^{1,a)}and Thomas P. Witelski

^{2,b)}

### Abstract

We consider the inertially driven, time-dependent biaxial extensional motion of inviscid and viscous thinning liquid sheets. We present an analytic solution describing the base flow and examine its linear stability to varicose (symmetric) perturbations within the framework of a long-wave model where transient growth and long-time asymptotic stability are considered. The stability of the system is characterized in terms of the perturbation wavenumber, Weber number, and Reynolds number. We find that the isotropic nature of the base flow yields stability results that are identical for axisymmetric and general two-dimensional perturbations. Transient growth of short-wave perturbations at early to moderate times can have significant and lasting influence on the long-time sheet thickness. For finite Reynolds numbers, a radially expanding sheet is weakly unstable with bounded growth of all perturbations, whereas in the inviscid and Stokes flow limits sheets are unstable to perturbations in the short-wave limit.

This research was supported by the National Science Foundation Grant Nos. DMS-0707755 and DMS-0968252.

I. INTRODUCTION

II. THE UNPERTURBED FLOW

III. STABILITY OF A RADIALLY EXPANDING LIQUID SHEET

A. Scalings and long-wave model

B. Axisymmetric perturbations

1. Inviscid problem

2. Viscous problem

C. Comparisons with “frozen-time” analysis

D. General two-dimensional perturbations

IV. CONCLUSIONS

### Key Topics

- Viscosity
- 45.0
- Sheet flows
- 42.0
- Extensional flows
- 34.0
- Perturbation methods
- 16.0
- Kinematics
- 13.0

## Figures

Schematic of a radially expanding, thinning liquid sheet with free surfaces located at z = ±h(t).

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.

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.

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.

The numerically computed amplification factor α(k) in the low-viscosity limit β → 0 (solid curves) and the estimate given by Eq. (31) (dashed curve). The amplification factor was computed from Eqs. (15a) and (15b) with We = 1, Re = 1000, and initial data H(T 0) = U(T 0) = 1.

The numerically computed amplification factor α(k) in the low-viscosity limit β → 0 (solid curves) and the estimate given by Eq. (31) (dashed curve). The amplification factor was computed from Eqs. (15a) and (15b) with We = 1, Re = 1000, and initial data H(T 0) = 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).

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 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).

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(T) perturbations in the high viscosity limit with Re = 1: (a) for k = 15 > k 3 showing decay (α < 1) and (b) for k = 8 < k 3 showing growth (α > 1). The dashed curves compare Eq. (35) with the results computed from Eqs. (15a) and (15b) with H(T 0) = U(T 0) = 1 and We = 1.

Evolution of H(T) perturbations in the high viscosity limit with Re = 1: (a) for k = 15 > k 3 showing decay (α < 1) and (b) for k = 8 < k 3 showing growth (α > 1). The dashed curves compare Eq. (35) with the results computed from Eqs. (15a) and (15b) with H(T 0) = U(T 0) = 1 and We = 1.

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) .

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.

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 ).

(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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