The feed-out process: Rayleigh-Taylor and Richtmyer-Meshkov instabilities in thin, laser-driven foils Page: 54 of 193
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39
namic property in Eqs. (7) with two quantities, Eq. (8)
p = 9+p
V = v + (8)
P = Po+ P
The variables with the "0" subscripts represent equilibrium and are the "zeroth-
order" aspect of the fluid motion. They satisfy Eqs. (7) by themselves, so one may
write
aPo
+ V*(po0o) = 0
(9)
pv + p(3e " V)O = -Vp0+ p0g.
The quantities with the subscript "n" in Eqs. (8) represent the perturbation being
placed on the equilibrium properties and are very small. They are sometimes
referred to as the "first-order" quantities.
After substituting Eqs. (8) into Eqs. (7), Eqs. (9) are subtracted, and the
result is linearized. The process of linearization involves the elimination of any
products of perturbed quantities, as these will be much smaller than terms that
are linear in the first order terms. The linearized equations are
+ V*(p"v0 + p0v") = 0
(10)
Pt0 + Oat+p() e OV50+0. v W)+p(v " Vo) = -Vp+ p.
Now an initial condition is applied to Eqs. (10). For the Rayleigh-Taylor
instability problem, one assumes the two fluids are initially at rest, implying,
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Smitherman, D.P. The feed-out process: Rayleigh-Taylor and Richtmyer-Meshkov instabilities in thin, laser-driven foils, report, April 1, 1998; New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc699271/m1/54/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.