Interdisciplinary Applied Mathematics

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We note that despite the geometric differences between a sphere and a cylinder, a substantial drag force due to the viscous normal stress is observed for both geometries for increased Kn values.

6.4-2 Sphere-in-a-Pipe

For a sphere located along the axis of the pipe, we consider the cases of a moving sphere in a stationary pipe, and Poiseuille flow past a stationary sphere in a pipe. The flow is axisymmetric for both cases. Considering the sphere diameter D and pipe diameter H, the moving sphere case results in the following total drag force (Haberman and Sayre, 1958):


_QTrpURjl- 0.75857(g)5)_

(l- 2.1050f + 2.0865(f)3 — 1.7068(f)5 + 0.72603(f)6)’    ‘ while the pressure-driven flow past a stationary sphere-in-a-pipe results in

p =_6тгМ7Д(1 — |(f)2 — 0.20217(f)5)_

°    (l- 2.1050f + 2.0865(f)3 — 1.7068(f)5 + 0.72603(f)6)’ 1 ‘    

where U is the sphere velocity in equation (6.22), and it is the pipe maximum velocity in equation (6.23).

FIGURE 6.20.    Variation    of    the    normalized    total    drag    coefficient    for    a    confined

sphere-in-a-pipe as a function of the Knudsen number in the slip flow regime. Results for various blockage ratios (H/D) are presented. (Courtesy of R.W. Barber and D.R. Emerson.)

Flow past a stationary sphere in a pipe has been investigated by (Liu et al., 1998) using p,Flow and DSMC simulations. In a later study, Barber and Emerson (2001) presented numerical results for the same problem using continuum-based slip models. Figure 6.20 presents the drag coefficient variation as a function of the Knudsen number for various H/D ratios. For convenience, the results are normalized by the drag coefficient of the external Stokes flow in equation (6.20). The drag force increases with decreasing H/D, due to the increased blockage effects. In the continuum flow limit, there is a tenfold increase in the drag force for H/D = 2. However, the drag force increases by a factor of two for H/D = 40. Numerical results in (Barber and Emerson, 2002) for Kn = 0.1 flow at H/D = 2 show about 50% drag reduction due to the slip flow effects. However, for H/D = 40, drag reduction due to velocity slip is about 10%.

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