Contents Menu Expand Light mode Dark mode Auto light/dark, in light mode Auto light/dark, in dark mode Skip to content
multiFlowKobe
multiFlowKobe
  • 1. Local instantaneous equations
    • 1.1. Governing equations
    • 1.2. Jump conditions
    • 1.3. Interface tracking
  • 2. Void fraction
    • 2.1. Averaged quantities
    • 2.2. Drift-flux model
  • 3. Two-phase pressure drop
    • 3.1. Global balance in steady upward flow
    • 3.2. Frictional pressure drop: Homogeneous model
    • 3.3. Frictional pressure drop: Lockhart-Martinelli correlation
  • 4. Annular flow
    • 4.1. Upward co-current annular flow
    • 4.2. Counter-current annular flow
    • 4.3. Counter-current limitation
  • 5. Averaged equations
    • 5.1. Averaging basic equations
    • 5.2. Interfacial momentum transfer
  • 6. Analysis of bubble rise motion
    • 6.1. Dimensionless groups
    • 6.2. Bubble velocity correlations
    • 6.3. Shape deformation
  • 7. Virtual mass
  • 8. Drag force acting on spherical body
    • 8.1. Drag acting on a solid sphere in Stokes flow (LL)
    • 8.2. Drag acting on a fluid sphere in Stokes flow
    • 8.3. Drag acting on a spherical bubble at high \(Re\)
  • 9. Vorticity-related problems
    • 9.1. Drag acting on a solid sphere in Stokes flow (GKB)
    • 9.2. Drag-enstrophy relation
    • 9.3. Drag-vorticity relation
    • 9.4. Lift acting on spherical body in inviscid flow with weak shear
    • 9.5. Basset (history) force on a solid sphere in Stokes flow
  • 10. Deformed bubbles
    • 10.1. Large bubbles in liquid
    • 10.2. Wave analogy for bubble rise velocity
    • 10.3. Lift reversal
  • 11. Taylor bubble: bullet-shaped bubbles in pipe
    • 11.1. Order-of-magnitude analysis for bubble velocity
    • 11.2. Lubrication of liquid film
    • 11.3. Bretherton problem
  • 12. Contaminated system
    • 12.1. Isotherm and surfactant kinetics
    • 12.2. Surfactant transport
    • 12.3. Contaminated drops in Stokes flow
    • 12.4. Bretherton problem with surfactant
  • Mathematics tips
    • Some useful identities
    • Polar and axial vectors
    • Biot-Savart law
    • \(\nabla^{2} (1/r)\) behaves as Dirac’s delta
  • Fluid mechanics fundamentals
    • Continuity and NS equations in polar coordinate systems
    • Viscous dissipation
    • Potential flow about sphere
    • Water wave of infinitesimal amplitude
    • Vorticity equation
    • Helmholtz’s law
    • Vorticity in terms of drift function
    • Biot-Savart field by uniform line vorticity
  • References
Back to top
View this page

9. Vorticity-related problems¶

Summary

  • Subject: Drag, lift and history forces in relation with vorticity

  • Key idea Vorticity \(\nabla \times \mathbf{v}\) plays key roles in hydrodynamic forces acting on particles.

  • References:

    • Stone [Sto93]

    • Legendre [Leg07]

    • Auton [Aut87]

    • Landau and Lifshitz [LL87]

  • 9.1. Drag acting on a solid sphere in Stokes flow (GKB)
  • 9.2. Drag-enstrophy relation
  • 9.3. Drag-vorticity relation
  • 9.4. Lift acting on spherical body in inviscid flow with weak shear
  • 9.5. Basset (history) force on a solid sphere in Stokes flow
Next
9.1. Drag acting on a solid sphere in Stokes flow (GKB)
Previous
8.3. Drag acting on a spherical bubble at high \(Re\)
Copyright © 2026, Kosuke Hayashi@Kobe University
Made with Sphinx and @pradyunsg's Furo