theorem 31.37 The Navier–Stokes equations

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:1056 · p. 1053

Rests on

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 30.17 Traction ¶
      - depends_on definition 13.5 Tensor under orthogonal transformations ¶
      - proves proof ch:13-continuum-elasticity@proof-7 ¶
  - 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 ¶
      - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.99 Differentiability at a point ¶
      - depends_on definition 7.102 Differential of a function ¶
      - proves proof ch:05-real-analysis@proof-63 ¶
    - proves proof ch:14-fluid-dynamics@proof-1 ¶
  - depends_on theorem 7.133 Gauss ¶
    - depends_on definition 7.127 Simple regions ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 6.9 Compact set ¶
    - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on definition 7.125 Multiple integral ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - 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 ¶

Supports

depends_on definition A.741 The statistical setting ¶
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.766 Nondimensionalisation ¶
  - depends_on proposition A.765 The roll equations ¶ ↺
depends_on definition 31.43 Creeping flow ¶
- depends_on lemma A.716 The creeping-flow equation for $\psi$ ¶
  - depends_on lemma A.717 The radial solution ¶
    - depends_on theorem A.718 Stokes' drag law with its coefficient ¶
- depends_on proposition 31.44 Kinematic reversibility ¶
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 phenomenon 31.41 The fourth-power law of pipe flow ¶
- 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 31.42 The entry length ¶
depends_on phenomenon 31.49 Dynamical similarity ¶
- depends_on definition 31.51 The other similarity groups ¶
  - depends_on definition A.760 The convecting layer ¶ ↺
  - depends_on phenomenon 31.78 Convective onset at a computed threshold ¶ ↺
  - depends_on proposition A.766 Nondimensionalisation ¶ ↺
  - 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 31.50 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 ¶
depends_on proposition A.761 The Boussinesq equations ¶ ↺
depends_on proposition 31.40 Circular Couette flow ¶
- depends_on proposition 31.80 Rayleigh's circulation criterion ¶
  - depends_on remark 31.81 Where the threshold comes from, and what it costs ¶ ↺
- depends_on remark 31.79 Bénard's own cells, and Taylor's cylinders ¶ ↺
depends_on proposition 31.39 Plane Couette flow ¶
- depends_on proposition 31.40 Circular Couette flow ¶ ↺
depends_on remark 31.48 What is actually proved ¶
- depends_on remark 31.87 What a theory of turbulence would have to do ¶
depends_on remark 31.38 Whose equations, and the Stokes hypothesis ¶
depends_on theorem A.745 Kármán–Howarth–Monin relation ¶
- depends_on proposition A.746 The Kármán–Howarth equation in classical form ¶
- depends_on theorem A.749 Kolmogorov's equation and the four-fifths law ¶
… 3 more

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Edges

type	direction	node	provenance	where
`depends_on`	→	The Newtonian constitutive relation	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1074
`depends_on`	→	Euler's equations of motion	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1074
`depends_on`	←	The statistical setting	declared	appendices/A-long-proofs.tex:35945
`depends_on`	←	The convecting layer	declared	appendices/A-long-proofs.tex:37085
`depends_on`	←	Creeping flow	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1304
`depends_on`	←	Convective onset at a computed threshold	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2608
`depends_on`	←	The fourth-power law of pipe flow	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1228
`depends_on`	←	Dynamical similarity	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1476
`depends_on`	←	The Boussinesq equations	declared	appendices/A-long-proofs.tex:37111
`depends_on`	←	Circular Couette flow	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1185
`depends_on`	←	Plane Couette flow	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1151
`depends_on`	←	What is actually proved	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1456
`depends_on`	←	Whose equations, and the Stokes hypothesis	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1126
`depends_on`	←	Kármán–Howarth–Monin relation	declared	appendices/A-long-proofs.tex:36166
`depends_on`	←	Boundary-layer thickness	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2262
`depends_on`	←	The Reynolds-averaged equations	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2788
`depends_on`	←	The vorticity transport equation	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1832
`proves`	←	ch:14-fluid-dynamics@proof-18	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1077