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arxiv: 2605.19286 · v1 · pith:DWMJWIONnew · submitted 2026-05-19 · ⚛️ physics.flu-dyn

Prescribed Wall-Heat-Flux Control of Blockage and Impulse in a Rarefied Micro-Nozzle

Amirmehran Mahdavi , Ehsan Roohi This is my paper

Pith reviewed 2026-05-20 03:02 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords rarefied gas dynamicsmicro-nozzlewall heat fluxspecific impulseblockagethrust augmentationviscous-thermal compressionDSMC simulation
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The pith

Prescribed wall heating in a rarefied micro-nozzle raises specific impulse from 156 s to 201 s because thermal and pressure thrust gains outweigh the mass-flow penalty.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates how a controlled heat flux applied to the diverging wall of a small converging-diverging nozzle in rarefied gas conditions alters the internal flow and overall performance. Simulations scale the wall heat flux against the inlet kinetic-energy flux to span cooling through strong heating, revealing strong temperature stratification between the wall and the bulk flow. Heating shrinks the effective mass-carrying core, which raises aerodynamic blockage and lowers the mass flow rate. Despite this, the added thermal energy and pressure forces produce a net rise in specific impulse. The internal compression region transforms into a viscous-thermal zone whose response to changing heat flux stays low-dimensional.

Core claim

The central claim is that prescribed wall heat flux supplies an active control method for rarefied micro-nozzle flows through the coupled wall-bulk thermal response. With the nondimensional heat flux Q_w/E ranging from -10.5 percent to 97.3 percent, heating produces wall temperatures more than five times the inlet value while the bulk temperature rises more slowly. This contracts the mass-carrying core, increases blockage, and reduces mass flow, yet strong heating lifts specific impulse from 156 s to 201 s because thermal and pressure-thrust augmentation exceeds the mass-flow penalty. The original internal compression feature develops into a finite viscous-thermal compression zone whose heat

What carries the argument

The nondimensional wall heat flux Q_w/E, which parametrizes the coupled wall-bulk thermal response and drives the transition of the internal compression feature into a viscous-thermal compression zone.

Load-bearing premise

The imposed heat flux scaled by inlet kinetic-energy flux accurately represents the coupled wall-bulk thermal response in the simulations across the full range without unaccounted real-world effects such as surface accommodation or radiation.

What would settle it

An experiment that measures thrust and mass flow rate in a physical micro-nozzle under the same strong-heating wall-flux condition and obtains a specific impulse at or below 156 s would falsify the claim that thermal and pressure augmentation outweighs the mass-flow penalty.

Figures

Figures reproduced from arXiv: 2605.19286 by Amirmehran Mahdavi, Ehsan Roohi.

Figure 1
Figure 1. Figure 1: Converging–diverging micro-nozzle and thermal boundary conditions. All pre [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Combined numerical verification of grid density, time step, and particle-per-cell [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Coupled wall–bulk temperature response on the heated/cooled diverging wall. [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Normalized near-wall tangential slip response on Wall-2. The plotted quantity is [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Signed numerical-schlieren field ∂(ρ/ρ0)/∂(x/L) for representative heat-flux cases. Cooling produces a sharper shock-cell-like compression signature, whereas heating weakens and spreads the compression into a broader viscous–thermal zone [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Composite gradient-length Knudsen-number field for the heat-flux sweep. The [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Effective aerodynamic blockage, 1 − βm, along the nozzle. Heating produces a progressive increase in blockage throughout the diverging section, demonstrating contrac￾tion of the effective mass-carrying core. The same mechanism is shown from the complementary perspective of the mass-flux thickness in [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Effective mass-flux thickness, βm = hm/hgeo, along the nozzle. The reduction of βm downstream of the throat under heating provides direct evidence that wall heat flux contracts the mass-carrying core. 4.6. Thermal–aerodynamic trade-off The blockage mechanism is connected to the integrated performance met￾rics in [PITH_FULL_IMAGE:figures/full_fig_p031_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Thermal–aerodynamic trade-off induced by prescribed wall heat flux. (a) Vari [PITH_FULL_IMAGE:figures/full_fig_p033_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Wall–bulk thermal conditioning and local Nusselt-type response on Wall [PITH_FULL_IMAGE:figures/full_fig_p036_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Coupled viscous–thermal scaling on the thermally forced diverging wall. Panel [PITH_FULL_IMAGE:figures/full_fig_p038_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: POD energy spectrum of the signed numerical-schlieren field. The first mode [PITH_FULL_IMAGE:figures/full_fig_p040_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: First four POD modes of the signed numerical-schlieren field. The modes are [PITH_FULL_IMAGE:figures/full_fig_p042_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Four-mode POD reconstruction of representative cooling, adiabatic and strong [PITH_FULL_IMAGE:figures/full_fig_p043_14.png] view at source ↗
read the original abstract

