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In the preceding chapters, the discussion has been restricted to the hydrodynamic behavior of single particles, with occasional reference to the motion of ensembles, in a variety of non-Newtonian fluids. In this chapter, the singleand two-phase flow of non-Newtonian fluids through porous media, a subject of overwhelming practical significance, is considered. Porous media has often been simulated using packed columns of well-characterized particles of regular shape (spheres and cylinders, for instance) as well as naturally occurring rocks, stones, etc. The flow of fluids in a porous medium is encountered literally everywhere, both in nature and in technology. The phenomena of the uptake of water and the flow of sap and gums in trees, the flow of blood in blood vessels and in various organs with porous boundaries, oxygenation of blood, for instance, immediately come to mind. Further examples are found in the flow of fluids through textile fabrics, pulp fibers, and woven and nonwoven mats (see Pierce, 1947; Cheikhrouhou and Sigli, 1988; Picaro and van de Ven, 1995; Epps and Leonas, 1997; Termonia, 1998; Caputo and Pelagagge, 1999; Brasquet and Le Cloirec, 2000) and in coffee and cigarette filters. Likewise, in the technology world, porous media flow applications abound. The classical examples include the groundwater hydraulics (Springer et al., 1998; Barr, 2001a, 2001b), flow in fractured rocks (Zimmerman, 2000) and their contamination by non-Newtonian oil pollutants (Theodoropoulou et al., 2001; Gioia and Urciuolo, 2004), flow in saturated aquifers (Li and Helm, 1998), aerosol filtration (Brown, 1998; Raynor, 2002), filtration of slurries, sludges and polymer melts using sand pack beds, screens, and metallic filters (Churaev and Yashchenko, 1966; Lorenzi, 1975; Khamashta and Virto, 1981; Kolodziej, 1986; Kozicki, 1988; Kiljanski and Dziubinski, 1996; Khuzhayorov et al., 2000; Auriault et al., 2002; Fadili et al., 2002; Chase and Dachavijit, 2003, etc.). Of course, the widespread use of packed and fluidized beds to carry out diverse operations in chemical, food, and biochemical process engineering applications may also be mentioned in passing. For instance, drying of wheat and other agricultural products is
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carried out by passing a hot gas through a bed of grains (Giner and Denisienia, 1996), cotton seeds (Tabak et al., 2004), coffee beans (Agullo and Marenya, 2005). The polymer and ceramic processing industry are inundated with numerous steps wherein a liquid permeates and flows through a porous matrix, for example, during consolidation of warp-knitted reinforced laminates (Rozant et al., 2001; Nielsen and Pitchumani, 2002; Idris et al., 2004), impregnation of compressible fiber mats with a resin (Parnas and Phelan, 1991; Lin et al., 1994; Phelan and Wise, 1996; Choi et al., 1998; Michaud and Manson, 2001; Michaud and Mortensen, 2001; Abrate, 2002), and in the production of fiber composites via the liquid metallurgy route (Reed, 1993; Bhat et al., 1995; Nagelhout et al., 1995). Last but not the least, the flow of polymer solutions on their own and together with a gas is extensively encountered at various stages of the drilling and enhanced oil recovery operations (Burcik, 1968, 1969; Jennings et al., 1971; Dabbous, 1977; Dreher and Gogarty, 1979; Schramm, 1996; Ates and Kelkar, 1998; Baca et al., 2003; Grattoni et al., 2004, etc.). This limited list of porous media flows is concluded by mentioning the use of packed beds as a calming section to eliminate turbulence in a fluid stream prior to entering a test section (Schmid et al., 1999), or to disentangle polymer melts prior to processing (Done et al., 1983; Bourgeat and Mikelic, 1996; Goutille and Guillet, 2002). The flow of fluids in ligaments, tendons, tumors, and biological tissues (Chen et al., 1998; Khaled and Vafai, 2003), in paper sheets (Sayegh and Gonzalez, 1995; Reverdy-Bruas et al., 2001), and in gel permeation chromatography columns (Hoagland and Prud’homme, 1989) provide additional applications of porous media flows. In view of such a wide occurrence of porous media flows, it will be no exaggeration to say that the phenomenon is ubiquitous!