theorem D.2 Buckingham's $\Pi$ theorem

open in the book · appendices/E-dimensionless-constants.tex:80 · p. 3216

Rests on

depends_on equation D.3 eq:E-kernel-condition ¶
proves proof app:E-dimensionless@proof-1 ¶

Supports

depends_on definition 31.51 The other similarity groups ¶
- depends_on definition A.760 The convecting layer ¶
  - depends_on proposition A.761 The Boussinesq equations ¶
    - depends_on definition A.763 Two-dimensional rolls ¶
      - depends_on proposition A.765 The roll equations ¶
    - depends_on proposition A.765 The roll equations ¶ ↺
- depends_on phenomenon 31.78 Convective onset at a computed threshold ¶
  - depends_on proposition A.761 The Boussinesq equations ¶ ↺
  - depends_on remark 31.79 Bénard's own cells, and Taylor's cylinders ¶
  - depends_on remark 31.81 Where the threshold comes from, and what it costs ¶
  - depends_on theorem A.767 Marginal stability of the conducting state ¶
    - depends_on remark A.768 Two conventions for the wavenumber ¶
    - depends_on theorem A.769 The Lorenz system ¶
      - depends_on remark A.771 What was assumed, and where it fails ¶
      - depends_on remark A.770 What the three variables and three parameters are ¶
- depends_on proposition A.766 Nondimensionalisation ¶
  - depends_on theorem A.767 Marginal stability of the conducting state ¶ ↺
  - depends_on theorem A.769 The Lorenz system ¶ ↺
- depends_on remark 31.89 The Mach number and the regimes it separates ¶
  - depends_on phenomenon 31.93 The Mach cone ¶
depends_on phenomenon 31.84 The five-thirds spectrum ¶
- depends_on proposition 31.85 The Kolmogorov scales ¶
- depends_on remark 31.86 The one exact result, and the split attribution ¶
depends_on proposition 31.85 The Kolmogorov scales ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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Edges

type	direction	node	provenance	where
`cites`	→	On Physically Similar Systems; Illustrations of the Use of Dimensional Equations	derived	appendices/E-dimensionless-constants.tex:86
`depends_on`	→	eq:E-kernel-condition	declared	appendices/E-dimensionless-constants.tex:87
`depends_on`	←	The other similarity groups	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1563
`depends_on`	←	The five-thirds spectrum	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2870
`depends_on`	←	The Kolmogorov scales	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2931
`proves`	←	app:E-dimensionless@proof-1	declared	appendices/E-dimensionless-constants.tex:104