Interdisciplinary Applied Mathematics

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+e


~[ЦMoKno ^ (f) (UUyx)y + Uxy + Vy Y-KhqMq ^ ((^) (PxVUyy + PxUUxy)


+ (t “ T + t) РжУЛ)


+e


(Y)PyUa


Neglecting the O(e) terms, the x- and у-momentum equations are reduced to (in dimensional form)


dp


dx



dp


dy



1 — (^2/3 + we/12)



+ (^e/122^2/3)



d


dy



p2


U2_


p



du2

dy)


du2


dy)



d2u

d0^2+O(e)’


O(e).


Furthermore, assuming a Maxwellian gas model for which the coefficients 12,Шб) = (10/3, 2, 8) (from (Schamberg, 1947)) and nondimensional-izing with the reference exit conditions (p0,u0), we obtain

Px



127П Kn0 m0



Po


p



2



u2



Uyy + O(e).


Similarly, for the у-momentum equation we obtain


Py



P0



2



2



p



4


3



VtV^Mo Kn0



Po


P



Uy Uyy + O(e),


where the nondimensionalized variables are denoted by capital letters.


It is clear that the Mq Kn0(p/p0)2 term is relatively small for low Mach number flows in the early transition regime (i.e., Kn < 1). In this case, for flow in a very long channel the Burnett equations reduce to


Px = Uyy,    (4.17a)


Py=m° (?)UyUyy (4л7ь)


Therefore, the streamwise Burnett equation is reduced to the form obtained in the Navier-Stokes limit. On the other hand, the cross-flow momentum equation (4.17b) shows that the pressure gradient in that direction is balanced by the Burnett normal stresses, which in the case of continuum are identically zero for a flat surface. The cross-flow momentum equation agrees with the simplified set of equations for Couette flow in the transition regime obtained in (Schamberg, 1947). Results obtained from simulations employing the full Burnett equations do not show any significant deviations from the semianalytical results we have obtained for microchannel flows (Balakrishnan et al., 1999; Agarwal et al., 2001; Agarwal and Yun, 2002; Xu, 2003).

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