Fundamental solutions of an extended hydrodynamic model in two dimensions: Derivation, theory, and applications

TL;DR

This paper develops a numerical framework using fundamental solutions to solve an extended hydrodynamic CCR model for rarefied gas flows in two dimensions, capturing effects that classical models miss.

Contribution

It derives the two-dimensional fundamental solutions for the CCR model and applies the method of fundamental solutions to simulate rarefied gas flows with validated results.

Findings

The method efficiently simulates quasi two-dimensional rarefied flows.

The CCR model accurately describes rarefaction effects like transpiration and thermal stress.

Fundamental solutions are essential for low-cost, accurate simulations in microscale gas flows.

Abstract

The inability of the Navier-Stokes-Fourier equations to capture rarefaction effects motivates us to adopt the extended hydrodynamic equations. In the present work, a hydrodynamic model, which consists of the conservation laws closed with the recently propounded coupled constitutive relations (CCR), is utilized. This model is referred to as the CCR model and is adequate for describing moderately rarefied gas flows. A numerical framework based on the method of fundamental solutions is developed to solve the CCR model for rarefied gas flow problems in quasi two dimensions. To this end, the fundamental solutions of the linearized CCR model are derived in two dimensions. The significance of deriving the two-dimensional fundamental solutions is that they cannot be deduced from their three-dimensional counterparts that do exist in literature. As applications, the developed numerical framework based on the derived fundamental solutions is used to simulate (i) a rarefied gas flow between two coaxial cylinders with evaporating walls and (ii) a temperature-driven rarefied gas flow between two non-coaxial cylinders. The results for both problems have been validated against those obtained with the other classical approaches. Through this, it is shown that the method of fundamental solutions is an efficient tool for addressing quasi two-dimensional multiphase microscale gas flow problems at a low computational cost. Moreover, the findings also show that the CCR model solved with the method of fundamental solutions is able to describe rarefaction effects, like transpiration flows and thermal stress, generally well.