Navier-Stokes equations solution procedures

At larger Re and for more marked deformation, theoretical approaches have had limited success. There have been no numerical solutions to the full Navier-Stokes equation for steady flow problems in which the shape, as well as the flow, has been an unknown. Savic (S3) suggested a procedure whereby the shape of a drop is determined by a balance of normal stresses at the interface. This approach has been extended by Pruppacher and Pitter (P6) for water drops falling through air and by Wairegi (Wl) for drops and bubbles in liquids. The drop or bubble adopts a shape where surface tension pressure increments, hydrostatic pressures, and hydrodynamic pressures are in balance at every point. Thus... 

Kelkar, K.M. and Patankar, S.V. (1989), Development of generalized block correction procedure for the solution of discretized Navier-Stokes Equation, Comput. Phys. Commun., 53, 329-336. 

Tryggvason and co-workers [227, 228, 127, 128, 170, 224, 71, 72, 225] solve the Navier-Stokes equations using a projection method very similar to the MAC method [96]. The interface is tracked explicitly by connected marker points. In the solution procedure a fixed structured grid is used for the transport equations, and a moving grid of lower dimension marks the boundary between the bulk phases. This moving grid is called the front. 

However, an exact solution to the problem of convective diffusion to a solid surface requires first the solution of the hydrodynamic equations of motion of the fluid (the Navier-Stokes equations) for boundary conditions appropriate to the mainstream velocity of flow and the geometry of the system. This solution specifies the velocity of the flrrid at any point and at any time in both tube and yam assembly. It is then necessary to substitute the appropriate values for the local fluid velocities in the convective diffusion equation, which must be solved for boundary cortditiorts related to the shape of the package, the mainstream concentration of dye and the adsorptions at the solid surface. This is a very difficrrlt procedure even for steady flow through a package of simple shape. " ... 

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