theorem A.745 Kármán–Howarth–Monin relation

open in the book · appendices/A-long-proofs.tex:36152 · p. 3156

Rests on

depends_on lemma A.744 Vanishing of the pressure–velocity correlation ¶
- depends_on corollary 31.13 Incompressible flow ¶
  - depends_on theorem 31.12 Conservation of mass ¶
    - depends_on lemma 31.11 Transport theorem for a material volume ¶
      - depends_on definition 31.4 Material derivative ¶
      - depends_on theorem 7.133 Gauss ¶
      - proves proof ch:14-fluid-dynamics@proof-4 ¶
    - proves proof ch:14-fluid-dynamics@proof-5 ¶
  - proves proof ch:14-fluid-dynamics@proof-6 ¶
- depends_on lemma A.742 Isotropic representation ¶
  - depends_on definition 14.26 The rotation group ¶
  - depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
    - depends_on lemma 5.33 Determinant through the Levi–Civita symbol ¶
      - depends_on definition 5.32 Levi–Civita symbol; cross product ¶
      - depends_on equation 5.19 eq:lin-leibniz-det ¶
      - proves proof ch:03-linear-algebra-representations@proof-8 ¶
    - depends_on lemma 5.132 Parity constraint ¶
      - depends_on definition 5.131 Isotropic Cartesian tensor ¶
      - depends_on equation 5.162 eq:rep-special-orthogonal ¶
      - proves proof ch:03-linear-algebra-representations@proof-57 ¶
    - depends_on proposition 5.130 An orthogonal transformation is an isometry ¶
      - depends_on definition 5.18 Inner product ¶
      - depends_on equation 5.44 eq:lin-norm-assoc ¶
      - depends_on proposition 5.49 Injective, surjective, invertible ¶
      - proves proof ch:03-linear-algebra-representations@proof-56 ¶
    - proves proof ch:03-linear-algebra-representations@proof-58 ¶
  - proves proof app:A-long-proofs@proof-432 ¶
- proves proof app:A-long-proofs@proof-434 ¶
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 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 app:A-long-proofs@proof-435 ¶

Supports

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 ¶

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`	→	Vanishing of the pressure–velocity correlation	declared	appendices/A-long-proofs.tex:36166
`depends_on`	→	The Navier–Stokes equations	declared	appendices/A-long-proofs.tex:36166
`depends_on`	←	The Kármán–Howarth equation in classical form	declared	appendices/A-long-proofs.tex:36251
`depends_on`	←	Kolmogorov's equation and the four-fifths law	declared	appendices/A-long-proofs.tex:36422
`proves`	←	app:A-long-proofs@proof-435	declared	appendices/A-long-proofs.tex:36169