RESEARCH PAPER Predicting microscale gas ﬂows and rarefaction effects through extended Navier–Stokes–Fourier equations from phoretic transport considerations Nishanth Dongari • Franz Durst • Suman Chakraborty Received: 3 December 2009 / Accepted: 20 January 2010 Ó Springer-Verlag 2010 Abstract We test an extended continuum-based approach for analyzing micro-scale gas ﬂows over a wide range of Knudsen number and Mach number. In this approach, additional terms are invoked in the constitutive relations of Navier–Stokes–Fourier equations, which originate from the considerations of phoretic motion as triggered by strong local gradients of density and/or temperature. Such aug- mented considerations are shown to implicitly take care of the complexities in the ﬂow physics in a thermo-physically consistent sense, so that no special boundary treatment becomes necessary to address phenomenon such as Knudsen paradox. The transition regime gas ﬂows, which are other- wise to be addressed through computationally intensive molecular simulations, become well tractable within the extended quasi-continuum framework without necessitating the use of any ﬁtting parameters. Rigorous comparisons with direct simulation Monte Carlo (DSMC) computations and experimental results support this conjecture for cases of isothermal pressure driven gas ﬂows and high Mach number shock wave ﬂows through rectangular microchannels. Keywords Extended Navier–Stokes–Fourier equations  Phoretic mass diffusion  Rareﬁed gas ﬂows  Shock waves  DSMC  Effective mean free path 1 Introduction Rapid advancements in micro and nano fabrication tech- nologies over the past few years have triggered the intro- duction of intricate small-scale devices in several emerging applications (Karniadakis et al. 2005). The primary dis- tinction between ﬂuid ﬂows in the micro-scale and the conventional devices originates from the pertinent surface area to volume ratios. This profoundly affects the mass, momentum and energy transport; and leads to additional effects like slip ﬂow, rarefaction, etc. in gas ﬂows. In highly rareﬁed system, deviations from local thermody- namic equilibrium are signiﬁcant. The extent of this devi- ation is not merely dictated by the mean free path (k) in an absolute sense, but also its comparability with the charac- teristic system length scale (L) that describes the relative importance of rarefaction in the system and the ratio of these two, known as the Knudsen number (Kn = k/L). It has commonly been proposed that the no-slip boundary condition may fail to be applicable when the Kn becomes greater than of the order of 10 -3 , although bulk continuum considerations still appear to hold appropriate within that limit. Experimental investigations by Arkilic et al. (1997), Harley et al. (1995), Maurer et al. (2003) and Ewart et al. (2007) have conﬁrmed that the conventional forms of the Navier–Stokes equations together with the no-slip boundary condition may under-predict the experi- mentally observed mass ﬂow rates, and the concerned discrepancies tend to become more severe for higher values of the Kn (Maurer et al. 2003; Ewart et al. 2007). Several authors have theoretically addressed the phe- nomenon of enhanced mass ﬂow rates by employing the continuum equations in conjunction with the applica- tion of the Maxwell slip boundary condition (Maxwell 1879). Identical considerations were also invoked by N. Dongari Department of Mechanical Engineering, University of Strathclyde, Glasgow G1 1XJ, UK e-mail: nishanth.dongari@strath.ac.uk F. Durst FMP Technology GmbH, 91058 Erlangen, Germany S. Chakraborty (&) Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India e-mail: suman@mech.iitkgp.ernet.in 123 Microﬂuid Nanoﬂuid DOI 10.1007/s10404-010-0604-5