definition 31.1 Fluid

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:50 · p. 1041

ground object -- no derivation owed

Rests on

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 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 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 definition 30.2 Representative volume element; continuum field ¶
    - depends_on definition 7.97 Partial derivative; gradient ¶ ↺
- depends_on definition 13.5 Tensor under orthogonal transformations ¶
  - depends_on definition 13.4 Orthogonal coordinate transformation ¶
    - depends_on definition 13.3 Orthogonal coordinate system ¶
- proves proof ch:13-continuum-elasticity@proof-7 ¶

Supports

depends_on definition 31.21 Ideal fluid ¶
- depends_on theorem 31.22 Euler's equations of motion ¶
  - 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 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 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 31.43 Creeping flow ¶
      - depends_on lemma A.716 The creeping-flow equation for $\psi$ ¶
      - 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 phenomenon 31.41 The fourth-power law of pipe flow ¶
      - depends_on phenomenon 31.53 Transition to turbulence in a pipe ¶
      - depends_on remark 31.42 The entry length ¶
    - depends_on phenomenon 31.49 Dynamical similarity ¶
      - depends_on definition 31.51 The other similarity groups ¶
      - 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 ¶
depends_on definition 31.2 Knudsen number ¶
- depends_on remark 31.3 When the continuum picture is allowed ¶
depends_on definition 31.32 Newtonian fluid; dynamic and kinematic viscosity ¶
- depends_on remark 31.35 Non-Newtonian fluids are out of scope ¶
- depends_on remark 31.33 Reading the two viscosities ¶
- 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 proposition 31.16 Hydrostatic balance ¶
- depends_on example 31.19 The barometric distribution ¶
- depends_on phenomenon 49.8 The white-dwarf luminosity function has an edge ¶
- depends_on phenomenon 31.18 Archimedes' principle ¶
  - depends_on example 31.46 Terminal velocity, and Millikan's droplets ¶
- depends_on proposition 52.1 Hydrostatic equilibrium of a self-gravitating sphere ¶
  - depends_on phenomenon 52.12 The mass–luminosity relation ¶
    - depends_on phenomenon 52.17 The Hertzsprung–Russell diagram ¶
  - depends_on proposition 49.3 Lane–Emden form of hydrostatic equilibrium; restated from the stellar-structure chapter ¶
    - depends_on theorem 49.7 The Chandrasekhar mass ¶
      - depends_on phenomenon 49.6 A maximum mass for a cold degenerate star ¶
  - depends_on theorem 52.2 Virial theorem for a self-gravitating sphere ¶
    - depends_on phenomenon 52.4 The Sun has shone for billions of years ¶
      - depends_on phenomenon 52.23 The Sun's energy budget audited reaction by reaction ¶
    - depends_on proposition 52.18 Schönberg–Chandrasekhar limit ¶

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`	→	Cauchy: the stress tensor exists	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:60
`depends_on`	←	Ideal fluid	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:582
`depends_on`	←	Knudsen number	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:77
`depends_on`	←	Newtonian fluid; dynamic and kinematic viscosity	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:920
`depends_on`	←	Hydrostatic balance	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:416