Prescribed wall heat flux provides an active route for controlling rarefied micro-nozzle flows, but its effect is governed by the coupled wall--bulk thermal response rather than by the imposed flux alone. This work uses direct simulation Monte Carlo (DSMC) simulations to study nitrogen flow in a converging--diverging micro-nozzle with cooling, adiabatic, and heating applied on the diverging wall. The imposed heat flux is scaled by the inlet kinetic-energy flux, $E=0.5\rho_i U_i^3$, giving $Q_w/E$ from $-10.5\%$ to $97.3\%$; this range spans moderate cooling, weak-to-intermediate heating, and a near-unity thermal-forcing regime. Wall and mass-flux-weighted bulk temperature profiles, film-temperature-based Nusselt and local-viscosity Brinkman-type diagnostics, gradient-length Knudsen indicators, mass-flux thickness, thrust decomposition, and proper orthogonal decomposition (POD) of signed numerical schlieren are analyzed. The results show that heating creates strong wall--bulk stratification: the wall temperature exceeds five times the inlet value, while the bulk temperature responds more gradually. Cooling cases contain locations where $T_w-T_b$ changes sign, making the local Nusselt-type response singular; the raw singular behavior is retained for diagnosis and a validity mask is used only for comparative plotting. Heating contracts the effective mass-carrying core, increasing aerodynamic blockage and reducing mass flow rate. However, strong heating increases the specific impulse from $156$ s to $201$ s because thermal and pressure-thrust augmentation outweigh the mass-flow penalty. The internal compression feature evolves into a finite viscous--thermal compression zone, and its heat-flux-parametric response remains low-dimensional, with the first two POD modes capturing more than $97\%$ of the fluctuation energy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript employs direct simulation Monte Carlo (DSMC) to examine nitrogen flow through a converging-diverging micro-nozzle under prescribed wall heat flux applied to the diverging section. The heat flux is normalized by the inlet kinetic-energy flux E = 0.5 ρ_i U_i^3, producing Q_w/E values from -10.5% to 97.3%. Diagnostics include wall and bulk temperature profiles, Nusselt and Brinkman-type numbers, gradient-length Knudsen indicators, thrust decomposition, and POD of signed numerical schlieren. The central claim is that strong heating increases specific impulse from 156 s to 201 s because thermal and pressure-thrust gains exceed the mass-flow penalty arising from increased aerodynamic blockage, while the internal compression feature remains low-dimensional (first two POD modes capture >97% of fluctuation energy).

Significance. If the boundary-condition implementation and quantitative thrust results hold, the work demonstrates an active thermal-control mechanism for rarefied micro-nozzle performance that could inform micro-propulsion design. The observation that the compression-zone response is captured by only two POD modes supplies a concrete, low-dimensional characterization of parametric sensitivity. The explicit thrust decomposition and retention of singular Nusselt behavior for diagnosis are methodologically transparent strengths.

