# Interdisciplinary Applied Mathematics

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In Figure 6.21 we present the normalized total drag coefficient variation as a    function    of    the    blockage    ratio    H/D    for    a sphere    confined in a pipe

in the continuum (Kn = 0) and slip flow Kn = 0.1 regimes (Barber and Emerson, 2004). The analytical solution of (Haberman and Sayre, 1958) is also shown in the figure. The results clearly indicate that velocity slip reduces the blockage effects, and drag reduction due to the rarefaction is more dominant for large blockage ratios (small H/D).

Wen and Lai (2003) presented analytical solutions for the cases of a sphere moving along the centerline of a micropipe, and a sphere moving parallel to the centerline of a micropipe. Analytical solutions of the Stokes equations with first-order slip boundary condition are obtained using the

FIGURE 6.21.    Variation    of    the    normalized    total    drag    coefficient    for    a    confined

sphere in a    pipe    as    a    function    of    the    blockage    ratio    (H/D).    Results    for no-slip

and Kn = 0.1 flows are shown. (Courtesy of R.W. Barber and D.R. Emerson.)

streamfunction method for the former case, while the method of reflections is utilized for the latter case. This enabled analytical expressions for the resultant force on the sphere, pressure drop over the sphere, and terminal velocity of the sphere (Wen and Lai, 2003).

##### 6.5 Microfilters

Gas microfilter systems can be used for filtering and detection of airborne biological and chemical entities and for environmental monitoring applications. Recent advancements in microfabrication technologies enabled development of sufficiently thin filters that are strong enough to provide useful flowrates under large pressure drops (Chu et al., 1999). Motivated by these developments, several research groups have investigated gas flows through microfilter systems (Yang et al., 1999b; Yang et al., 2001; Mott et al., 2001). Analysis of gas flows through microfilters requires consideration of rarefaction, compressibility, and geometric complexity effects.

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