the effects of viscous shear forces are greatest close to solid boundaries. Fluid actually in contact with a surface usually has no relative motion the so called no-slip condition. There is therefore a region extending from the surface to the bulk of the fluid within which the velocity changes from zero to the bulk value. This region is known as the boundary layer. 

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