major comments (2)
  1. [Boundary-condition description and results on impulse] The normalization Q_w/E uses a single inlet value E = 0.5 ρ_i U_i^3 for the entire range -10.5% to 97.3%. Because the flow accelerates and density drops along the diverging wall, the local kinetic-energy flux rises; consequently the same numerical Q_w represents a progressively smaller fractional energy input downstream. This global scaling is central to the interpretation that thermal/pressure augmentation outweighs the reported mass-flow reduction to produce the Isp rise from 156 s to 201 s, yet no local energy-balance check or alternative normalization is presented.
  2. [Numerical methods and thrust-decomposition results] The quantitative headline result (Isp increasing from 156 s to 201 s) rests on the DSMC implementation of the heat-flux boundary condition and the subsequent thrust decomposition. The manuscript provides no grid-convergence data, validation against a known rarefied-nozzle case, or accommodation-coefficient sensitivity study, all of which are load-bearing for the claimed magnitude of the impulse shift.
minor comments (3)
  1. [Abstract] The abstract states that cooling cases produce locations where T_w - T_b changes sign, rendering the local Nusselt response singular, yet the precise definition of the film temperature used for the Nusselt number is not supplied.
  2. [Results] The range of Q_w/E is given as -10.5% to 97.3%, but the corresponding wall-temperature ratios (T_w up to >5× inlet temperature) are stated without an accompanying plot or table that directly links each Q_w/E value to the resulting T_w/T_i.
  3. [POD analysis] The POD analysis reports that the first two modes capture more than 97% of the fluctuation energy, but the manuscript does not indicate whether the POD was performed on the full domain or on a masked sub-region containing the compression zone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our work. We address the major comments point by point below, indicating the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: The normalization Q_w/E uses a single inlet value E = 0.5 ρ_i U_i^3 for the entire range -10.5% to 97.3%. Because the flow accelerates and density drops along the diverging wall, the local kinetic-energy flux rises; consequently the same numerical Q_w represents a progressively smaller fractional energy input downstream. This global scaling is central to the interpretation that thermal/pressure augmentation outweighs the reported mass-flow reduction to produce the Isp rise from 156 s to 201 s, yet no local energy-balance check or alternative normalization is presented.

    Authors: The inlet-based normalization was selected to maintain a uniform reference scale for the imposed heat flux relative to the fixed inlet conditions, facilitating consistent parametric comparison across the range of Q_w/E values. This choice emphasizes the overall energy input relative to the nozzle inlet rather than local variations. We agree that examining the local energy balance would be beneficial for interpreting the downstream effects. In the revised version, we will add a local energy-flux analysis along the wall to demonstrate that the thermal and pressure contributions remain dominant despite the flow acceleration. revision: partial

  2. Referee: The quantitative headline result (Isp increasing from 156 s to 201 s) rests on the DSMC implementation of the heat-flux boundary condition and the subsequent thrust decomposition. The manuscript provides no grid-convergence data, validation against a known rarefied-nozzle case, or accommodation-coefficient sensitivity study, all of which are load-bearing for the claimed magnitude of the impulse shift.

    Authors: We acknowledge the importance of verifying the numerical accuracy for the reported quantitative results. Although the methods section describes the DSMC setup, including the boundary condition implementation, we will enhance the manuscript by including explicit grid-convergence studies for the key quantities such as mass flow rate and thrust components. Additionally, we will provide validation results against a standard rarefied nozzle test case and perform a sensitivity analysis on the wall accommodation coefficient to confirm the robustness of the specific impulse increase from 156 s to 201 s. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results from direct DSMC simulation

full rationale

The paper reports outcomes from DSMC simulations with externally imposed wall heat-flux boundary conditions normalized by the fixed inlet quantity E = 0.5 ρ_i U_i^3. The reported specific-impulse increase (156 s to 201 s), blockage evolution, and POD energy capture (>97 % in first two modes) are direct numerical outputs under these boundary conditions rather than quantities obtained by fitting parameters to the target data or by self-referential definitions. No equations or claims reduce the central results to their own inputs by construction, and the analysis employs standard post-processing diagnostics without load-bearing self-citation chains for the primary findings.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The paper is a parametric DSMC study; its central claims rest on the standard assumptions of the DSMC method for rarefied nitrogen and on the chosen nondimensional heat-flux scaling. No new particles or forces are postulated.

free parameters (1)
  • Q_w/E
    Heat flux imposed on the diverging wall is nondimensionalized by inlet kinetic-energy flux E=0.5ρ_i U_i^3 and varied from -10.5% to 97.3% to span cooling to near-unity heating regimes.
axioms (1)
  • domain assumption DSMC with the selected collision model and boundary conditions accurately captures the coupled wall-bulk thermal response and mass-flux-weighted quantities in the rarefied regime.
    All reported temperature profiles, Nusselt diagnostics, and thrust decompositions presuppose the validity of the DSMC solver for this geometry and Knudsen range.

