Introduction to Microscale Flows and Mathematical Modelling

Derivation of extended-OBurnett and super-OBurnett equations and their analytical solution to plane Poiseuille flow at non-zero Knudsen number

“…2008; De Groot & Mazur 2013; Agrawal et al. 2020), as confirmed in Singh et al. (2017), Agrawal et al.…”

“…The linear forms of the Wood, conventional and super-Burnett equations are unstable when the wavenumber is larger than a critical value (Bobylev 1982(Bobylev , 2006Welder et al 1993;Agrawal et al 2020), while the proposed equations are unconditionally stable at Burnett as well as a super-Burnett level, as shown in § 4. Additionally, in comparison with one-dimensional flow, the value of the critical Knudsen number for the onset of instability becomes smaller in the 2-D flow for the original and conventional Burnett equations (Bao & Lin 2005), whereas there is no such limitation in the proposed equations.…”

“…The improvement of manufacturing technology and computational capacity has led to increasing attention to the importance of rarefied gas flows in various practical applications such as high-speed hypersonic flows (Ivanov & Gimelshein 1998), miniaturized microchannel flows (Agrawal, Kushwaha & Jadhav 2020; Akhlaghi, Roohi & Stefanov 2023) and rarefied flow conditions (Akintunde & Petculescu 2014). In such flow situations, the Knudsen number (), which is the ratio of the mean free path () to the characteristic length scale (), plays a crucial role in governing the flow physics.…”

“…However, the N-S equations can capture the flow physics in the early-slip flow regime by employing a -dependent slip and temperature jump boundary conditions (Agrawal et al. 2020); such modifications still yield inaccurate predictions of the pressure drop, shear stress, heat flux and mass flow rate in the late-slip and transition flow regimes. Thus, improved flow modelling capabilities are needed to accurately describe flows, particularly in the transition flow regime.…”

“…In such flow situations, the Knudsen number (Kn = λ/H), which is the ratio of the mean free path (λ) to the characteristic length scale (H), plays a crucial role in governing the flow physics. For the case of a microchannel, the flow generally spans the slip (10 −3 < Kn < 10 −1 ) or transition (10 −1 < Kn < 10) regimes for which the well-known Navier-Stokes (N-S) equations based on the continuum formulation fail to predict the primary variables correctly (García-Colín, Velasco & Uribe 2008;Agrawal et al 2020). Further, several analytical and numerical studies have demonstrated that the N-S equations do not capture, even qualitatively, non-equilibrium phenomena occurring in the plane Couette and Poiseuille flows and lid-driven cavity problems in the slip flow regime (Tij & Santos 1994;Mansour, Baras & Garcia 1997;Zheng, Garcia & Alder 2002;Benzi, Gu & Emerson 2010, 2013.…”