# Interdisciplinary Applied Mathematics

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For    incompressible    flows    over    flat    no-slip    surfaces    the    viscous    normal

stress components тп are identically zero. However,

in    slip    surfaces    the    viscous    normal    stresses achieve    finite    values    and

increase substantially proportional to the Knudsen number.

This effect is demonstrated in Figure 6.19, where we plot the viscous normal stress distribution around the cylinder periphery. In this case, the viscous normal stresses for no-slip flow are nonzero due to the curvilinear boundary. However, they are considerably smaller compared to the slip flow even for this relatively small value of Knudsen number (Kn = 0.015). Levels of the pressure distribution are also reduced compared to the no-slip case in agreement    with    the results    of    (Gampert,    1978).    The    reduction    in    the pressure levels is balanced by the increase in the viscous normal stresses. Therefore:

• The total normal stresses (i.e., combined pressure and viscous normal stresses) do not vanish as the rarefaction effects increase (Gampert, 1978).

##### 6.4 Flow Past a Sphere: Stokes Flow Regime

In this section, we present analytical and numerical results for flow past a sphere in the Stokes flow regime (Re ^ 0). We will examine both the unbounded external flow and the sphere-in-a-pipe flow problems. The results in the continuum flow regime are equally applicable to the gas and liquid flows. However, the slip flow results are relevant to the gas flows. The external flow problem may be useful in estimating the drag force on airborne microparticles; see Section 14.3.2. The sphere-in-a-pipe problem is a prototype rarefied gas dynamics problem. Large-scale, low-pressure applications of this flow configuration are commonly used in spinning rotor gauge (SRG) devices (Fremerey, 1985), where damping of an electromagnetically supported rotating sphere is used for pressure, viscosity, molecular weight, and rate of molecular impingement measurements (Fremerey, 1982). A spherical particle moving in a pipe is also observed in Coulter counter devices (Ito et al., 2003). However, particle surface charge and electric double layer play an important role in these cases (see Section 7.5 for details).

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