Revisiting the Taylor-Culick approximation. Part 2 : retraction of a viscous sheet
Résumé
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.
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