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Newtonian pizza: spinning a viscous sheet

Published online by Cambridge University Press:  02 August 2010

PETER D. HOWELL ,
BENOIT SCHEID  and
HOWARD A. STONE
PETER D. HOWELL
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St Giles, Oxford OX1 3LB, UK
BENOIT SCHEID*
Affiliation:
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA TIPs – Fluid Physics unit, Université Libre de Bruxelles, C.P. 165/67, 1050 Brussels, Belgium
HOWARD A. STONE
Affiliation:
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email address for correspondence: bscheid@ulb.ac.be
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Abstract

We study the axisymmetric stretching of a thin sheet of viscous fluid driven by a centrifugal body force. Time-dependent simulations show that the sheet radius R(t) tends to infinity in finite time. As time t approaches the critical time t*, the sheet becomes partitioned into a very thin central region and a relatively thick rim. A net momentum and mass balance in the rim leads to a prediction for the sheet radius near the singularity that agrees with the numerical simulations. By asymptotically matching the dynamics of the sheet with the rim, we find that the thickness h in the central region is described by a similarity solution of the second kind, with h ∝ (t* − t)α where the exponent α satisfies a nonlinear eigenvalue problem. Finally, for non-zero surface tension, we find that the exponent increases rapidly to infinity at a critical value of the rotational Bond number B = 1/4. For B > 1/4, surface tension defeats the centrifugal force, causing the sheet to retract rather than to stretch, with the limiting behaviour described by a similarity solution of the first kind.

JFM classification

Low-Reynolds-number flows: Lubrication theory Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 659 , 25 September 2010 , pp. 1 - 23
Copyright
Copyright © Cambridge University Press 2010

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