theorem 31.58 The vorticity transport equation

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:1818 · p. 1062

Rests on

depends_on equation 31.5 eq:fluid-curl-cross ¶
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-27 ¶

Supports

depends_on phenomenon 31.84 The five-thirds spectrum ¶
- depends_on proposition 31.85 The Kolmogorov scales ¶
- depends_on remark 31.86 The one exact result, and the split attribution ¶
depends_on phenomenon 31.61 The vortex street and its tone ¶
- depends_on remark 31.62 What the spacing calculation settles, and what it does not ¶
depends_on theorem 31.59 Helmholtz's vortex theorems ¶
- depends_on corollary 31.65 Lagrange: an irrotational flow stays irrotational ¶
  - depends_on remark 31.69 The Kutta condition, and where the circulation comes from ¶
- depends_on example 31.60 Vortex rings ¶
depends_on theorem 31.77 Rayleigh's inflexion-point criterion ¶

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`	→	eq:fluid-curl-cross	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1832
`depends_on`	→	The Navier–Stokes equations	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1832
`depends_on`	←	The five-thirds spectrum	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2870
`depends_on`	←	The vortex street and its tone	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1965
`depends_on`	←	Helmholtz's vortex theorems	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1896
`depends_on`	←	Rayleigh's inflexion-point criterion	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2540
`proves`	←	ch:14-fluid-dynamics@proof-27	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1835