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By R. K. Shah

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DIFFERENTIAL EQUATIONS AND BOUNDARY CONDITIONS "^t,e /"tW V =U-wall 1—! Ϋ///////////////////4 te wall 2 first kind ^ second Kind x=0 111{Ί11 (11Ί11Ί/i11'^-L_wg|| v te q"=0 /■tw \ te S É = 3 = S — ^ w a l l 2-κ= V7 ^ third kind fourth a fifth kinds FIG. 8. Five kinds of fundamental boundary conditions for doubly connected ducts. for any combination of these boundary conditions can be obtained by super­ position techniques [7,41,42]. The fundamental boundary conditions of the fifth kind are added here to Kays' classification because they are mathe­ matically most amenable for fully developed flow through doubly connected ducts.

F. Reynolds Number Re This is defined as Re = ρηΜμ = GDJß (131) For internal flow, Re is proportional to the ratio of flow momentum rate ("inertia force") to viscous force for a specified duct geometry. For example, it can be shown that the ratio of flow momentum rate to viscous force for fully developed laminar flow through a circular tube and parallel plates of length L is (Dh/24L) Re and (Dh/40L) Re, respectively. Thus, Reynolds num­ ber is a flow modulus. If it is the same for two systems that are geometrically and kinematically similar, then dynamic similarity is also realized irrespec­ tive of the fluid.

Sandali and Hanna [58] are the first investigators who devised an approxi­ mate analytical method to determine L& H1 for a laminar or turbulent flow of a power law non-Newtonian fluid flowing through a circular or noncircular duct. They considered constant fluid properties and hydrodynamically de­ veloped flow at the entrance of a duct. They made an overall energy balance between two duct cross sections, x = — L and x = L th , where — L is the upstream duct length at which axial heat conduction effects are no longer B.

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