The Mathematical Institute, University of Oxford, Eprints Archive

Newtonian pizza: spinning a viscous sheet

Howell, P. D. and Scheid, B. and Stone, H. A. (2010) Newtonian pizza: spinning a viscous sheet. Journal of Fluid Mechanics, 659 . pp. 1-23. ISSN 0022-1120

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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.

Item Type:Article
Subjects:D - G > Fluid mechanics
Research Groups:Oxford Centre for Industrial and Applied Mathematics
ID Code:1557
Deposited By:Peter Howell
Deposited On:11 Jul 2012 07:22
Last Modified:11 Jul 2012 07:22

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