Interdisciplinary Applied Mathematics

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first plot, the difference in pressure between the measurements and Grad’s method with no-slip boundary conditions is shown. On the second plot, the


pressure difference is shown again but with the theoretical result obtained with Grad’s 13-moment method and slip boundary condition. The deviations start    at    about    2/3    of    the    channel,    where Kn increases    from    0.1 to


0.15. It is clear that the Grad method captures accurately the deviation due to velocity slip at the wall. For Grad’s method no information about the accommodation coefficient is needed, and this is one of its great advantages.


Another approach is the Chapman—Enskog method, where the distribution function is expanded into a perturbation series with the Knudsen number being the small parameter (see Chapter 2):


f = f0 + Kn f(1) + Kn2 f(2) + ••• .


Here, the first term corresponds to the local Maxwellian distribution, i.e.,


f0 = floc.


Successive high-order equations are obtained by substituting this expansion into the Boltzmann equation (15.5).


From the equation for f(1) we obtain the Newtonian Navier-Stokes equations and Fourier’s law of conduction. Furthermore, assuming a model for the molecular interaction we can obtain explicit expressions for the transport coefficients of the momentum and energy equations. For example, for the hard sphere molecules model the dynamic viscosity is


5 л/тгтквТ ^    16    7Г d2    ’


with d the molecular diameter.


Approximate Forms of Boltzmann Equation


Approximate formulations of the Boltzmann equation can be obtained by simplifying the collision integral. In the limit of very high Knudsen number, i.e., the free-molecular flow, the collision integral is zero, but for arbitrary rarefaction, simplifications are needed to make the Boltzmann equation computationally tractable.

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