# Pressure-Driven Nitrogen Flow in Divergent Microchannels with Isothermal Walls
## 1 Introduction  
Gas flow in micro- and nano-channels with nonuniform cross-sections offers opportunities to develop small devices with novel applications (see, for example, [1, 2, 3, 4, 5, 6]). Understanding heat and fluid flow in microchannels is essential to engineer novel micro-electromechanical systems (MEMS) [7, 8, 9]. However, this is a challenging task since the rate of collisions between the gas molecules and solid walls reduces with decreasing characteristic length-scale of the fluid flow or the gas density, affecting the random movement of molecules, which is commonly known as nonequilibrium effects [10, 11]. Nonequilibrium molecular transport processes cause velocity slip and temperature jump at walls, and influence thermal and fluid flow fields [12, 13]. The Knudsen number (Kn), which is the ratio of the molecular mean free path \(\lambda\) to a characteristic length scale \(\mathcal{L}\), is often employed as an indicator of deviation from the equilibrium condition. The higher the Knudsen number, the higher the deviation from the equilibrium condition. Four different rarefaction regimes have been defined based on the Knudsen number [14], and are commonly named the free molecular (Kn \(>10\)), transition (\(10^{-1}\leq\text{Kn}\leq 10\)), slip (\(10^{-3}\leq\text{Kn}\leq 10^{-1}\)), and continuum (Kn \(\leq 10^{-3}\)) regimes. Micro-devices usually operate in the slip and transition rarefaction regimes at standard pressure and temperature [12]. It is widely acknowledged that the standard Navier-Stokes-Fourier (NSF) equations fail to accurately predict thermal and fluid flow fields when the nonequilibrium effects are significant [15; 16; 17]. The Boltzmann equation governs fluid flow at micro-scales in the entire range of Knudsen numbers [18]. However, the direct numerical solution of the Boltzmann equation is generally expensive with available computational capabilities, and is limited to simple applications because of the high dimensionality of the Boltzmann equation and the complexity of the collision integral. Moreover, the application of the Boltzmann equation is limited to dilute gasses due to the assumption of binary intermolecular collisions [19]. The direct simulation Monte Carlo (DSMC) method is a numerical technique that has been developed based on the kinetic theory to approximate the solution of the Boltzmann equation [18]. [PERSON] et al. [20] compared the numerical predictions obtained from DSMC simulations with the solutions of the NSF equations for a Poiseuille gas flow between two parallel plates, and reported quantitative and qualitative differences between the predictions. Deviations between numerical predictions obtained using the DSMC method and the NSF equations increases with increasing Knudsen number [20; 21; 22; 23]. Using the regularised 13-moment equations, [PERSON] and [PERSON] [24] achieved better agreement with the DSMC results up to \(\text{Kn}\approx 1\) in comparison with the NSF-based solutions. [PERSON] et al. [25] studied Poiseuille gas flow in channels with finite lengths using the DSMC method, and reported that the nonequilibrium effects are the most significant factor affecting rarefied gas flow in channels. The majority of the published literature on subsonic internal gas flows at micro-scales is limited primarily to flow passages with uniform cross-sections [26; 27; 28]. Moreover, previous studies mostly focused on characterising pressure loss, mass flow rate, and velocity field [29]. There is a crucial lack of detailed thermal field analyses of gas flow in micro- and nanochannels with nonuniform cross-sections in which nonequilibrium effects play an important role [30]. Using the NSF equations with the first-order slip boundary condition, [PERSON] et al. calculated temperature variations in divergent [31] and convergent [32] microchannels for both continuum and slip flow regimes. Using the same approach, [PERSON] et al. [33; 34] calculated temperature distribution in convergent microchannels in the slip and early-transition rarefaction regimes.