Jeffery-Hamel flow

There is a class of flow situations, first identified by Jeffery [201] and Hamel [163], for which the flow has self-similar behavior. To realize the similar behavior leading to ordinary-differential-equation boundary-value problems, the analysis is restricted to steady-state, incompressible, constant property flows. After first discussing the classic analysis, 

Jeffery-Hamel analysis leads to either full or local similarity of the velocity profiles in certain channels and ducts. Since it is based on incompressible flow, there are certainly limitations on its applicability. Nevertheless, given the very significant mathematical reductions, the analysis can be used effectively to provide some important insights about channel flows. 

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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. 

Fig. 5.2 Illustration of the geometry for a Jeffery-Hamel flow between two long inclined plates. The flow either originates or terminates in a line source or sink.

Fig. 5.3 Nondimensional velocity distribution for the Jeffery-Hamel flow between two plates inclined at a = 10°. A negative Re indicates radially inward flow. Separation occurs at approximately Re 4 for the outward flow.

One can analyze the Jeffery-Hamel flow using a nondimensional velocity scaled by the maximum velocity at a radial location [429]. This approach permits determination of the local shape of the velocity profiles but still requires an integral mass-flow constraint to determine the local maximum velocity and hence the specific velocity profiles (i.e., in m/s). Nevertheless, using this approach and the limit of small angle, but large Reynolds number, permits the determination of the separation point as a function of the combined parameter Rea2 alone. 

Figure 5.7 illustrates a spherical variation of the Jeffery-Hamel flow. Here the flow either originates or terminates in a point source or sink. As in the wedge flow (Section 5.2) the analysis here considers steady, incompressible, constant-property flow. 

Problem 10-11. Jeffrey-Hamel Flow. Consider converging flow (that is, flow inward toward the vertex) between two infinite plane walls with an included angle 2a, as shown in the figure. This is known as Jeffery-Hamel flow. 

Bararnia, H., Ganji, Z.Z., Ganji, D.D., Moghimi, S.M., 2010. Numerical and analytical approaches to MHD Jeffery-Hamel flow in a porous channel. Int. J. Numer. Methods Heat Fluid Flow 22, 491-502. 

As an illustration of the mass-transfer behavior in a Jeffery-Hamel channel, consider the following problem Assume that a portion of the wall of a long Jeffery-Hamel channel r < r < r0) is chemically active. For this illustration, assume a channel angle of a = 10° and a Schmidt number of Sc = 0.5. The fluid flows inward (converging direction) at two Reynolds numbers, Re = —10 and Re = —100. The nondimensional radius spans 1 < r < 1.5. Figure 5.4 shows the normalized mass-fraction profiles at several locations in the channel. The nondimensional velocity profiles / for these cases are shown in Fig. 5.3. 

Consider the steady-state, fully developed, incompressible flow between parallel disks, such as illustrated in Fig. 5.9. In concert with the Jeffery-Hamel assumptions that were made in the previous configurations, one can assume that only the radial velocity is nonzero. As a consequence the continuity and momentum equations reduce to the following ... 

For the given geometry and flow conditions, develop a nondimensional representation of the velocity field based on the Jeffery-Hamel wedge flow. Remember that the flow is radially inward, so the Reynolds number is negative. Use the properties of the helium carrier gas. 

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