Viscous cross-flow simple model

The results from the more complex eight layer cross-section, and the previous Brent Sands model, have confirmed that the same recovery mechanisms operate as were discussed in detail for the simple two layer model above. In particular, the importance of the polymer in changing the viscous forces in the oil displacement process is emphasised and results in viscous cross-flow of fluids between layers in heterogeneous formations of this type. The cross-sectional examples also show the importance of carrying out some simple scoping calculations in order to establish the mechanism... 

In Chapter 1, the types of gas flow that could be established in vacuum systems were defined. This chapter deals with the quantification of viscous and molecular flow for simple, model systems (pipelines of constant, circular cross-section, orifices and apertures, etc.). These are nevertheless useful, and worked examples are presented to encourage users to quantify existing or proposed systems and to provide reassurance that calculations are not only relatively straightforward but very useful indeed. 

The numerical value of the conductance of a component in a vacuum system depends on the type of flow in the system. Gas flow in simple, model systems (e.g. tubes of constant circular cross-section, orifices, apertures) was considered for viscous flow (Examples 2.6-2.8) and molecular flow (Examples 2.9-2.11). The chapter concluded with two illustrations (Examples 2.13, 2.14) of Knudsen (intermediate) flow through a tube. 

Hooke s law, the direct proportionality between stress and strain in tension or shear, is often assumed such that the constitutive equations for a purely elastic solid are o = fjs for unidirectional extension and x = qy in simple shear flow. The latter expression is recognized from Chapter 7 as the constitutive relationship for a Newtonian fluid and, in analogy to Hooke s law for elastic solids, is sometimes termed Newton s law of viscosity. For cross-linked, amorphous polymers above 7, a nonlinear relationship can be derived theoretically. For such materials v = 0.5. When v is not 0.5, it is an indication that voids are forming in the sample or that crystallization is taking place. In either case, neither the theoretical equation nor Hooke s law generally applies. Before turning to one of the simplest mathematical models of viscoelasticity, it is important to recall that the constitutive equations of a purely viscous fluid are a = fj for elongational flow and x = qy for shear flow. 

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