theorem 31.82 The Reynolds-averaged equations

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:2772 · p. 1073

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-41 ¶

Supports

depends_on remark 31.83 The closure problem, stated exactly ¶
- depends_on remark 31.87 What a theory of turbulence would have to do ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`cites`	→	On the dynamical theory of incompressible viscous fluids and the determination of the criterion	derived	parts/03-classical-mechanics/14-fluid-dynamics.tex:2787
`depends_on`	→	The Navier–Stokes equations	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2788
`depends_on`	←	The closure problem, stated exactly	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2837
`proves`	←	ch:14-fluid-dynamics@proof-41	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2791