example 31.60 Vortex rings

Rests on

• depends_on theorem 31.59 Helmholtz's vortex theorems ¶
  • depends_on theorem 31.64 Kelvin's circulation theorem ¶
    • depends_on definition 31.63 Circulation ¶
      • 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 theorem 7.132 Stokes ¶
        • depends_on proposition 7.104 Chain rule in several variables ¶
        • depends_on proposition 7.105 Clairaut–Schwarz ¶
        • depends_on theorem 7.131 Green ¶
        • proves proof ch:05-real-analysis@proof-78 ¶
    • 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 theorem 7.133 Gauss ¶
        • 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-29 ¶
  • depends_on theorem 31.58 The vorticity transport equation ¶
    • 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 proposition 31.9 Cauchy–Stokes decomposition ¶
        • proves proof ch:14-fluid-dynamics@proof-17 ¶
      • depends_on theorem 31.22 Euler's equations of motion ¶ ↺
      • proves proof ch:14-fluid-dynamics@proof-18 ¶
    • proves proof ch:14-fluid-dynamics@proof-27 ¶
  • proves proof ch:14-fluid-dynamics@proof-28 ¶

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