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Newtonian pizza: spinning a viscous sheet

  • PETER D. HOWELL (a1), BENOIT SCHEID (a2) (a3) and HOWARD A. STONE (a2) (a4)

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.

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Corresponding author

Email address for correspondence: bscheid@ulb.ac.be

References

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Newtonian pizza: spinning a viscous sheet

  • PETER D. HOWELL (a1), BENOIT SCHEID (a2) (a3) and HOWARD A. STONE (a2) (a4)

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