Interdisciplinary Applied Mathematics

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3.2.2 Shear Stress Model


The shear stress for Couette flows exhibits two distinct behaviors in the continuum and free-molecular flow regimes. Using the classical constitutive laws employed in the Navier-Stokes equations, the shear stress for plane Couette flow is given by





where the viscosity p = (2RTw/п)1/2р0Х does not depend on pressure, and the minus sign is due to direction of the shear stress on the fluid. So, in the hydrodynamic approximation the shear stress is proportional to the velocity gradient, and this representation is also valid in the slip flow regime with the appropriate velocity slip corrections. In the free-molecular flow regime,


the shear stress is proportional to the density and relative velocity of the plates, and it is given by (Kogan, 1969)


Too =    (3.10)


In the free-molecular flow regime, the shear stress is due to the tangential momentum exchange between the two plates that interact via the impinging and reflecting molecules. We note that this behavior is independent of (du/dy); hence, a free-molecular shear stress exists even if du/dy ^ 0.


An analytical expression was derived by (Cercignani, 1963) for the shear stress using different molecular interaction models, i.e.,





(3.11)

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