Ghost Effect and Bifurcation II: Ghost Effect of Infinitesimal Curvature and Bifurcation of the Plane Couette Flow
According to the asymptotic analysis for small Knudsen numbers in Chapter 3, the fluid-dynamic-type equations that describe the behavior of a gas in the continuum limit are classified depending on the size of the parameters in the problem under consideration (see, especially, Section 3.6). There is an important class of problems where the classical fluid dynamics, the Euler or Navier—Stokes system, is inapplicable and an infinitesimal flow velocity plays a decisive role to determine the behavior of the gas (the ghost and non-Navier-Stokes effects). Various examples of the ghost and non-Navier—Stokes effects are given in Chapter 8, in addition to that in Chapter 3. A geometrical parameter can be a source of the ghost effect. We will show an important case of this in this chapter. That is, taking a gas between two rotating coaxial circular cylinders, we consider the behavior of the gas in the limit that the Knudsen number and the inverse of radius (the curvature) of the inner cylinder tend to zero simultaneously, keeping the difference of the radii of the two cylinders fixed. The limiting behavior depends on the relative speed of decay of the two parameters. This singular character in the continuum limit introduces the ghost effect of infinitesimal curvature. Bifurcation of the plane Couette flow, a long-lasting problem of the classical fluid dynamics, occurs owing to the effect of infinitesimal curvature.
KeywordsMach Number Boltzmann Equation Bifurcation Point Continuum Limit Knudsen Number
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