Planar Wedge Channel

As illustrated in Fig. 5.2, the classic Jeffery-Hamel flow concerns two-dimensional radial flow in a wedge-shaped region between flat inclined walls. The flow may be directed radially outward (as illustrated) or radially inward. The flow is assumed to originate in a line source or terminate in a line sink. Velocity at the solid walls obeys a no-slip condition. In practice, there must be an entry region where the flow adjusts from the line source to the channel-confined flow with no-slip walls. The Jeffery-Hamel analysis applies to the channel after this initial adjustment is accomplished. 

An essential assumption of the Jeffery-Hamel flow is that only the radial velocity is nonzero. However, there remain both radial and circumferential variations of axial velocity. 

Moreover the analysis retains both radial and circumferential pressure variations. Under these circumstances the continuity and Navier-Stokes equations reduce to 

From the continuity equation, it is apparent that the product vr can at most be a function of 0, 

The Jeffery-Hamel analysis seeks solutions for the radial velocity v in a separable form as 

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Consider the Jeffrey-Hamel flow in a planar wedge channel as discussed in Section 5.2, where a similar solution for f 6) is developed. The ordinary differential equation that describes the scaled velocity / does not directly involve pressure. The objective of this exercise is to recover the pressure field from the similar solution. 

For the more general case when surface tension and viscous effects are appreciable, there are few data available. Grigorev and Krokhin (GIO) presented some results for the rise of bubbles in thin rectangular slits and wedge-shaped channels, while Schad and Bishop (S2) investigated bubble rise in thin annular and planar gaps. 

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