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Works this paper leans on

34 extracted references · 34 canonical work pages · 1 internal anchor

  1. [1]

    Computers & Fluids 127, 78–101

    A novel algorithm for implementing a specified wall heat flux in dsmc: Application to micro/nano flows and hypersonic flows. Computers & Fluids 127, 78–101. doi:10.1016/j.compfluid.2015. 12.008. Akhlaghi, H., Roohi, E., Stefanov, S.,

    work page doi:10.1016/j.compfluid.2015 2015

  2. [2]

    International Journal of Thermal Sciences 59, 111–125

    A new iterative wall heat flux specifying technique in dsmc for heating/cooling simulations of mems/nems. International Journal of Thermal Sciences 59, 111–125. doi:10.1016/j.ijthermalsci.2012.04.002. Balaj, M., Roohi, E., Akhlaghi, H.,

    work page doi:10.1016/j.ijthermalsci.2012.04.002 2012

  3. [3]

    International Journal of Heat and Mass Transfer 83, 69–74

    Effects of shear work on non- equilibrium heat transfer characteristics of rarefied gas flows through mi- cro/nanochannels. International Journal of Heat and Mass Transfer 83, 69–74. doi:10.1016/j.ijheatmasstransfer.2014.11.087. Balaj, M., Roohi, E., Akhlaghi, H., Myong, R.S.,

    work page doi:10.1016/j.ijheatmasstransfer.2014.11.087 2014

  4. [4]

    International Journal of Heat and Mass Transfer 71, 633–638

    Investigation of con- vective heat transfer through constant wall heat flux micro/nano channels using dsmc. International Journal of Heat and Mass Transfer 71, 633–638. doi:10.1016/j.ijheatmasstransfer.2013.12.053. Bird, G.A.,

    work page doi:10.1016/j.ijheatmasstransfer.2013.12.053 2013

  5. [5]

    ThePhysics of Fluids 13, 2676–2681

    Directsimulationandtheboltzmannequation. ThePhysics of Fluids 13, 2676–2681. doi:10.1063/1.1692849. Darbandi, M., Roohi, E.,

    work page doi:10.1063/1.1692849

  6. [6]

    Microflu- idics and Nanofluidics 10, 321–335

    Study of subsonic–supersonic gas flow throughmicro/nanoscalenozzlesusingunstructureddsmcsolver. Microflu- idics and Nanofluidics 10, 321–335. doi:10.1007/s10404-010-0671-7. Groll, R., Frieler, T.,

    work page doi:10.1007/s10404-010-0671-7

  7. [7]

    Frontiers in Mechani- cal Engineering 9, 1217645

    Validation of dsmc mass flow modeling for transsonic gas flows in micro-propulsion systems. Frontiers in Mechani- cal Engineering 9, 1217645. doi:10.3389/fmech.2023.1217645. Hao, P.F., Ding, Y.T., Yao, Z.H., He, F., Zhu, K.Q.,

    work page doi:10.3389/fmech.2023.1217645 2023

  8. [8]

    Journal of Micromechanics and Microengineering 15, 2069–2073

    Size effect on gas flow in micro nozzles. Journal of Micromechanics and Microengineering 15, 2069–2073. doi:10.1088/0960-1317/15/11/019. 54 Horisawa, H., Sawada, F., Onodera, K., Funaki, I.,

    work page doi:10.1088/0960-1317/15/11/019 2069

  9. [9]

    Vac- uum 83, 52–56

    Numerical simula- tion of micro-nozzle and micro-nozzle-array flowfield characteristics. Vac- uum 83, 52–56. doi:10.1016/j.vacuum.2008.03.097. Ivanov, M.S., Markelov, G.N., Ketsdever, A.D., Wadsworth, D.C.,

    work page doi:10.1016/j.vacuum.2008.03.097 2008

  10. [10]

