Interdisciplinary Applied Mathematics

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Po » Po » h P0+ hP0


where ДР/P,o represents the nondimensional pressure drop. Concentrating on the term M/hPo(pouo)/Po and using the continuity equation (uo — uipi/po) and the equation of state for an ideal gas (pi/poPi/Po, assuming isothermal conditions) , we obtain


ДР _ L i_ р0и0щ f P_ _Л Po ~ h PQ P0P0    )■


Since PopoRT, and c2pRT, where cs is the speed of sound, the above equation can be simplified as


ДР



4y + -‘MMl7T’



(4.2)


where M denotes the Mach number at respective locations. Rearranging, we obtain


д p    т t


-^(I-OM.MO.2—.


Without further simplification we see that the inertial terms in the momentum equation (right-hand side of equation (4.1)) can be neglected if pMoMi ^ 1. To this end, we note that:


1.    In microchannels with « 103 to 104, relatively large pressure drops can be sustained for small Mach number flows.


2. Since the Mach number in microflows is usually small, the inertial effects are small. Therefore, we expect semianalytic formulas based on    balancing the pressure    drop    with    drag    on    the    channel    walls    to


work reasonably well. (This is not true for micronozzles; see Section


6.6.)


3. If the diffusion term is simplified by approximating the wall shear stress as т to цп/h and recognizing ^/Po to X/cs, we obtain


д p    T


— (1-7М0М)«2-М0Кп0.    (4.3)


The above relation indicates the relative importance of compressibility effects in the slip flow regime.

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