phenomenon 31.49 Dynamical similarity

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:1462 · p. 1058

phenomenon-tested: no experiment declares a \tests edge to it and it is not listed in untested.txt

Rests on

depends_on theorem 31.37 The Navier–Stokes equations ¶
- depends_on theorem 31.36 The Newtonian constitutive relation ¶
  - depends_on definition 31.32 Newtonian fluid; dynamic and kinematic viscosity ¶
    - depends_on definition 31.1 Fluid ¶
      - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
    - depends_on definition 31.8 Rate of strain and vorticity ¶
      - depends_on definition 13.91 Symmetric and antisymmetric parts ¶
      - depends_on equation 7.126 eq:ana-nabla-curl ¶
  - depends_on proposition 31.9 Cauchy–Stokes decomposition ¶
    - depends_on definition 31.8 Rate of strain and vorticity ¶ ↺
    - proves proof ch:14-fluid-dynamics@proof-3 ¶
  - proves proof ch:14-fluid-dynamics@proof-17 ¶
- depends_on theorem 31.22 Euler's equations of motion ¶
  - depends_on definition 31.21 Ideal fluid ¶
    - depends_on definition 31.1 Fluid ¶ ↺
  - depends_on lemma 31.11 Transport theorem for a material volume ¶
    - depends_on definition 31.4 Material derivative ¶
      - depends_on equation 7.124 eq:ana-nabla-gradient ¶
      - depends_on proposition 7.104 Chain rule in several variables ¶
      - proves proof ch:14-fluid-dynamics@proof-1 ¶
    - depends_on theorem 7.133 Gauss ¶
      - depends_on definition 7.127 Simple regions ¶
      - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof ch:05-real-analysis@proof-79 ¶
    - proves proof ch:14-fluid-dynamics@proof-4 ¶
  - depends_on theorem 30.21 Cauchy's equation of motion ¶
    - depends_on theorem 7.133 Gauss ¶ ↺
    - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶ ↺
    - proves proof ch:13-continuum-elasticity@proof-9 ¶
  - proves proof ch:14-fluid-dynamics@proof-11 ¶
- proves proof ch:14-fluid-dynamics@proof-18 ¶
proves proof ch:14-fluid-dynamics@proof-24 ¶

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 31.53 Transition to turbulence in a pipe ¶
- depends_on remark 31.54 This one is not a gap in the writing ¶
depends_on remark 31.50 Reading the Reynolds number ¶
depends_on remark 31.52 Similarity is only as good as the control of the data ¶
depends_on remark 31.62 What the spacing calculation settles, and what it does not ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`cites` (evidence)	→	Sur la résistance des sphères dans l'air en mouvement	derived	parts/03-classical-mechanics/14-fluid-dynamics.tex:1473
`cites` (evidence)	→	An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels	derived	parts/03-classical-mechanics/14-fluid-dynamics.tex:1473
`depends_on`	→	The Navier–Stokes equations	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1476
`depends_on`	←	The other similarity groups	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1563
`depends_on`	←	Transition to turbulence in a pipe	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1596
`depends_on`	←	Reading the Reynolds number	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1537
`depends_on`	←	Similarity is only as good as the control of the data	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1579
`depends_on`	←	What the spacing calculation settles, and what it does not	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1996
`proves`	←	ch:14-fluid-dynamics@proof-24	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1479