Interdisciplinary Applied Mathematics

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TNS


Tij



du



+



dui



dxi    dx



2 dum c    *dum c


I /’ 7. T.    67 .у    C~    67



3 dxm



’ dxm



ij,



(2.17)


where p and Z are the dynamic (first coefficient)) and bulk (second coefficient) viscosities of the fluid, and Sij is the Kronecker delta. The heat flux is determined from Fourier’s law (equation (2.3)). This level of conservation equations defines the compressible Navier-Stokes eqnations.


In the slip flow regime, the Navier-Stokes equations (2.16), (2.17) are solved subject to the velocity slip and temperature jump boundary conditions given by


2    TV



1



us    uw


T — T


T s    Tw



av p(2RTw /n)i/2



3 Pr(7 — 1).    .


Ts + — ,    ^



2<jt


Tt



2(Y — 1)



4 YpRTw 1



Y +1



Rp(2RTw / n)1/2



(    qn),



(2.18)


(2.19)


where qn,qs are the normal and tangential heat flux components to the wall.    Also,    ts    is    the    viscous    stress    component    corresponding to    the    skin


friction, y is the ratio of specific heats, uw and Tw are the reference wall velocity and temperature, respectively. Here Pr is the Prandtl number


pr =Cpl1


к


Equation (2.19) was proposed by Maxwell in 1879. The second term in (2.19) is associated with the thermal creep (transpiration) phenomenon, which can be important in causing pressure variation along channels in the presence of tangential temperature gradients (see Section 5.1). Since the fluid motion in a rarefied gas can be started with tangential temperature variations along the surface, the momentum and energy equations are coupled through the thermal creep effects. In addition, there are other thermal stress terms that are omitted in classical gas dynamics, but they may be present in rarefied microflows, as we discuss in Section 5.1. Equation (2.19) is due to von Smoluchowski (Kennard, 1938); it models temperature jump effects. Here avare the tangential momentum and energy accommodation coefficients, respectively (see Section 2.2.2). After nondimensionalization with a reference velocity and temperature, the slip conditions are written as follows: where the capital letters are used to indicate nondimensional quantities. Also, n and s denote the outward normal (unit) vector and the tangential (unit) vector.

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