Interdisciplinary Applied Mathematics

Po » Po » h P₀⁺ hP₀ ’

where ДР/P,o represents the nondimensional pressure drop. Concentrating on the term M/hP_o — (p_ou_o)/P_o and using the continuity equation (u_o — u_ip_i/p_o) and the equation of state for an ideal gas (p_i/p_o — P_i/P_o, assuming isothermal conditions) , we obtain

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

Since P_o — p_oRT, and c² — pRT, where c_s is the speed of sound, the above equation can be simplified as

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 pM_oM_i ^ 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 ^/P_o to X/c_s, 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.

Скачать в pdf «Interdisciplinary Applied Mathematics»