Interdisciplinary Applied Mathematics

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In order to identify the relative importance of inertial terms in the momentum equation compared to the diffusion terms, we compare their respective magnitudes:


РиШ _ Iм2!L =    (h = Rp fh


~ Pu/h2 P L)    L) ‘


A similar estimate can be obtained by taking the ratio of inertial terms to diffusion terms in equation (4.3) as


^(7М0М;) _ M; h АР _ M; h AP _ hReAP ^M0Kn0    KnQL P0 Kn;L Pi    L 6Pi


where we have used


Kno



^Knb


Po



and



Kn



M


Re’


in order to arrive at the third and the fourth equations, respectively. Therefore, the above two estimates are similar, with the exception of the term


AP = Р,- P0


Pi Pi


which is always smaller than unity. This analysis verifies that for relatively low Re flows (Re < O(1)) in large aspect ratio channels (L/h ^ 1) the inertial effects in the momentum equation can be neglected. Under such conditions the momentum equation in the streamwise direction is reduced to the familiar form


This equation results in the following analytical solution for the streamwise velocity profile:


У    y 2    &v (    2    17    


l?-h + ^T(KnKn)



dP



д 4 5i



dx ^ dy5



(4.4)


,2    _


……. »    ‘    (4-5) where we have used the high-order boundary condition (equation (2.26)). Notice that the second-order correction, which is typically omitted in other works, has the opposite sign compared to the first-order term; its contribution may be significant, especially for high Kn flows. We will examine this in Section 4.1.3.

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