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By F. Oort

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The elimination of the vorticity leads to the pure streamfunction formulation ∂ψ ∂ ∂ψ ∂ ∂ (∆ψ) + (∆ψ) − (∆ψ) = ν∆2 ψ . 24) ∂t ∂y ∂x ∂x ∂y The extension of these approaches to three-dimensional flows requires the introduction of a second streamfunction or the use of a vector potential. The appropriate equations for the former can be found in Murdock (1986) and those for the latter in Brosa and Grossman (2002). 4 Vorticity–Velocity Equations Another approach to eliminating the pressure from the incompressible Navier– Stokes equations is to take the curl of the momentum equation.

A gas is a compressible fluid that not only conforms to the shape of its container but also expands to occupy the full container. Plasma is ionized gas. The various phases of a substance are represented by a phase diagram, which marks the domains in pressure–temperature where each phase exists. 1 illustrates the well-known phase diagram for water. A rigorous mathematical description of the states of matter appears implausible. Goodstein (1975) states: “Precisely what do we mean by the term liquid?

64) as pressure plays a dynamic role rather than a thermodynamic one. 13), and then letting Mref → 0 yields the nondimensional compressible flow equations at zero Mach number: ∂ρ∗ + ∇∗ · (ρ∗ u∗ ) = 0 , ∂t∗ ∂(ρ∗ u∗ ) 1 ∗ ∗ ∇ ·τ , + ∇∗ · (ρ∗ u∗ u∗T ) + ∇∗ p˜ = ∗ ∂t Re ∂(ρ∗ E ∗ ) 1 ∇∗ · (κ∗ ∇∗ T ∗ ) . 4 Incompressible Fluid Dynamics Equations The equations described in the preceding section all have special forms in the incompressible limit. There are several different formulations of the incompressible Navier–Stokes equations.

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