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Physical Review Letters
Volumn 69, Issue 1, 1992, Pages 208-211
Geometry and foams: 2D dynamics and 3D statics
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EID: 0000195664     PISSN: 00319007     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevLett.69.208     Document Type: Article
Times cited : (37)
References (24)
  • 13
    • 84927846399 scopus 로고    scopus 로고
    • The relation of the von Neumann law for planar foam to the Gauss-Bonnet Theorem was noted in passing by V. E. Fradkov and D. G. Udler, ``Normal Grain Growth in 2D Polycrystals,'' Institute of Solid State Physics, Academy of Sciences of the USSR (unpublished).
  • 17
    • 84927846398 scopus 로고    scopus 로고
    • The equilibrium configurations on a closed surface of genus g satisfy V - 2F = 4 ( g - 1 ), where V is the total number of vertices and F the total number of bubbles. This may be viewed as a version of the Euler relation for (polygonlike) bubbles. This relation is obtained by summing Eq. (2) over all bubbles and using the fact that over the entire surface tint K d A = 4 π ( 1 - g ).
  • 18
    • 0003032419 scopus 로고
    • presents a 3D analog to von Neumann's law for the growth of average bubbles which is based on the Lewis law. Such a model is distinct from the foam considered here.
    • (1983) Philos. Mag. B , vol.47 , pp. L45
    • Rivier, N.1
  • 19
    • 84927846397 scopus 로고    scopus 로고
    • Note that in 3D the analog of Eq. (1), σ Δ P = 2 H, with H the mean curvature, is not a term in Gauss-Bonnet formula, Eq. (2) (which involves only the Gaussian and geodesic curvatures).
  • 20
    • 84927846396 scopus 로고    scopus 로고
    • Note that this equation is distinct from the well-known relation n bar = 6 - 12 /f¯. Here, $< n > = tsumBtsumFn (F) / tsumBtsumF1, where the first sum is over all the bubbles in the foam and the second over the faces of a given bubble, whereas n bar = VT/ FT, where VT and FT are the total number of vertices and faces in the foam, respectively. It is simple to show that < n > obeys < n > = 6 - 12 < 1 / f > . For isobaric foam, this gives an upper bound < n > < 6, while Eq. (5) gives a lower bound < n > > 5.104.
  • 21
    • 0042565166 scopus 로고
    • The Three-Dimensional Shape of Bubbles in Foam-An Analysis of the Role of Surface Forces in Three-Dimensional Cell Shape Determination
    • (1946) American Journal of Botany , vol.33 , pp. 58
    • Matzke, E.B.1

* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.