Book contents
- Frontmatter
- Contents
- Preface
- List of Abbreviations
- 1 Introduction
- 2 Historical notes
- 3 Boundary conditions for viscous fluids
- 4 Helmholtz decomposition coupling rotational to irrotational flow
- 5 Harmonic functions that give rise to vorticity
- 6 Radial motions of a spherical gas bubble in a viscous liquid
- 7 Rise velocity of a spherical cap bubble
- 8 Ellipsoidal model of the rise of a Taylor bubble in a round tube
- 9 Rayleigh–Taylor instability of viscous fluids
- 10 The force on a cylinder near a wall in viscous potential flows
- 11 Kelvin–Helmholtz instability
- 12 Energy equation for irrotational theories of gas–liquid flow: viscous potential flow, viscous potential flow with pressure correction, and dissipation method
- 13 Rising bubbles
- 14 Purely irrotational theories of the effect of viscosity on the decay of waves
- 15 Irrotational Faraday waves on a viscous fluid
- 16 Stability of a liquid jet into incompressible gases and liquids
- 17 Stress-induced cavitation
- 18 Viscous effects of the irrotational flow outside boundary layers on rigid solids
- 19 Irrotational flows that satisfy the compressible Navier–Stokes equations
- 20 Irrotational flows of viscoelastic fluids
- 21 Purely irrotational theories of stability of viscoelastic fluids
- 22 Numerical methods for irrotational flows of viscous fluid
- Appendix A Equations of motion and strain rates for rotational and irrotational flow in Cartesian, cylindrical, and spherical coordinates
- Appendix B List of frequently used symbols and concepts
- References
- Index
12 - Energy equation for irrotational theories of gas–liquid flow: viscous potential flow, viscous potential flow with pressure correction, and dissipation method
Published online by Cambridge University Press: 09 October 2009
- Frontmatter
- Contents
- Preface
- List of Abbreviations
- 1 Introduction
- 2 Historical notes
- 3 Boundary conditions for viscous fluids
- 4 Helmholtz decomposition coupling rotational to irrotational flow
- 5 Harmonic functions that give rise to vorticity
- 6 Radial motions of a spherical gas bubble in a viscous liquid
- 7 Rise velocity of a spherical cap bubble
- 8 Ellipsoidal model of the rise of a Taylor bubble in a round tube
- 9 Rayleigh–Taylor instability of viscous fluids
- 10 The force on a cylinder near a wall in viscous potential flows
- 11 Kelvin–Helmholtz instability
- 12 Energy equation for irrotational theories of gas–liquid flow: viscous potential flow, viscous potential flow with pressure correction, and dissipation method
- 13 Rising bubbles
- 14 Purely irrotational theories of the effect of viscosity on the decay of waves
- 15 Irrotational Faraday waves on a viscous fluid
- 16 Stability of a liquid jet into incompressible gases and liquids
- 17 Stress-induced cavitation
- 18 Viscous effects of the irrotational flow outside boundary layers on rigid solids
- 19 Irrotational flows that satisfy the compressible Navier–Stokes equations
- 20 Irrotational flows of viscoelastic fluids
- 21 Purely irrotational theories of stability of viscoelastic fluids
- 22 Numerical methods for irrotational flows of viscous fluid
- Appendix A Equations of motion and strain rates for rotational and irrotational flow in Cartesian, cylindrical, and spherical coordinates
- Appendix B List of frequently used symbols and concepts
- References
- Index
Summary
Viscous potential flow
We obtained the effects of viscosity on irrotational motions of spherical cap bubbles, Taylor bubbles in round tubes, and RT and KH instabilities described in previous chapters by evaluating the viscous normal stress on potential flow. In gas–liquid flows, the viscous normal stress does not vanish and it can be evaluated on the potential. It can be said that, in the case of gas–liquid flow, the appropriate formulation of the irrotational problem is the same as the conventional one for inviscid fluids with the caveat that the viscous normal stress is included in the normal stress balance. This formulation of VPF is not at all subtle; it is the natural and obvious way to express the equations of balance when the flow is irrotational and the fluid viscous.
In this chapter we use the acronym VPF, viscous potential flow, to stand for the irrotational theory in which the viscous normal stresses are evaluated on the potential.
In gas–liquid flows we may assume that the shear stress in the gas is negligible so that no condition need be enforced on the tangential velocity at the free surface, but the shear stress must be zero. The condition that the shear stress be zero at each point on the free surface is dropped in irrotational approximations.
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- Potential Flows of Viscous and Viscoelastic Liquids , pp. 126 - 133Publisher: Cambridge University PressPrint publication year: 2007