Wave Motions in Stratified Pipe Flows

A fundamental account of unsteady motion in a dense suspension requires knowledge of phase interactions and origins of wavy stratified motion [Zhu, 1991 Zhu et al., 1996]. Flow stratification involves a dense phase of solids flowing along the bottom of a horizontal pipe with wave motions, or a layer of the dense phase of solids flowing near the wall of a vertical pipe with wave motions. The former was analyzed, in part, on the basis of the theoretical work of de Crecy (1986) on stratified liquid-vapor flows. 

To simplify the analysis of wave motions in stratified pipe flows, several assumptions can be made  

An instantaneous regional average of phase j (j = gas or solids) in the region k is defined as 

From Delhaye (1981), the instantaneous regional-averaged continuity equation projected along the tube axis can be expressed as 

The integrals in the preceding equations can be simplified. It is noted that (1) C(z, t) represents the common boundary of interface A with the cross section plane (2) Ck(z, t) + C(z, t) is the enclosed curve in the cross section and Ck is on the wall of the pipe and (3) kc. located in the cross section plane and directed away from region k, is the unit vector normal to C. The mass transfer rate through the interface is expressed by 

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