theorem 31.22 Euler's equations of motion

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:585 · p. 1047

Rests on

depends_on definition 31.21 Ideal fluid ¶
- 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 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 ¶
    - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on definition 7.99 Differentiability at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on definition 5.37 Linear transformation ¶
      - depends_on equation 6.12 eq:top-euclidean-metric ¶
    - depends_on definition 7.102 Differential of a function ¶
      - depends_on definition 7.99 Differentiability at a point ¶ ↺
      - depends_on definition 7.97 Partial derivative; gradient ¶
    - proves proof ch:05-real-analysis@proof-63 ¶
  - proves proof ch:14-fluid-dynamics@proof-1 ¶
- depends_on theorem 7.133 Gauss ¶
  - depends_on definition 7.127 Simple regions ¶
    - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶ ↺
    - depends_on definition 6.9 Compact set ¶
      - depends_on definition 6.5 Open cover ¶
  - depends_on remark 7.128 What the derivations below take as given ¶
    - depends_on definition 7.125 Multiple integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
    - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶ ↺
      - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - proves proof ch:05-real-analysis@proof-23 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
  - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
    - depends_on definition 7.41 Antiderivative ¶
      - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
    - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
      - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
      - depends_on theorem 7.24 Extreme value theorem ¶ ↺
      - proves proof ch:05-real-analysis@proof-24 ¶
    - proves proof ch:05-real-analysis@proof-25 ¶
  - 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 ¶

Supports

depends_on lemma A.712 The force is a far-field integral ¶
depends_on phenomenon 31.88 Sound travels at the adiabatic speed ¶
- depends_on remark 31.89 The Mach number and the regimes it separates ¶
  - depends_on phenomenon 31.93 The Mach cone ¶
- depends_on remark 31.90 Why a compression wave must steepen ¶
depends_on proposition A.797 The outer limit loses a boundary condition ¶
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 proposition 52.24 The p-mode cavity ¶
- depends_on phenomenon 52.25 The Sun rings, and its interior can be inverted ¶
  - depends_on phenomenon 52.22 The solar-neutrino deficit, and its resolution ¶
depends_on remark 31.23 Boundary conditions, and what the ideal model omits ¶
depends_on theorem 31.24 Bernoulli ¶
- depends_on corollary A.799 The impressed pressure gradient ¶
- depends_on lemma 31.67 Blasius' force formula ¶
  - depends_on theorem 31.68 Kutta–Joukowski ¶
    - depends_on proposition 31.70 The lift-curve slope of a thin aerofoil ¶
    - depends_on remark 31.69 The Kutta condition, and where the circulation comes from ¶
    - depends_on theorem A.737 The lift-curve slope of the flat plate ¶
- depends_on phenomenon 31.25 Pressure falls where the flow speeds up ¶
  - depends_on example 31.26 The Venturi meter ¶
  - depends_on remark 31.27 What the 1738 source contains ¶
- depends_on theorem A.707 D'Alembert's paradox ¶
- depends_on theorem 31.74 The interface dispersion relation ¶
  - depends_on corollary 31.76 Kelvin–Helmholtz instability ¶
  - depends_on corollary 31.75 Rayleigh–Taylor instability ¶
  - depends_on theorem 31.94 Gravity–capillary waves on a free surface ¶
    - depends_on phenomenon 31.96 The solitary wave ¶
    - depends_on phenomenon 31.95 The observed dispersion of water waves ¶
depends_on theorem 31.64 Kelvin's circulation theorem ¶
- 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 remark 31.66 Only three things make circulation ¶
- depends_on theorem 31.59 Helmholtz's vortex theorems ¶
  - depends_on corollary 31.65 Lagrange: an irrotational flow stays irrotational ¶ ↺
  - depends_on example 31.60 Vortex rings ¶
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 31.53 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 31.79 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
depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶
- depends_on phenomenon 31.93 The Mach cone ¶ ↺
- depends_on remark 31.92 Entropy selects the physical branch ¶
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
`cites`	→	Principes généraux du mouvement des fluides	derived	parts/03-classical-mechanics/14-fluid-dynamics.tex:595
`depends_on`	→	Ideal fluid	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:596
`depends_on`	→	Transport theorem for a material volume	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:596
`depends_on`	→	Cauchy's equation of motion	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:596
`depends_on`	←	The force is a far-field integral	declared	appendices/A-long-proofs.tex:34418
`depends_on`	←	Sound travels at the adiabatic speed	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3035
`depends_on`	←	The outer limit loses a boundary condition	declared	appendices/A-long-proofs.tex:39026
`depends_on`	←	Rayleigh's circulation criterion	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2677
`depends_on`	←	The p-mode cavity	declared	parts/05-general-relativity-cosmology/11-stellar-structure-nucleosynthesis.tex:1883
`depends_on`	←	Boundary conditions, and what the ideal model omits	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:639
`depends_on`	←	Bernoulli	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:660
`depends_on`	←	Kelvin's circulation theorem	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2019
`depends_on`	←	The Navier–Stokes equations	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1074
`depends_on`	←	Rankine–Hugoniot jump conditions	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3132
`depends_on`	←	Rayleigh's inflexion-point criterion	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2540
`proves`	←	ch:14-fluid-dynamics@proof-11	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:600