Viscous Continuum Flow

A in cm ) For continuum flow (viscous flow) the follow/ing equations (after Prandtl) apply to air at 20 °C where pj/p = 5 ... 

This value is close to that for viscous flow given in Table 1.4. Consequently, supersonic continuum flow will occur through the orifice. As the pressure in chamber 2 is below the critical pressure (4.9 mbar, see Example 6.17), a maximum gas throughput will result ... 

We have presented in this chapter an account on the development of diffusion theory. Various modes of flow are identified Knudsen, viscous, continuum diffusion and surface diffusion. Constitutive flux equations are presented for all these flow mechanisms, and they can be readily used in any mass balance equations for the solution of concentration distribution and fluxes. Treatment of systems containing more than two species will be considered in a more systematic approach of Stefan-Maxwell in the next chapter. 

The polymer chains are considered in (4.1) as non-Unear elastic dumbbells embedded in a viscous continuum subjected to the flow deformation. Time... 

The mean free path of molecules in air at atmospheric pressure is /free — 1 /(Niiyg), where Nl 2.69 10 cm is the number density of gas molecules and cTg 10 " cm is the cross section for elastic collisions of molecules. These numbers result in /free — 3.7 10 cm, or 37 nm. The mean pore radius of the GDL is in the order of 10 pm, which means that the flow in the GDL pores occurs in a continuum regime. Thus, pressure-driven oxygen transport in a dry porous GDL can be modeled as a viscous Hagen-Poiseuille flow in an equivalent duct. However, determination of the equivalent duct radius and the dependence of this radius on the GDL porosity is a nontrivial task (Tamayol et al., 2012). Much workhas recently been done to develop statistical models of porous GDLs and to calculate viscous gas flows in these systems using Navier-Stokes equations (Thiedmann et al., 2012). 

Continuum theory has also been applied to analyse tire dynamics of flow of nematics [77, 80, 81 and 82]. The equations provide tire time-dependent velocity, director and pressure fields. These can be detennined from equations for tire fluid acceleration (in tenns of tire total stress tensor split into reversible and viscous parts), tire rate of change of director in tenns of tire velocity gradients and tire molecular field and tire incompressibility condition [20]. 

The Knudsen number Kn is the ratio of the mean free path to the channel dimension. For pipe flow, Kn = X/D. Molecular flow is characterized by Kn > 1.0 continuum viscous (laminar or turbulent) flow is characterized by Kn < 0.01. Transition or slip flow applies over the range 0.01 < Kn < 1.0. 

Slip Flow In the transition region between molecular flow and continuum viscous flow, the conductance for fully developed pipe flow is most easily obtained by the method of Brown, et al. (J. Appl. Phys., 17, 802-813 [1946]), which uses the parameter... 

The subject of this chapter is single-phase heat transfer in micro-channels. Several aspects of the problem are considered in the frame of a continuum model, corresponding to small Knudsen number. A number of special problems of the theory of heat transfer in micro-channels, such as the effect of viscous energy dissipation, axial heat conduction, heat transfer characteristics of gaseous flows in microchannels, and electro-osmotic heat transfer in micro-channels, are also discussed in this chapter. 

The frequency correlation time xm corresponds to the time it takes for a single vibrator to sample all different cavity sizes. The fluctuation-dissipation theorem (144) shows that this time can be found by calculating the time for a vertically excited v = 0 vibrator to reach the minimum in v = 1. This calculation is carried out by assuming that the solvent responds as a viscoelastic continuum to the outward push of the vibrator. At early times, the solvent behaves elastically with a modulus Goo. The push of the vibrator launches sound waves (acoustic phonons) into the solvent, allowing partial expansion of the cavity. This process corresponds to a rapid, inertial solvent motion. At later times, viscous flow of the solvent allows the remaining expansion to occur. The time for this diffusive motion is related to the viscosity rj by Geo and the net force constant at the cavity... 

Nagatani [57] arrived at a similar equation he presented a generalized theory describing slow viscous flow of suspensions in terms of statistical continuum mechanics. 

Viscous flow of a pure fluid (Kn 1). The gas acts as a continuum fluid driven by a pressure gradient, and both molecule-molecule collisions and molecule-wall collisions are important. This is sometimes called convective or bulk flow. 

Soo and Radke (11) confirmed that the transient permeability reduction observed by McAuliffe (9) mainly arises from the retention of drops in pores, which they termed as straining capture of the oil droplets. They also observed that droplets smaller than pore throats were captured in crevices or pockets and sometimes on the surface of the porous medium. They concluded, on the basis of their experiments in sand packs and visual glass micromodel observations, that stable OAV emulsions do not flow in the porous medium as a continuum viscous liquid, nor do they flow by squeezing through pore constrictions, but rather by the capture of the oil droplets with subsequent permeability reduction. They used deep-bed filtration principles (i2, 13) to model this phenomenon, which is discussed in detail later in this chapter. 

The gas is applied as a mixture to the retentate (high pressure) side of the membrane, the components of the mixture diffuse with different rates through the membrane under the action of a total pressure gradient and are removed at the permeate side by a sweep gas or by vacuum suction. Because the only segregative mechanisms in mesopores are Knudsen diffusion and surface diffusion/capillary condensation (see Table 9.1), viscous flow and continuum (bulk gas) diffusion should be absent in the separation layer. Only the transition state between Knudsen diffusion and continuum diffusion is allowed to some extent, but is not preferred because the selectivity is decreased. Nevertheless, continuum diffusion and viscous flow usually occur in the macroscopic pores of the support of the separation layer in asymmetric systems (see Fig. 9.2) and this can affect the separation factor. Furthermore the experimental set-up as shown in Fig. 9.11 can be used vmder isobaric conditions (only partial pressure differences are present) for the measurement of diffusivities in gas mixtures in so-called Wicke-Callenbach types of measurement. 

Isobaric applications in the continuum regime, making use of molecular bulk diffusion and/or some viscous flow are found in catalytic membrane reactors. The membrane is used here as an intermediating wall or as a system of microreactors [29,46]. For this reason some attention will be paid to the general description of mass transport, which will also be used in Sections 9.4 and 9.5. 

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