lemma 31.11 Transport theorem for a material volume

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:279 · p. 1044

Rests on

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 3.43 Map ¶
  - depends_on definition 7.99 Differentiability at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
    - 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 ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - 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 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.97 Partial derivative; gradient ¶ ↺
  - depends_on definition 6.9 Compact set ¶
    - depends_on definition 6.5 Open cover ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
- 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 axiom 7.1 Completeness of $\R$ ¶ ↺
  - 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 axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on proposition 7.22 Sequential characterization ¶ ↺
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
      - proves proof ch:05-real-analysis@proof-10 ¶
    - 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 theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-20 ¶
    - 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 ¶

Supports

depends_on theorem 31.12 Conservation of mass ¶
- depends_on corollary 31.13 Incompressible flow ¶
  - depends_on corollary A.743 Two scalar functions instead of a tensor field ¶
    - depends_on lemma A.748 The second-order structure function ¶
      - depends_on theorem A.749 Kolmogorov's equation and the four-fifths law ¶
    - depends_on lemma A.747 The third-order structure functions ¶
      - depends_on theorem A.749 Kolmogorov's equation and the four-fifths law ¶ ↺
    - depends_on proposition A.746 The Kármán–Howarth equation in classical form ¶
  - depends_on example 31.15 The stream function of a plane flow ¶
    - depends_on definition A.715 Stokes stream function ¶
      - depends_on lemma A.716 The creeping-flow equation for $\psi$ ¶
  - depends_on lemma A.744 Vanishing of the pressure–velocity correlation ¶
    - 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 ¶ ↺
  - depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
    - depends_on example 31.29 Flow past a sphere ¶
    - depends_on lemma A.777 The tangential field above a bending plate ¶
      - depends_on corollary A.778 The powder collects at the antinodes ¶
    - depends_on lemma A.733 The circle with incidence and circulation ¶
      - depends_on lemma A.735 Transfer of the flow ¶
    - depends_on proposition A.797 The outer limit loses a boundary condition ¶
    - depends_on proposition 31.70 The lift-curve slope of a thin aerofoil ¶
    - 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 remark 31.14 When incompressibility is legitimate ¶
- 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 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 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 31.22 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 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 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 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 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 remark A.768 Two conventions for the wavenumber ¶
      - depends_on theorem A.769 The Lorenz system ¶
  - 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 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 ¶ ↺
  - … 3 more
- depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶ ↺
- depends_on theorem 31.77 Rayleigh's inflexion-point criterion ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Material derivative	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:291
`depends_on`	→	Gauss	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:291
`depends_on`	←	Conservation of mass	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:322
`depends_on`	←	Euler's equations of motion	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:596
`proves`	←	ch:14-fluid-dynamics@proof-4	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:294