Viscous-dominated flows

For viscous dominated flows, it can be assumed that the gas inertia and the gas gravitational forces are negligible. By dropping the gas inertia and gravity time from the gas momentum equation and simplifying the dimensionless drag coefficient to the linear viscous term, the set of dimensionless equations does not include gas-to-solid density ratio as a parameter. 

The ratio between the bed and particle diameters and the Reynolds number based on bed diameter, superficial velocity, and solid density appear only in the modified drag expression, in which they are combined, see Eq. (40). These parameters form a single parameter, as discussed by Glicksman (1988) and other investigators. The set of independent parameters controlling viscous dominated flow are then... 

From a superficial point of view, this chapter simply represents the generalization of the theory of viscous dominated flows to consider three-dimensional problems. However, it also introduces much more powerful and convenient mathematical methods, many of which can be used in other applications. Sections A-C are particularly important. Other sections represent more advanced (and thus elective) topics for coverage in class. 

Viscous dominated flows lubrication, injection molding, wire coating, volcanoes, and continental drift... 

In a series of publications, Glicksman et al. (16-18) divided the scale-up into two regimes, namely, inertia-dominated and viscous-dominated flow regimes. In viscous-dominated flow regime, where particle Reynolds number based on fluid density is <4, i.e., when, the (dpupa/p<4, dimensionless numbers that need to be kept constant are... 

The majority of polymer flow processes are characterized as low Reynolds number Stokes (i.e. creeping) flow regimes. Therefore in the formulation of finite element models for polymeric flow systems the inertia terms in the equation of motion are usually neglected. In addition, highly viscous polymer flow systems are, in general, dominated by stress and pressure variations and in comparison the body forces acting upon them are small and can be safely ignored. 

The facdor K would be 1 in the case of full momentum recoveiy, or 0.5 in the case of negligible viscous losses in the portion of flow which remains in the pipe after the flow divides at a takeoff point (Denn, pp. 126-127). Experimental data (Van der Hegge Zijnen, Appl. Set. Re.s., A3,144-162 [1951-1953] and Bailey, ]. Mech. Eng. ScL, 17, 338-347 [1975]), while scattered, show that K is probably close to 0.5 for discharge manifolds. For inertiaUy dominated flows, Ap will be negative. For return manifolds the recovery factor K is close to 1.0, and the pressure drop between the first hole and the exit is given by... 

The resulting induced flow may be laminar (usually at small temperature differences and/or viscous fluids) or turbulent, or often in a transition or mixed laminar and turbulent regime. Correlations may be based on analytical or experimental studies, and numerical methods are now available. A major limitation to analytical solutions is that constant viscosity is generally assumed, whereas the variation of viscosity with temperature is likely to have a major effect upon the velocity gradient and the dominant flow regime near the heat-transfer surface. 

C third (viscous dominant to inertial dominant flow transition)... 

If the particles are not spherical, even in the very dilute limit where the translational Brownian motion would still be unimportant, rotational Brownian motion would come into play. This is a consequence of the fact that the rotational motion imparts to the particles a random orientation distribution, whereas in shear-dominated flows nonspherical particles tend toward preferred orientations. Since the excess energy dissipation by an individual anisotropic particle depends on its orientation with respect to the flow field, the suspension viscosity must be affected by the relative importance of rotational Brownian forces to viscous forces, although it should still vary linearly with particle volume fraction. 

Separate from particle/droplet size and the Knudsen number, there is another reason that aerosol sedimentation does not always follow Stokes law. As the flow regime goes from laminar flow (viscous dominated Reynolds number, Af u < 1) to turbulent flow (inertia dominated Reynolds number, Nr > 1000), things change (see Section 6.1 and Equation 6.6 for more on the Reynolds number). 

Figure 2.24 Illustration of flow regimes in aerosol sedimentation, as the flow regime goes from laminar flow (viscous dominated Reynolds number, Nf, < 1) to turbulent flow (inertia...

Since the pressures of the two ends of the medium are finite, viscous (Darcy) flow might be operating in addition to the Knudsen flow. To restrict the flow to only Knudsen diffusion, we must maintain the conditions of the experiment such that the Knudsen mechanism is dominating. This is possible when the pressure is... 

The Reynolds number. Re, is the ratio of inertial forces to viscous forces. If Uand L denote the characteristic velocity and length scales, respectively. Re = UL/v, where v is the kinematic viscosity. Small values of Re (i.e. less than 1000) correspond to laminar (viscosity-dominated) flows and large values of Re to turbulent flows. 

