Interdisciplinary Applied Mathematics

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where nxy is the shear stress normalized with the free molecular shear stress. The values of the coefficients a1, a2, a3, and a4 in equation (3.11) depend on the molecular interaction model used, and here we present the coefficients corresponding to the hard-sphere molecules. (Sone et al., 1990) have obtained accurate numerical solutions for shear stress in the entire Knudsen regime. They also developed the following analytical solution valid for Kn ^ 0, using perturbation expansions


nxy



Yin Kn


2(1 + 2(71 Kn)’ 71



1.270042,



Ci — 1.111.


(Bahukudumbi and Beskok, 2003) presented a shear stress model, similar in form to the result of (Cercignani, 1963), as follows:


a



nxy


0.5297,



aKn2 +26 Kn a Kn2 +c Kn +6’ b — 0.6030, c



(3.12)



1.6277,


where the coefficients a, b, and c are obtained by a least squares fit to the linearized Boltzmann solution of (Sone et al., 1990). It is important to test the model for uniform convergence to the correct continuum (Kn ^ 0) and free-molecular (Kn ^ ж) limits. A Taylor series expansion of the new model in the above-mentioned limits is given by


т


ТЖ



nxy



1 | 26-c 1    |-b-(^)c


a Kn    a



1



Kn2



+ O(Kn-3)



as Kn ^ ж, (3.13)



т



Tcont



nxy



1 +



a — 2c 26



Kn +



)c~2a


26



Kn2 +O(Kn3)


as Kn ^ 0,    (3.14)


where the coefficients of the O(Kn*) terms (i — …, —2, —1,1, 2,…) are corrections to the shear stress due to different orders of Kn dependence. It can be seen that the coefficient of the O(Kn) term in the expansion for П is = —2.2601 « —2Ci = 2.222. Comparing equation (3.14) with the asymptotic theory in (Sone et al., 1990), for the continuum limit, we obtain that the two representations are similar up to second-order terms in Kn.

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