Schematic of a radially expanding, thinning liquid sheet with free surfaces located at z = ±h(t).
Evolution of perturbation amplitude in the inviscid short-wave case for k > k 1: (a) The oscillatory form of H(T) for early times from initial conditions H(T 0) = 1, U(T 0) = 1, and T 0 = 1/110 (corresponding to k = 110 at We = 1); (b) perturbation magnitude on a log-log plot to illustrate the O(T 1/2) transient growth and saturation to the asymptotic amplification factor, for two wavenumbers, α(110) ≈ 14.4 and α(109.9886) ≈ 0.106, obtained from numerical solution of Eqs. (15a) and (15b) with β = 0.
The amplification factor α(k) for an inviscid radially expanding sheet subject to axisymmetric perturbations with dashed asymptotes indicating the long- and short-wave limits. For k < k 1 the sheet is weakly stable, and for k 1 < k < ∞ the sheet is weakly unstable due to transient growth of the perturbations. The plot for α(k) uses definition (21) with H(T) determined by solving Eqs. (15a) and (15b) with β = 0, We = 1, and initial conditions H(T 0) = 1, U(T 0) = 1.
Evolution of the magnitude of the surface perturbations, |H(T)|, from Eqs. (15a) and (15b) in the low-viscosity limit, Re = 1000, for (a) k = 15 < k 2 showing overall growth (α > 1) and (b) k = 50 > k 2 showing overall decay (α < 1) with H(T 0) = U(T 0) = 1 and We = 1. The dashed curve shows the amplitude from Eq. (30) and vertical dotted-dashed lines indicate T 0 and T 1 (T 0 < T 1).
The critical timescale for transition from oscillatory to non-oscillatory evolution for the viscous problem: T 1 for β < β1 and T 2(β) from Eq. (32) for larger β.
The amplification factor α(k) for (a) Re = 1 and (b) Re = 2.5 (both with We = 1) compared with Eq. (36) for β > β1 (dashed curve).
Evolution of |H| in the moderate viscosity regime with Re = 5 and k = 8 > k osc, the threshold given by Eq. (37) . The oscillations, occurring on T 0 ⩽ T < T 2 (T 0, T 2 indicated by vertical dotted-dashed lines), are well approximated by Eq. (30) (dashed curve). After T 2, there is an intermediate range where Eq. (35) applies leading to the long-time behavior Eq. (19) .
Maximum amplification factor for axisymmetric perturbations as a function of Reynolds number with asymptotes derived from Eqs. (31) and (36) and with We fixed.
(a) The growth rates Real(λ±) predicted by the frozen-time analysis, and (b) the short-time evolution of the perturbation amplitude from Eqs. (15a) and (15b) (solid curve) compared with the prediction from frozen-time system (38) (dashed) for Re = 5, We = 1, and k = 8 (as shown in Fig. 9 ).
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