Exploring external rarefied gas flows through the method of fundamental solutions

TL;DR

This work addresses the challenge of Stokes' paradox in two-dimensional rarefied gas flows by employing the linearized CCR model and a meshfree method of fundamental solutions (MFS). An analytic CCR solution for flow around a circular disk is derived and validated against the MFS, with an artificial outer boundary used to bound the far field and avoid logarithmic divergence. The approach is then extended to flow past a semicircular cylinder to demonstrate applicability to non-symmetric shapes, revealing temperature polarization and anti-Fourier heat transfer that NSF cannot capture. Drag predictions are compared with classical results, and the study demonstrates that the framework can handle arbitrary 2D shapes, albeit with limitations near walls for higher Knudsen numbers. The work provides a robust, efficient tool for analyzing rarefied gas flows in micro/nano and space contexts and points to future extensions that incorporate convective terms to remove the need for an artificial boundary.

Abstract

The well-known Navier--Stokes--Fourier equations of fluid dynamics are, in general, not adequate for describing rarefied gas flows. Moreover, while the Stokes equations -- a simplified version of the Navier--Stokes--Fourier equations -- are effective in modeling slow and steady liquid flow past a sphere, they fail to yield a non-trivial solution to the problem of slow and steady liquid flow past an infinitely long cylinder (a two-dimensional problem essentially); this is referred to as Stokes' paradox. The paradox also arises when studying these problems for gases. In this paper, we present a way to obtain meaningful solutions for two-dimensional flows of rarefied gases around objects by circumventing Stokes' paradox. To this end, we adopt an extended hydrodynamic model, referred to as the CCR model, consisting of the balance equations for the mass, momentum and energy and closed with the coupled constitutive relations. We determine an analytic solution of the CCR model for the problem and compare it with a numerical solution based on the method of fundamental solutions. Apart from addressing flow past a circular cylinder, we aim to showcase the capabilities of the method of fundamental solutions to predict the flow past other objects in two dimensions for which analytic solutions do not exist or are difficult to determine. For that, we investigate the problem of rarefied gas flow past an infinitely long semicircular cylinder.

Figures (20)

• Figure 1: Schematic of Stokes flow past an infinite circular cylinder of radius $R$, where the fluid is moving transversely to the axis of the cylinder.
• Figure 2: Cross-sectional view of the problem of a rarefied gas flow past an infinitely long cylinder. The solid circle represents the periphery of the cylinder while the dashed circle represents an artificial boundary far away from the cylinder.
• Figure 3: Placement of the collocation points (black dots) on the boundary and singularities (black stars) outside the confined domain. The blue and red arrows at each boundary node denote the unit tangent and normal vectors, respectively.
• Figure 4: Speed of the gas varying with the radial position in different directions for $\mathrm{Kn}=0.1$, $0.5$ and $1$. The solid lines represent the results obtained from the MFS applied to the CCR model and the symbols represent the analytic solutions. The other parameters are $N_{b_1}=N_{s_1}=50$, $N_{b_2}=N_{s_2}=100$, $R_1=1$, $R_2=10$, $R_1^\prime=0.5$, $R_2^\prime=20$.
• Figure 5: Velocity streamlines over speed contours obtained from the MFS applied on the CCR model for the Knudsen numbers $\mathrm{Kn}=0.1$, $0.5$ and $1$. The other parameters are the same as those for Fig. \ref{['fig:speed']}.