Treatments of Micro-channel Flows Revisited: Continuum Versus Rarified Gas Considerations
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Treatments of Micro-channel Flows Revisited: Continuum Versus Rarified Gas Considerations F. Durst1
•
D. Filimonov1 • R. Sambasivam2
Received: 26 May 2020 / Accepted: 27 May 2020 Ó The Institution of Engineers (India) 2020
Abstract There are numerous treatments of micro-channel flows available that point out the breakdown of the Navier– Stokes equations under molecular flow conditions when continuum conditions should still apply. The wrong conclusion regarding the validity of the Navier–Stokes equations comes about from the missing mass diffusion terms in the continuity equation. This term also affects the Navier– Stokes equations and yields improved theoretical results for flows with strong pressure and temperature gradients. Rarified gas treatments are often applied, although the fluid mechanics conditions are still such, that continuum equations should work. In the paper, the missing mass diffusion term is added, and it is shown that those extended fluid mechanics equation (EFME) allows micro-channel flows to be treated in the so-called slip regime. Hence, the continuum approach for flow treatment holds in micro-channel flows, in the flow regimes where modeling of wall interactions is applied these days. The paper also describes the treatments of micro-channel flows in the slip-flow regime by the rarified gas flow treatment method of Shapiro and Seleznev, and the results are compared with the corresponding results of the EFME. Good agreement was obtained, but differences exist regarding the wall interactions, which are explained, and suggested to use both methods to obtain a deeper insight into molecule–wall interactions in micro-channel flows. The EFME claim that large pressure and temperature gradients are the reasons for the differences between experimental and theoretical & F. Durst [email protected] 1
FMP Technology GmbH, Am Weichselgarten 34, 91058 Erlangen, Germany
2
Petronas Holdings, 50088 Kuala Lumpur, Malaysia
results of micro-channel flows. Such differences also exist in other fluid flows with strong property gradients, e.g., in shock waves. Flows of this kind are also treated in this paper in a way to show that the EFME have a wide range of applications. Keywords Extended Navier–Stokes equation Micro-channel flows Rarified gas flows Shock wave flows List of Symbols D Coefficient of mass diffusion h Half-height of the parallel plate microchannel ‘ Molecular mean free path L Length of the micro-channel and microcapillary MD Diffusive mass flow rate mM Molecular mass MT Total mass flow rate P Pressure Pc Characteristic pressure R Gas constant t Time T Temperature UT, UC, UD Total, convective and diffusive fluid velocity in the x-direction, respectively Ui Velocity in the i-direction U; V Velocity components in the x- and ydirections, respectively u Velocity vector w Width of the micro-channel x; y; r Components of rectangular and cylindrical coordinates
123
J. Inst. Eng. India Ser. C
Greek Symbols l Viscosity of the fluid k Heat conductivity q Density o
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