Exploring external rarefied gas flows through the method of fundamental solutions

TL;DR

This paper develops an analytic and numerical approach to model two-dimensional rarefied gas flows around objects, overcoming Stokes' paradox by using an extended CCR hydrodynamic model and the method of fundamental solutions.

Contribution

It introduces a novel application of the method of fundamental solutions to solve rarefied gas flow problems with an extended CCR model, addressing limitations of classical equations.

Findings

Analytic solutions for flow past a circular cylinder and semicircular cylinder.

Numerical solutions align well with analytic results, validating the approach.

Demonstrates the method's capability to handle complex geometries without existing analytic solutions.

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.