Asymptotic methods for the eigenvalues of the Rayleigh equation for the linearized Rayleigh–Taylor instability

Abstract

In this paper, we study the spectrum of the Rayleigh equation, which models the Rayleigh–Taylor instability in a fluid of variable density ρ(x), where ρ(x) goes to ρ₋ at −∞ and ρ₊ at +∞: $-h^{2}{\frac{\mathrm{d}}{\mathrm{d}x}}\bigl(\rho(x){\frac{\mathrm{d}u}{\mathrm{d}x}}\bigr)+(\rho(x)+\delta \rho'(x))u=0,\quad u\in L^{2}(\mathbb{R} ).$ The behavior of the smallest value δ(h) of δ>0 for which there exists a nontrivial solution u is investigated in terms of h for the two regimes h→0 (called the semi‐classical regime) and h→+∞. The so‐called growth rate of the Rayleigh–Taylor instability is $\sqrt{h/\delta(h)}$ .

When ρ−ρ₋∈L²( $\mathbb{R} $ ⁻) and ρ−ρ₊∈L²( $\mathbb{R} $ ⁺), we prove that δ(h)/h→(ρ₊+ρ₋)/(ρ₋−ρ₊), as h→+∞, generalizing the result of Lord Rayleigh [23].

We then investigate the expansion of δ(h)/h in terms of 1/h and properties of ρ. In particular, we identify the number of terms n of the expansion in function of the behavior of ρ(x)−ρ_± at ±∞, extending the results of Cherfils, Lafitte and Raviart [6].