proposition 31.9 Cauchy–Stokes decomposition

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:222 · p. 1043

Rests on

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 ¶
proves proof ch:14-fluid-dynamics@proof-3 ¶

Supports

depends_on remark 31.10 Vorticity is local rotation, not curvature of the path ¶
depends_on theorem 31.36 The Newtonian constitutive relation ¶
- 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 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 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 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 31.53 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

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
`depends_on`	→	Rate of strain and vorticity	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:233
`depends_on`	←	Vorticity is local rotation, not curvature of the path	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:274
`depends_on`	←	The Newtonian constitutive relation	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1020
`proves`	←	ch:14-fluid-dynamics@proof-3	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:236