theorem 31.36 The Newtonian constitutive relation

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:1007 · p. 1052

Rests on

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 7.126 Line and surface integrals ¶
      - depends_on definition 30.2 Representative volume element; continuum field ¶
    - depends_on definition 13.5 Tensor under orthogonal transformations ¶
      - depends_on definition 13.4 Orthogonal coordinate transformation ¶
    - 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 ¶

Supports

depends_on theorem A.718 Stokes' drag law with its coefficient ¶
depends_on theorem 31.37 The Navier–Stokes equations ¶
- 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.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 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 31.50 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`	→	Newtonian fluid; dynamic and kinematic viscosity	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1020
`depends_on`	→	Cauchy–Stokes decomposition	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1020
`depends_on`	←	Stokes' drag law with its coefficient	declared	appendices/A-long-proofs.tex:34825
`depends_on`	←	The Navier–Stokes equations	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1074
`proves`	←	ch:14-fluid-dynamics@proof-17	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1023