Revisiting the Taylor-Culick approximation. II. Retraction of a viscous sheet

We study the retraction of a viscous liquid sheet of finite length with negligible effect of the ambient medium. Using the long-wavelength model we derive the scaling laws and similarity solution for the interface profile of the retracting sheet. Far from the tip, the similarity solution for the interface profiles converges to an asymptotic value of 1/4. Direct numerical simulations are performed to compare the theoretical results with the simulations. When the inertia is negligible, the interface profiles remain flat during the retraction process which is in agreement with the self-similar solution. Using this similarity solution we derive the expression for the temporal variation of the tip speed for finite liquid sheets. We demonstrate that unlike an infinite sheet where the sheet retracts with a steady speed (known as Taylor-Culick speed), the tip speed decreases as a function of time for a finite liquid sheet. This is true when the viscous effects are larger than or of the same order with the inertia effects. Otherwise, the sheet retracts with the formation of a bulbous tip whose speed reaches a value closer to the Taylor-Culick speed.

Figure 1

(a) The self-similar interface profile and (b) the self-similar velocity profile. The numerical results for $L_0/h_0=10$ are shown by the solid line, while the circles are the results for $L_0/h_0=20$. The tip of the self-similar interface profile can be represented by a semicircle with good accuracy [the dashed-dot line is a reference circle in (a)]. In the velocity profile, the dashed line represents the theoretical solution $U=2C-\eta$.

Figure 2

(a) Schematic of a bursting soap film in air (midsection view) and (b) schematic of a bursting bubble at a liquid-air interface (midsection view).

Figure 3

(a) Interface profiles of the retracting sheet at different time instants. The profiles are plotted at time interval $\mathrm{\Delta }t/t_v=1$. (b) Temporal variation of the thickness of the sheet at $x=0$. The dimensionless parameters are $\mathrm{Oh}=40$ and $L_0/h_0=10$.

Figure 4

Comparison of the retraction speed of the liquid sheet obtained from numerical simulations (solid lines) with the theoretical prediction (dashed lines). (a) The numerical results are compared with the early stage retraction theory of Savva and Bush [14]. (b) Comparison of the numerical results with the present theory [Eq. (20)] for the later stage of retraction. The results are shown for $\mathrm{Oh}=40,L_0/h_0=10$ (red) and $\mathrm{Oh}=40,L_0/h_0=20$ (blue). A zoomed view at the later stage when the self-similar solution is no longer valid is shown in the inner image for $\mathrm{Oh}=40,L_0/h_0=10$.

Figure 5

(a) Interface profiles of the retracting sheet at different time instants. The profiles are shown at a time interval $\mathrm{\Delta }t/t_v=1$. (b) The zoomed view of the interface profiles is shown in (a). (c) Temporal variation of the thickness of the sheet at $x/h_0=10$. The dimensionless parameters are $\mathrm{Oh}=40$ and $L_0/h_0=10$.

Figure 6

Comparison of the retraction speed of the liquid sheet obtained from numerical simulations (solid lines) with the early stage retraction theory of Savva and Bush [14] (dashed line). The results are shown for $\mathrm{Oh}=40,L_0/h_0=10$ (red) and $\mathrm{Oh}=40,L_0/h_0=20$ (blue).

Figure 7

(a) Velocity at the junction between the sheet and the initially circular arc ($x/h_0=10$). (b) Comparison of the retraction speed of the liquid sheet obtained from numerical simulations (solid lines) with the theoretical prediction of Eq. (23) (dashed line). The results are shown for $\mathrm{Oh}=40,L_0/h_0=10$ (red) and $\mathrm{Oh}=40,L_0/h_0=20$ (blue).

Figure 8

Comparison of the retraction speed of the liquid sheet obtained from numerical simulations (solid lines) with the theoretical predictions (dashed lines) for the early stage (Savva and Bush [14]) and later stage [Eqs. (20) and (23)] of retraction. Here (a) and (b) show the comparison of the numerical results and theory for the soap film (symmetric retraction) in the early stage and later stage of retraction, respectively. Panels (c) and (d) show the comparison of the numeric and theory in the early stage and later stage of retraction for the case of bursting bubble at the interface of a pool. The results are shown for $\mathrm{Oh}=20,L_0/h_0=20$ (red) and $\mathrm{Oh}=40,L_0/h_0=40$ (blue).

Figure 9

Interface profiles at different time instants for the retraction of a symmetric film (a) and a bursting bubble at the liquid-air interface (b). The profiles are plotted at time interval $\mathrm{\Delta }t/t_v=1$ for $\mathrm{Oh}=20$ and $L_0/h_0=80$. A comparison of the retraction speed of the liquid sheet obtained from numerical simulations (solid lines) with the theoretical prediction (dashed lines) given by Eqs. (20) and (23) are shown in (c) and (d), respectively. The purple dashed line is the theoretical prediction of Savva and Bush [14] for the early stage of retraction.