# A phenomenological and extended continuum approach for modelling non-equilibrium flows

## Summary (2 min read)

### 1 Introduction

- Micro-electro-mechanical systems (MEMS) have found many applications in industrial and process systems, biomedical devices, environmental control devices, micro-processor cooling, and high precision printing.
- Rapid progress in micro-engineering has not been matched by an increased understanding of the fundamental physics occurring in such small-scale domains.
- The Navier–Stokes–Fourier (NSF) fluid dynamic equations with conventional no-slip boundary conditions are no longer valid when the system characteristic length scale approaches the molecular mean free path of the gas [5].
- In discrete molecular methods, the fluid is modelled using a microscopic formalism, i.e., as a collection of moving molecules which interact through collisions or very close proximity potentials.
- Extended gas dynamic continuum models can be derived by either perturbation methods, commonly known as the Chapman–Enskog expansion [4], or moment methods [6,9].

### 2 Extended hydrodynamics: the moment method

- The angular brackets <> indicate the traceless part of a tensor.
- Two options exist to derive relationships for these higher-order moments; (i) derive their transport equations, or (ii) obtain a constitutive relationship in the form of a closure approximation in terms of the five lower-order moments, i.e., ρ, vi and T .
- −Pr qi τc , (3) where the collision frequency is defined as τc = µ/p. Higher-order moments ρ<i jk>, ρrr<ik> and ρrrss appear in the viscous stress and heat flux transport equations, Eqs. (2) and (3).
- A closure for the first 13 moments yields the Grad 13-moment equations (G13), whereby ρ<i jk> = 0, ρrr<ik> = 7RT τik and ρrrss = 15p2/ρ. The NSF equations can be derived from both moment equation sets described here.
- A fully coupled solution of any moment equation set larger than five requires additional boundary conditions, which may be derived from kinetic schemes [18].

### 3 A method of differential iteration using moment equations

- The specific structure of the moment equations lends itself to being decoupled into two sub-systems and solved without the need for any additional boundary conditions.
- System I is equivalent to the conservation laws together with constitutive relations for the viscous stress and the heat flux.
- It is assumed that the non-equilibrium fluid property fields are continuous up to the boundaries.
- The solution procedure switches between the two sub-systems until the L2 error norm, defined for the uncontrollable boundary data in system II over successive iterations, is considered to be sufficiently small [12].
- The mathematical details of this iterative procedure have been presented in [10–12].

### 4 Phenomenological variants of simple continuum models

- In continuum models, phenomenological techniques are often used to capture a particular physical behavior rather than modelling it from first principles.
- The subscript ‘wall’ indicates wall boundary conditions.
- Nevertheless, modifying such boundary conditions is not sufficient to fully capture many non-equilibrium effects in the transition regime.
- Modifying the coefficients α1, α2 and β1 in the velocity slip and temperature jump conditions of Eqs. (12) and (13) only changes the magnitude of the wall boundary values.

### 5 Capturing non-equilibrium effects: a combined technique

- Restricting the modelling of rarefied gas flows to the conservation equations and modifications thereof might hamper the possibility of capturing the correct physics occurring in the transition regime.
- The combination of conservation equations and scaling methods alone is not enough to capture all non-equilibrium flow effects.
- Nevertheless, this method can capture some of the stress/strain rate non-linearities occurring in the proximity of solid boundaries.
- We, therefore, propose a solution of the moment equations in conjunction with constitutive scaling methods within the Knudsen layer.
- Hence the same scaling function will be assumed for the momentum flux and thermal heat flux.

### 7 Results

- The G13 equations and their modifications have been solved using differential iteration, in which system I is the governing and constitutive equation set resolving ρ, v1, T , p, τ12 and q2.
- The production terms of the uncontrollable boundary data, i.e., τ11, τ22 and q1, are computed in system II and weighted values thereof appear as source terms in system I in the next iteration.
- In both these cases, the Mach number, defined as Ma = viwall/ √ 2RTwall, was approximately 0.3.
- The results are shown in half-space for clarity, and compare the DSMC predictions to the NSF solution with conventional slip/jump boundary conditions and the new solution to the G13 equations for both test cases.

