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Diffusive Boltzmann equation, its fluid dynamics, Couette flow and Knudsen layers
Fluid Dynamics · Submitted 7 Jan 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We discover that the full-fledged Knudsen velocity boundary layers develop with all tested closures when the viscosity and diffusivity are appropriately scaled in the vicinity of the walls.
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The abstract, sentence by sentence
In the current work we construct a multimolecule random process which leads to the Boltzmann equation in the appropriate limit, and which is different from the deterministic real gas dynamics process. We approximate the statistical difference between the two processes via a suitable diffusion process, which is obtained in the multiscale homogenization limit. The resulting Boltzmann equation acquires a new spatially diffusive term, which subsequently manifests in the corresponding fluid dynamics equations. We test the Navier-Stokes and Grad closures of the diffusive fluid dynamics equations in the numerical experiments with the Couette flow for argon and nitrogen, and compare the results with the corresponding Direct Simulation Monte Carlo (DSMC) computations. We discover that the full-fledged Knudsen velocity boundary layers develop with all tested closures when the viscosity and diffusivity are appropriately scaled in the vicinity of the walls. Additionally, we find that the component of the heat flux parallel to the direction of the flow is comparable in magnitude to its transversal component near the walls, and that the nonequilibrium Grad closure approximates this parallel heat flux with good accuracy.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In the current work we construct a multimolecule random process which leads to the Boltzmann equation in the appropriate limit, and which is different from the deterministic real gas dynamics process. | contribution 0.94 | 0.95 | 0.07 | asserted 1.00 |
| s1 | We approximate the statistical difference between the two processes via a suitable diffusion process, which is obtained in the multiscale homogenization limit. | method 0.93 | 0.93 | 0.06 | asserted 1.00 |
| s2 | The resulting Boltzmann equation acquires a new spatially diffusive term, which subsequently manifests in the corresponding fluid dynamics equations. | result 0.56 | 0.81 | 0.20 | asserted 1.00 |
| s3 | We test the Navier-Stokes and Grad closures of the diffusive fluid dynamics equations in the numerical experiments with the Couette flow for argon and nitrogen, and compare the results with the corresponding Direct Simulation Monte Carlo (DSMC) computations. | method 0.97 | 0.95 | 0.05 | asserted 0.85 |
| s4 | We discover that the full-fledged Knudsen velocity boundary layers develop with all tested closures when the viscosity and diffusivity are appropriately scaled in the vicinity of the walls. | result 1.00 | 0.96 | 0.32 | asserted 1.00 |
| s5 | Additionally, we find that the component of the heat flux parallel to the direction of the flow is comparable in magnitude to its transversal component near the walls, and that the nonequilibrium Grad closure approximates this parallel heat flux with good accuracy. | result 1.00 | 0.96 | 0.17 | asserted 1.00 |
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The paper
Original abstract
In the current work we construct a multimolecule random process which leads to the Boltzmann equation in the appropriate limit, and which is different from the deterministic real gas dynamics process. We approximate the statistical difference between the two processes via a suitable diffusion process, which is obtained in the multiscale homogenization limit. The resulting Boltzmann equation acquires a new spatially diffusive term, which subsequently manifests in the corresponding fluid dynamics equations. We test the Navier-Stokes and Grad closures of the diffusive fluid dynamics equations in the numerical experiments with the Couette flow for argon and nitrogen, and compare the results with the corresponding Direct Simulation Monte Carlo (DSMC) computations. We discover that the full-fledged Knudsen velocity boundary layers develop with all tested closures when the viscosity and diffusivity are appropriately scaled in the vicinity of the walls. Additionally, we find that the component of the heat flux parallel to the direction of the flow is comparable in magnitude to its transversal component near the walls, and that the nonequilibrium Grad closure approximates this parallel heat flux with good accuracy.
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How the literature uses it
1 arXiv paper cite this in 16 sentences
Papers citing it · 13 sentences without a confident label
Uses its method or tool 1 paper
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As in [ref] , for the computation of $E_1(x)$ we use the approximation proposed in [ref] .
Nonequilibrium diffusive gas dynamics: Poiseuille microflow · Near-wall scaling of viscosity and mass diffusivity · 0.91
Uses its data or values 1 paper
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* The adiabatic constant was set to $γ=1.4$, * The Prandtl number was set to $=0.69$, * The higher-order Prandtl numbers for the regularized Grad equations were set to the same values as for argon above [ref] , $_Q=3/2$, $_R=2/3$ and $_R=7/6$, * The molar mass $M$ was set to $2.801·10^-2$ kg/mol, * The viscosity $μ$ and empirically scaled mass diffusivity were as in [ref] above, with the reference temperature set to $T^*=288.15$ K, whereas the reference viscosity and empirical scaled mass diffusivity constants were set to $μ^*=1.74·10^-5$ kg/(m sec) [ref] , and $D_α^*=10^-6$ kg/(m sec), respectively.
Nonequilibrium diffusive gas dynamics: Poiseuille microflow · Nitrogen · 0.96
Cites it as background 1 paper
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The diffusive Boltzmann equation and the corresponding hierarchy of the diffusive moment closure equations were derived in a recent work [ref] , including the equilibrium (diffusive Navier-Stokes) and nonequilibrium (diffusive Grad) closures.
Nonequilibrium diffusive gas dynamics: Poiseuille microflow · Introduction · 0.84
What it cites · 2 sentences without a confident label
Uses the method or tool of 1 paper
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Then, we approximate the difference between the two dynamics via a multscale formalism in the homogenization time limit [ref] , which equips the Boltzmann equation and the corresponding fluid dynamics equations with an additional spatially diffusive term.
Deterministic homogenization for fast-slow systems with chaotic noise · Introduction · 0.92
Cites as support 1 paper
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We discover that if the viscosity and diffusivity are appropriately scaled in the vicinity of the walls [ref] , then all studied moment closures develop the full-fledged Knudsen velocity boundary layers in agreement with the DSMC computations.
Gas near a wall: a shortened mean free path, reduced viscosity, and the manifestation of a turbulent Knudsen layer in the Navier-Stokes solution of a shear flow · Introduction · 0.96
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