    Numerical study of cold gas micronozzle flows, in: 37th Aerospace Sciences Meeting and Exhibit, pp. 1–11. doi:10.2514/6.1999-166. Karniadakis, G.E., Beskok, A., Aluru, N.R.,

    work page doi:10.2514/6.1999-166 1999

  11. [11]

    Physics of Fluids 33, 082007

    Separation of a binary gas mixture outflowing into vacuum through a micronozzle. Physics of Fluids 33, 082007. doi:10.1063/5.0055879. Lijo, V., Setoguchi, T., Kim, H.D.,

    work page doi:10.1063/5.0055879

  12. [12]

    Anhichem, S

    Analysis of supersonic micronozzle flows. Journal of Propulsion and Power 31, 754–757. doi:10.2514/1. B35453. Lockerby, D.A., Reese, J.M., Emerson, D.R., Barber, R.W.,

    work page doi:10.2514/1

  13. [13]

    Physical Review E 70, 017303

    Velocity boundary condition at solid walls in rarefied gas calculations. Physical Review E 70, 017303. doi:10.1103/PhysRevE.70.017303. Louisos, W.F., Hitt, D.L.,

    work page doi:10.1103/physreve.70.017303

  14. [14]

    Journal of Spacecraft and Rockets 49, 51–58

    Viscous effects on performance of three- dimensional supersonic micronozzles. Journal of Spacecraft and Rockets 49, 51–58. doi:10.2514/1.53026. Lu, L., Jin, P., Pang, G., Zhang, Z., Karniadakis, G.E.,

    work page doi:10.2514/1.53026

  15. [15]

    Procedia CIRP72(March), 159–164 (2018) https://doi.org/10.1016/j

    A novel hybrid dsmc-fokker planck algorithm implemented to rarefied gas flows. Vacuum 181, 109736. doi:10.1016/j. vacuum.2020.109736. Maxwell, J.C.,

    work page doi:10.1016/j 2020

  16. [16]

    Philosophical Transactions of the Royal Society of London 170, 231–256

    On stresses in rarified gases arising from inequalities of temperature. Philosophical Transactions of the Royal Society of London 170, 231–256. doi:10.1098/rstl.1879.0067. 55 Peyvan, A., Kumar, V., Karniadakis, G.E.,

    work page doi:10.1098/rstl.1879.0067

  17. [17]

    Karniadakis

    Physics-informed neural networks: A deep learning framework for solving forward and inverse prob- lems involving nonlinear partial differential equations. Journal of Compu- tational Physics 378, 686–707. doi:10.1016/j.jcp.2018.10.045. Riabov, V.V.,

    work page doi:10.1016/j.jcp.2018.10.045 2018

  18. [18]

    doi:10.2514/6

    Shock interference in hypersonic rarefied-gas flows near a cylinder, in: AIAA 30th Fluid Dynamics Conference. doi:10.2514/6. 1999-3207. Roohi, E., Mahdavi, A., 2026a. Analysis of the rarefied flow at micro-step usingadeeponetsurrogatemodelwithaphysics-guidedzonallossfunction. Microfluidics and Nanofluidics 30,

    work page doi:10.2514/6 1999

  19. [19]

    Shock-Centered Low-Rank Structure and Neural-Operator Representation of Rarefied Micro-Nozzle Flows

    doi:10.1007/s10404-026-02899-8. Roohi, E., Mahdavi, A., 2026b. Shock-centered low-rank structure and neural-operator representation of rarefied micro-nozzle flows. arXiv preprint arXiv:2605.12723 . Rothe, D.E.,

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s10404-026-02899-8

  20. [20]

    AIAA Journal 9, 804–811

    Electron-beam studies of viscous flow in supersonic noz- zles. AIAA Journal 9, 804–811. doi:10.2514/3.49904. Saadati, S.A., Roohi, E.,

    work page doi:10.2514/3.49904

  21. [21]

    Aerospace Science and Technology 46, 236–255

    Detailed investigation of flow and thermal field in micro/nano nozzles using simplified bernoulli trial (sbt) collision scheme in dsmc. Aerospace Science and Technology 46, 236–255. doi:10. 1016/j.ast.2015.07.013. Sabouri, M., Darbandi, M.,

    work page 2015

  22. [22]

