Fluids with reduced dimensionality

For steady, two-dimensional velocity, thermal, and concentration boundary layers, the equations of change apply. These equations are developed elsewhere [8-10,13]. However, it is unusual for all the terms in the equations of change to be needed to resolve a given situation. Therefore, it is customary to simplify these equations by assuming fluids with constant physical properties (p, k, and ft), negligible body forces, no chemical reaction, and no heat generation. In addition, the usual botmdary layer approximations are incorporated to further reduce the equations of change [10] to... 

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

Geothermal reservoir rocks are typically fractured and therefore exhibit variable and anisotropic permeability. For that reason it is neither possible to predict with confidence how an injection well may perform with respect to its injectivity nor with respect to which way the injected fluid will flow once it is in the reservoir. Because of this complication, the success of injection varies between fields and it is anticipated that a special injection scheme must be developed for each field depending on its characteristics, mainly the three-dimensional distribution of permeability and the waste fluid composition. Injection may require drilling of special wells. Alternatively, wells drilled for the purpose of production may not have adequate yield but can be used successfully as injection wells. When this is the case, no special wells need to be drilled for injection purposes, which reduces road building and therefore scenery spoliation. 

Consider the two-dimensional flow in a channel formed by parallel plates, through which fluid may enter or leave the channel (Fig. 5.13). The similarity analysis of this situation is facilitated by assuming the form of the cross-channel velocity. With an assumed crossstream velocity, the axial-momentum equation can be reduced to an ordinary differential equation for a scaled axial velocity. 

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