Newtonian pizza: spinning a viscous sheet

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.

Evolution and Breakup of Viscous Rotating Drops

“…Once an equilibrium shape becomes unstable it undergoes evolution towards the formation of a singularity as discussed in [15]. The singularity is such that the drop evolves into a toroidal rim of fluid with a thin film inside (see Figure 5).…”

“…The singularity is such that the drop evolves into a toroidal rim of fluid with a thin film inside (see Figure 5). According to [15], the radius of the toroidal rim grows as r max = O((t 0 − t) …”

“…In fact, the fact that r max diverges follows from the following argument in [15]. Assume that the shape consists of a disc of radius β(t) and r-dependent thickness h(r, t), surrounded by a toroidal rim of radius of the tube R(t); Then, the evolution can be described by the following thin film equations (see [15]):…”

“…If the Reynolds number is small, then inertial terms can be neglected in comparison with both viscous forces and centrifugal forces and one arrives then to Stokes system subject to centrifugal forces. This is the approach adopted by Howell et al [15] to study the evolution and possible singularity formation of an axisymmetric drop rotating with constant angular velocity. Under a thin film approximation, Howell et al show that for sufficiently large angular velocities, drops may become very flat and thin at the center with a torus-like boundary (a pizza shape, in the notation of [15]).…”

“…This is the approach adopted by Howell et al [15] to study the evolution and possible singularity formation of an axisymmetric drop rotating with constant angular velocity. Under a thin film approximation, Howell et al show that for sufficiently large angular velocities, drops may become very flat and thin at the center with a torus-like boundary (a pizza shape, in the notation of [15]). Moreover, they find that the thickness of the center can become zero in finite time so that a hole develops.…”