Skip to main content
SpringerLink
Log in

Gradient Divergence of Fluid-Dynamic Quantities in Rarefied Gases on Smooth Boundaries

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

The behavior of fluid-dynamic (or macroscopic) quantities of rarefied gases is studied, with a special interest in its non-analytic feature near boundaries. It is shown that their gradients normal to the boundary diverge even if the boundary is smooth, irrespective of the value of the (nonzero) Knudsen number. The boundary geometry determines the diverging rate. On a planar or concave boundary, the logarithmic divergence \(\ln s\) should be observed, where s is the normal distance from the boundary. In other cases, the diverging rate is enhanced to be the inverse-power \(s^{-1/n}\), where \(n({\ge }2)\) is the degree of the dominant terms of the polynomial which locally represents the boundary. Some numerical demonstrations are given as well.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Notes

  1. The source S arises in the equation, if we put \(f=\left( 1+\phi +g\right) E\) with g being a given function. Then, S is related to g as \(S=-\zeta _i\frac{\partial g}{\partial x_i}+\mathcal {L}[g]\) and is usually analytic in both \(\varvec{x}\) and \(\varvec{\zeta }\). Such a formulation is common in treating, for instance, the Poiseuille and thermal creep (transpiration) flow problems. See, e.g., Refs. [10, 13].

  2. The assumption on \(\mathcal {K}[\phi ]\) becomes rigorous if we use the model kinetic equations like the Bhatnagar–Gross–Krook (BGK) model [1, 15].

  3. By definition, \(F_{1}(x)=\gamma +\ln (x)+\mathrm {E}_{1}(x)=\gamma -\int _{x}^{1}t^{-1}dt+\int _{x}^{\infty }t^{-1}e^{-t}dt=\gamma -\int _{x}^{1}t^{-1}\left( 1-e^{-t}\right) dt+\int _{1}^{\infty }t^{-1}e^{-t}dt\). It is easy to see that \(x-F_{1}(x)\) is a monotonically increasing function and \(F_1(0)=0\), so that \(x\ge F_{1}(x)\ge 0\) for \(x>0\). Since \(2\sqrt{x}\ge x\) for \(0<x\le 1\), the desired inequality holds in this interval. For \(x>1\), consider \(g(x)\equiv 2\sqrt{x}-F_{1}(x)\). Then, g is a monotonically increasing function, because \(g^{\prime }(x)=x^{-1}\left( \sqrt{x}-1+e^{-x}\right) \ge 0\) for \(x\ge 1\). Therefore, \(g(x)\ge g(1)=2-\gamma -\int _{1}^{\infty }t^{-1}e^{-t}dt\ge 2-\gamma -e^{-1}>0\), which shows the desired inequality to hold for \(x>1\) as well.

  4. Here, the value on the negative real axis is taken as the limit from above.

  5. Here \(x_{w}\) is the \(x_{1}\)-coordinate of the intersection point of the molecular trajectory with the boundary. Hence \(x_{w}\) is one of the two solutions of \(x_{w}^{2}+\left( x_{w}-x_{1}\right) ^{2}\tan ^{2}\alpha =1\) or \(x_{w}^{2}-2x_{w}x_{1}\sin ^{2}\alpha +x_{1}^{2}\sin ^{2}\alpha -\cos ^{2}\alpha =0.\) We adopt the larger of the solutions \(x_{w}=x_{1}\sin ^{2}\alpha \pm \cos \alpha \left( 1-x_{1}^{2}\sin ^{2}\alpha \right) ^{1/2}\), which is the one with a positive sign.

  6. Obviously \(G(x)\ge 0\) for \(x\ge 0\). If we introduce \(G_{n}(x)\equiv (1/n)x^{n}e^{x}-G(x)\) \((n>0)\), then we have \(G_{1}^{\prime }=x^{-1}\left\{ \left( x+x^{2}-1\right) e^{x}+1\right\} \equiv x^{-1}g_{1}(x)\). Because \(g_{1}^{\prime }=\left( 3x+x^{2}\right) e^{x}\ge 0\) and \(g_1(0)=0\), \(g_1(x)\) is non-negative for \(x\ge 0\). Therefore \(G_{1}\ge G_{1}(0)=0\). Now compare \(G_{1/2}\) with \(G_{1}\). Because \(G_{1/2}-G_{1}=\left( 2\sqrt{x}-x\right) e^{x}\), \(G_{1/2}\ge G_{1}\ge 0\) holds for \(x\le 1\). For \(x\ge 1\), \(G_{1/2}^{\prime }=x^{-1}\left\{ \left( x^{1/2}+2x^{3/2}-1\right) e^{x}+1\right\} \ge 0\) and thus \(G_{1/2}\ge G_{1/2}(1)\ge G_1(1)\ge 0\). Thus we conclude \(G_{1/2}(x)\ge 0\), i.e., \(0\le G(x)\le 2\sqrt{x}e^{x}\), for \(x\ge 0\).

