Home Technique • Potential Flows of Viscous and Viscoelastic Liquids by Daniel Joseph, Toshio Funada, Jing Wang

Potential Flows of Viscous and Viscoelastic Liquids by Daniel Joseph, Toshio Funada, Jing Wang

By Daniel Joseph, Toshio Funada, Jing Wang

The aim of this booklet is to teach how capability flows input into the overall concept of motions of viscous and viscoelastic fluids. frequently, the speculation of strength flows is assumed to use to idealized fluids with out viscosity. the following we exhibit tips to practice this idea to actual fluids which are viscous. the idea is utilized to difficulties of the movement of bubbles; to the decay of waves on interfaces among fluids; to capillary, Rayleigh-Taylor, and Kelvin-Hemholtz instabilities; to viscous results in acoustics; to boundary layers on solids at finite Reynolds numbers; to difficulties of stress-induced cavitation; and to the production of microstructures within the movement of viscous and viscoelastic beverages.

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Additional resources for Potential Flows of Viscous and Viscoelastic Liquids (Cambridge Aerospace Series, 21)

Sample text

This important problem has not yet been solved. 7 Viscous decay of free-gravity waves Flows that depend on only two space variables, such as plane flows or axisymmetric flows, admit a stream function. 6 Examples from hydrodynamics 25 potential function. Lamb (1932, §349) calculated an exact solution of the problem of the viscous decay of free-gravity waves as a free-surface problem of this type. 17) where ∇ 2 ␾ = 0, ∂␺ = ␯∇ 2 ␺ . 18) This decomposition is a Helmholtz decomposition; it can be said that Lamb solved this problem in the Helmholtz formulation.

Weinberger developed a much simpler approach to this problem, which leads to Lamb’s result (the text that follows is taken from a personal communication from Prof. 12) which holds for all vector fields, and is essentially Equation (5) of §335 [of Lamb (1932)]. As in §335, we first look at the case of constant pressure. In this case curl (curl u) = 0, and of course div (curl u) = 0, so that curl u is the gradient of some harmonic function m. As Lamb points out, the function l := r · u is also harmonic.

An initial-value problem must be solved, and disturbances cannot be restricted to radial symmetry. Simplifying features are lost. Results by Birkhoff (1954) and Plesset (1954) are stated with many qualifiers and require special assumptions. The problem of stability of a spherical gas bubble in an inviscid liquid was considered by Radial motions of a spherical gas bubble in a viscous liquid 41 Birkhoff (1954), who reduced the problem to the study of perturbations of the free surface expressed by Legendre polynomials: ∞ bn (t)Pn (cos ␸).

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