### 8 Discussion and conclusions

- Figures 2 and 3 show that the combination of the G13 equations with scaling functions achieves a better representation of the Knudsen layer in the near wall region than the standard NSF slip-flow solution.
- As shown in Fig. 3f, this is partly due to the extended constitutive terms in the G13 equations and further improved by the wall scaling function.
- (4) The proposed method captures some of the non-equilibrium effects which are otherwise absent from NSF solutions.
- Additionally, the trend for tangential heat flux, Q1 is better represented in the proposed method.
- Additionally, the authors would like to thank Y. Zheng and L. O’Hare for the informative discussions.

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##### Citations

68 citations

### Cites background from "A phenomenological and extended con..."

...A number of reports have studied the Knudsen layer with DSMC calculations (Bird 1977; Lockerby et al. 2005b), with several appearing in the past year (Gu & Emerson 2007; Lilley & Sader 2007; Mizzi et al. 2007; Struchtrup & Torrilhon 2007; Torrilhon & Struchtrup 2008)....

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...Importantly, these models do not provide a proper treatment of the boundary conditions at the wall (Gu & Emerson 2007; Mizzi et al. 2007; Struchtrup & Torrilhon 2007; Torrilhon & Struchtrup 2008), and indeed can only be formally valid in the outer part of the Knudsen layer since they are derived…...

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...The structure of the Knudsen layer has been studied extensively (Bardos et al. 1986; Cercignani 2000; Sone 2002; Gu & Emerson 2007; Mizzi et al. 2007; Struchtrup & Torrilhon 2007; Torrilhon & Struchtrup 2008)....

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...While Bird (1977), Lockerby et al. (2005b), Gu & Emerson (2007), Mizzi et al. (2007), Struchtrup & Torrilhon (2007) and Torrilhon & Struchtrup (2008) do not report the power-law velocity structure of the Knudsen layer, the DSMC solution of Lockerby et al. (2005b) does exhibit the power-law…...

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...…interest in the application of high-order hydrodynamic models to simulate rarefied flows (Reese et al. 2003; Guo et al. 2006; Gu & Emerson 2007; Mizzi et al. 2007; Struchtrup & Torrilhon 2007; Torrilhon & Struchtrup 2008), with the expectation that such models may be employed in computational…...

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##### References

6,395 citations

### "A phenomenological and extended con..." refers methods in this paper

...Extended gas dynamic continuum models can be derived by either perturbation methods, commonly known as the Chapman–Enskog expansion [ 4 ], or moment methods [6,9]....

[...]

5,311 citations

### "A phenomenological and extended con..." refers methods in this paper

...Such modelling can be performed using either statistical ensemble averages, e.g., direct simulation Monte Carlo (DSMC) [ 2 ], or deterministic methods, e.g., molecular dynamics (MD) [17]....

[...]

3,124 citations

### "A phenomenological and extended con..." refers methods in this paper

...Such modelling can be performed using either statistical ensemble averages, e.g., direct simulation Monte Carlo (DSMC) [2], or deterministic methods, e.g., molecular dynamics (MD) [ 17 ]....

[...]

2,747 citations

### "A phenomenological and extended con..." refers methods in this paper

...Extended gas dynamic continuum models can be derived by either perturbation methods, commonly known as the Chapman–Enskog expansion [4], or moment methods [6,9]....

[...]

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###### Q2. What have the authors stated for future works in "A phenomenological and extended continuum approach for modelling non-equilibrium flows" ?

As shown in Fig. 3f, this is partly due to the extended constitutive terms in the G13 equations and further improved by the wall scaling function. In particular, it can be shown that the proposed solution method for the G13 equations is incapable of capturing non-linearities of normal stress and tangential heat flux occurring in the vicinity of the wall.