    Physics of Fluids 31, 042004

    Numerical study of species separation in rarefied gas mixture flow through micronozzles using dsmc. Physics of Fluids 31, 042004. doi:10.1063/1.5083807. Sabouri, M., Zakeri, R., Ebrahimi, A.,

    work page doi:10.1063/1.5083807

  23. [23]

    Physica Scripta 99, 085213

    Improving computational effi- ciency in dsmc simulations of vacuum gas dynamics with a fixed number of particles per cell. Physica Scripta 99, 085213. doi:10.1088/1402-4896/ ad5a46. Schaaf, S.A., Chambré, P.L.,

    work page doi:10.1088/1402-4896/

  24. [24]

    Princeton Uni- versity Press

    Flow of Rarefied Gases. Princeton Uni- versity Press. doi:10.1515/9781400885800. 56 Sharipov, F.,

    work page doi:10.1515/9781400885800

  25. [25]

    Felix Sharipov and Denize Kalempa

    Data on the velocity slip and temperature jump on a gas-solid interface. Journal of Physical and Chemical Reference Data 40, 023101. doi:10.1063/1.3580290. Sukesan, M.K., Shine, S.R.,

    work page doi:10.1063/1.3580290

  26. [26]

    Journal of Micromechanics and Microengi- neering 31, 125001

    Geometry effects on flow characteristics of micro-scale planar nozzles. Journal of Micromechanics and Microengi- neering 31, 125001. doi:10.1088/1361-6439/ac2bac. Tatsios, G., et al.,

    work page doi:10.1088/1361-6439/ac2bac

  27. [27]

    Journal of Computational Physics 520, 113500

    A dsmc-cfd coupling method using surrogate mod- elling for low-speed rarefied gas flows. Journal of Computational Physics 520, 113500. doi:10.1016/j.jcp.2024.113500. Tsien, H.S.,

    work page doi:10.1016/j.jcp.2024.113500 2024

  28. [28]

    Journal of the Aeronautical Sciences 13, 653–664

    Superaerodynamics, mechanics of rarefied gases. Journal of the Aeronautical Sciences 13, 653–664. doi:10.2514/8.11476. Wagner, W.,

    work page doi:10.2514/8.11476

  29. [29]

    Journal of Statistical Physics 66, 1011–1044

    A convergence proof for bird’s direct simulation monte carlo method for the boltzmann equation. Journal of Statistical Physics 66, 1011–1044. doi:10.1007/BF01055714. Wang, M.R., Li, Z.X.,

    work page doi:10.1007/bf01055714

  30. [30]

    Microfluidics and Nanoflu- idics 1, 62–70

    Numerical simulations on performance of mems- based nozzles at moderate or low temperatures. Microfluidics and Nanoflu- idics 1, 62–70. doi:10.1007/s10404-004-0008-5. Wu, L., Li, Q., Liu, H., Ubachs, W.,

    work page doi:10.1007/s10404-004-0008-5

  31. [31]

    Journal of Fluid Mechanics 901, A23

    Extraction of the translational eucken factor from light scattering by molecular gas. Journal of Fluid Mechanics 901, A23. doi:10.1017/jfm.2020.568. Wu, L., White, C., Scanlon, T.J., Reese, J.M., Zhang, Y.,

    work page doi:10.1017/jfm.2020.568 2020

  32. [32]

    Jour- nal of Fluid Mechanics 763, 24–50

    A kinetic model of the boltzmann equation for non-vibrating polyatomic gases. Jour- nal of Fluid Mechanics 763, 24–50. doi:10.1017/jfm.2014.632. Xie, C.,

    work page doi:10.1017/jfm.2014.632 2014

  33. [33]

    Physics of Fluids 19, 037102

    Characteristics of micronozzle gas flows. Physics of Fluids 19, 037102. doi:10.1063/1.2709707. Zhang, S., Li, Y., Wang, X., Lu, S., Yu, Y., Yang, J.,

    work page doi:10.1063/1.2709707

  34. [34]

    doi:10.3390/mi15010022

    work page doi:10.3390/mi15010022