Rayleigh-Taylor instability in accelerated elastic-solid slabs

Abstract

We develop the linear theory for the asymptotic growth of the incompressible Rayleigh-Taylor instability of an accelerated solid slab of density ρ_{2}, shear modulus G, and thickness h, placed over a semi-infinite ideal fluid of density ρ_{1}<ρ_{2}. It extends previous results for Atwood number A_{T}=1 [B. J. Plohr and D. H. Sharp, Z. Angew. Math. Phys. 49, 786 (1998)ZAMPA80044-227510.1007/s000330050121] to arbitrary values of A_{T} and unveil the singular feature of an instability threshold below which the slab is stable for any perturbation wavelength. As a consequence, an accelerated elastic-solid slab is stable if ρ_{2}gh/G≤2(1-A_{T})/A_{T}.

Magneto-Rayleigh–Taylor instability in an elastic finite-width medium overlying an ideal fluid

“…These limit show that (4.23) describes several different behaviours depending on the values of β 0 and A T which are discussed below: (i) For A T = 1 and β 0 = 0 we retrieve, as expected, the pure elastic case with a cut-off given by (4.31) (Bakhrakh et al1997, Plohr & Sharp 1998, Piriz, Piriz & Tahir 2017a, 2017b [see figure 6(a), and (4.31)]…”

“…Figures 5(c) and 5(d) show the same cases as before but for the marginal stability wavenumber κ c as a function of dimensionless thickness α. The behaviour is qualitatively the same for any Atwood number except for the fact that, for the purely elastic case (β 3 = 0), there is no instability threshold when A T = 1 (Plohr & Sharp 1998, Piriz, Piriz & Tahir 2017a, 2017b.…”

“…This is also the scenario in some recent laboratory experiments (Adkins et al 2017) involving viscous fluids, a situation that has also been studied theoretically by and previously, for some particular limits, by Lister & Kerr (1989), and Wilcock & Whitehead (1991). For the case of finite-width elastic-media with no presence of rigid boundaries RT instability has been studied by Bakhrakh et al (1997), Plohr & Sharp (1998), and by Piriz, Piriz & Tahir (2017a, 2017b. When magnetic fields are present the instability is known as the magneto-Rayleigh-Taylor (MRT) instability.…”

“…provided that β 1 2(1 − A T )/(1 + A T ). If not, a critical value α cr exists below which the system is stable for any perturbation wavenumber: from which we retrieve the result by Piriz, Piriz & Tahir (2017b) when β 1 = 0 and no magnetic field is present.…”

“…It is to be noted that, although the enormous magnetic fields exist on the surface of magnetars, there are evidences of much stronger magnetic fields beneath the neutron star crust (Cooper & Kaplan 2010, Ryu et al 2012, Mereghetti, Pons & Melator 2015. Besides, it has been shown that in the absence of magnetic fields, the sole effect of the crust elasticity imposes an instability threshold that depends on the magnitude of the density inversion in the neutron star crust (Blaes et al1990, 1992, Piriz, Piriz & Tahir 2017b). This density inversion produced by pycnonuclear and electron capture reactions in the crust (Blaes et al1990, 1992, Mock & Joss.…”