Sisoev G.M. and Shkadov V. Ya. (1997). Dominant waves in a viscous liquid flowing in a thin sheet. Physics-Doklady, 42 (12), 683-686. 

The permeance of porous stmctures is often expressed as / = a + (3p to represent a combination of Knudsen and viscous flow behavior. While such an expression may have little physical significance, it often describes experimental permeance data very well and can be used to analyze membrane defects and the dominant flow mechanism. The dusty gas model (DGM) (Mason, 1983) provides a fairly accurate and comprehensive treatment for the transport of mixtures in meso- and macroporous structures for aU flow regimes and mechanisms. Solutions of DGM equations, however, require finite-element methods (Benes et al., 1999). 

Capillary flows occur (e.g., parallel slit or cylindrical) and are driven by surface tension. The flow regime is viscous-dominated (see above) with a possible exception at the very beginning of the flow [22]. The scaling length scale is associated with the channel width or radius. Capillary flows are very convenient for initial loading of micro- and nanofluidic devices. The fluid must wet the surface and then a curved air/solution interface is formed at the flow front (Figure 19.2). Then the pressure drop in a perfectly wetted circular capillary with radius R is [23]... 

The same procedure described above for low velocity, viscous flow has been applied to the other extreme of high velocity, inertia-dominated flow. In this case the tube-flow equation is expressed in terms of the dimensionless friction factor f, which in the inertial flow regime remains essentially constant for a given tube ... 

Finally we require a case in which mechanism (lii) above dominates momentum transfer. In flow along a cylindrical tube, mechanism (i) is certainly insignificant compared with mechanism (iii) when the tube diameter is large compared with mean free path lengths, and mechanism (ii) can be eliminated completely by limiting attention to the flow of a pure substance. We then have the classical Poiseuille [13] problem, and for a tube of circular cross-section solution of the viscous flow equations gives 2... 

As a consequence of this, i enever bulk dlffusional resistance domin ates Knudsen diffusional resistance, so that 1, it follows that fi 1 also, and hence viscous flow dominates Knudsen streaming. Thus when we physically approach the limit of bulk diffusion control, by increasing the pore sizes or the pressure, we must simultaneously approach the limit of viscous flow. This justifies a statement made in Chapter 5. 

The cross-sectional area of the wick is deterrnined by the required Hquid flow rate and the specific properties of capillary pressure and viscous drag. The mass flow rate is equal to the desired heat-transfer rate divided by the latent heat of vaporization of the fluid. Thus the transfer of 2260 W requires a Hquid (H2O) flow of 1 cm /s at 100°C. Because of porous character, wicks are relatively poor thermal conductors. Radial heat flow through the wick is often the dominant source of temperature loss in a heat pipe therefore, the wick thickness tends to be constrained and rarely exceeds 3 mm. 

Figure 6-40 shows power number vs. impeller Reynolds number for a typical configuration. The similarity to the friction factor vs. Reynolds number behavior for pipe flow is significant. In laminar flow, the power number is inversely proportional to Reynolds number, reflecting the dominance of viscous forces over inertial forces. In turbulent flow, where inertial forces dominate, the power number is nearly constant. 

Porous Media Packed beds of granular solids are one type of the general class referred to as porous media, which include geological formations such as petroleum reservoirs and aquifers, manufactured materials such as sintered metals and porous catalysts, burning coal or char particles, and textile fabrics, to name a few. Pressure drop for incompressible flow across a porous medium has the same quahtative behavior as that given by Leva s correlation in the preceding. At low Reynolds numbers, viscous forces dominate and pressure drop is proportional to fluid viscosity and superficial velocity, and at high Reynolds numbers, pressure drop is proportional to fluid density and to the square of superficial velocity. 

Basic Equations AU of the processes described in this sec tion depend to some extent on the following background theory. Substances move through membranes by several meoianisms. For porous membranes, such as are used in microfiltration, viscous flow dominates the process. For electrodialytic membranes, the mass transfer is caused by an elec trical potential resulting in ionic conduction. For aU membranes, Ficldan diffusion is of some importance, and it is of dom-... 

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 Chapter 2 when the Maxwell and Kelvin models were analysed, it was found that the time constant for the deformations was given by the ratio of viscosity to modulus. This ratio is sometimes referred to as the Relaxation or Natural time and is used to give an indication of whether the elastic or the viscous response dominates the flow of the melt. 

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