  7. In order to discriminate s against \(s\ln s\), the quantities near the outer boundary are shown by the semi-log plot of the difference from the values on the boundary divided by the distance s. If the profile for small s is horizontal line, the difference grows in proportion to s near the boundary. If the profile for small s is a straight line with non-zero slope, the difference grows in proportion to \(s\ln s\) near the boundary.

References

  1. Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525 (1954)

    Article  ADS  MATH  Google Scholar 

  2. Chen, I.K.: Boundary singularity of moments for the linearized Boltzmann equation. J. Stat. Phys. 153, 93–118 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Guo, Y., Kim, C., Tonon, D., Trescases, A.: BV-regularity of the Boltzmann equation in non-convex domains. Arch. Rat. Mech. Anal. 220, 1045–1093 (2016). doi:10.1007/s00205-015-0948-9

    Article  MathSciNet  MATH  Google Scholar 

  4. Guo, Y., Kim, C., Tonon, D., Trescases, A.: Regularity of the Boltzmann equation in convex domains. Inv. Math. 207, 115–290 (2016). doi:10.1007/s00222-016-0670-8

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Hattori, M., Takata, S.: Second-order Knudsen-layer analysis for the generalized slip-flow theory I. Bul. Inst. Math. Acad. Sin. New Series 10, 423–448 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Sone, Y.: New kind of boundary layer over a convex solid boundary in a rarefied gas. Phys. Fluids 16, 1422–1424 (1973)

    Article  ADS  MATH  Google Scholar 

  7. Sone, Y.: Highly rarefied gas around a group of bodies with various temperature distributions I. Small tempearature variation. J. Méc. Théor. Appl. 3, 315–328 (1984)

    MathSciNet  MATH  Google Scholar 

  8. Sone ,Y.: Highly rarefied gas around a group of bodies with various temperature distributions II. Arbitrary temperature variation. J. Méc. Théor. Appl. 4, 1–14 (1985)

  9. Sone, Y.: Kinetic Theory and Fluid Dynamics. Birkhäuser (2002)

  10. Sone, Y.: Molecular Gas Dynamics. Birkhäuser (2007). http://hdl.handle.net/2433/66098

  11. Sone, Y., Aoki, K.: Negative thermophoresis: thermal stress slip flow around a spherical particle in a rarefied gas. In: Fisher, S.S. (ed.) Rarefied Gas Dynamics, pp. 489–503. AIAA, New York (1981)

    Google Scholar 

  12. Sone, Y., Takata, S.: Discontinuity of the velocity distribution function in a rarefied gas around a convex body and the S layer at the bottom of the Knudsen layer. Transp. Theor. Stat. Phys. 21, 501–530 (1992)

    Article  ADS  MATH  Google Scholar 

  13. Takata, S., Funagane, H.: Singular behaviour of a rarefied gas on a planar boundary. J. Fluid Mech. 717, 30–47 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Takata, S., Yoshida, T., Noguchi, T., Taguchi, S.: Singular behavior of the macroscopic quantities in the free molecular gas. Phys. Fluids 28, 022002 (2016)

    Article  ADS  Google Scholar 

  15. Welander, P.: On the temperature jump in a rarefied gas. Ark. Fys. 7, 507–553 (1954)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank Yutaro Akahoshi and Katsuyuki Inui for their helps in double checking the numerical data in Sect. 4.3. The research note of Kazuo Aoki of Ref. [11] was helpful in assessing our insight on the internal structure of the S layer. We appreciate his courtesy.

Author information

Author notes

  1. Satoshi Taguchi

    Present address: Department of Advanced Mathematical Sciences, Kyoto University, Kyoto, 606-8501, Japan

Authors and Affiliations

  1. Department of Aeronautics and Astronautics, Kyoto University, Kyoto, 615-8540, Japan

    Shigeru Takata

  2. Department of Mechanical and Intelligent Systems Engineering, The University of Electro-Communications, Chofu, Tokyo, 182-8585, Japan

    Satoshi Taguchi

Authors
  1. Shigeru Takata
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Satoshi Taguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shigeru Takata.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Takata, S., Taguchi, S. Gradient Divergence of Fluid-Dynamic Quantities in Rarefied Gases on Smooth Boundaries. J Stat Phys 168, 1319–1352 (2017). https://doi.org/10.1007/s10955-017-1850-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-017-1850-7

Keywords

Search

Navigation