Singularities in droplet pinching with vanishing viscosity

A slender-jet model for the pinching of a liquid column is considered in the limit of vanishing viscosity. We find the model to develop a singularity in the gradients of the local radius and the velocity at a finite thread radius, so it does not describe breakup. However, the observed steepening of the profile corresponds to experiments and simulations with fluids at low viscosity. The singularity has similarity form, which we compute analytically. The result agrees well with numerical simulations of the model equations.Comment: 18 pages including 4 eps figures, revte

Controlling a leaky tap

We apply the Ott, Grebogy and Yorke mechanism for the control of chaos to the analytical oscillator model of a leaky tap obtaining good results. We exhibit the robustness of the control against both dynamical noise and measurement noise.A possible way of controlling experimentally a leaky tap using magnetic-field-produced variations in the viscosity of a magnetorheological fluid is suggested.Comment: 14 pages, 12 figures. Submitted to Physics Letters

A Compact Eulerian Representation of Axisymmetric Inviscid Vortex Sheet Dynamics

A classical problem in fluid mechanics is the motion of an axisymmetric vortex sheet evolving under the action of surface tension, surrounded by an inviscid fluid. Lagrangian descriptions of these dynamics are well-known, involving complex nonlocal expressions for the radial and longitudinal velocities in terms of elliptic integrals. Here we use these prior results to arrive at a remarkably compact and exact Eulerian evolution equation for the sheet radius $r(z,t)$ in an explicit flux form associated with the conservation of enclosed volume. The flux appears as an integral involving the pairwise mutual induction formula for vortex loop pairs first derived by Helmholtz and Maxwell. We show how the well-known linear stability results for cylindrical vortex sheets in the presence of surface tension and streaming flows [A.M. Sterling and C.A. Sleicher, $J.~Fluid~Mech.$ ${\bf 68}$, 477 (1975)] can be obtained directly from this formulation. Furthermore, the inviscid limit of the empirical model of Eggers and Dupont [$J.~Fluid~Mech.$ $\textbf{262}$ 205 (1994); $SIAM~J.~Appl.~Math.$ ${\bf 60}$, 1997 (2000)], which has served as the basis for understanding singularity formation in droplet pinchoff, is derived within the present formalism as the leading order term in an asymptotic analysis for long slender axisymmetric vortex sheets, and should provide the starting point for a rigorous analysis of singularity formation.This work was supported in part by Established Career Fellowship EP/M017982/1 from the EPSRC (REG & AIP). REG and AIP are grateful to the I.H.E.S., and especially Patrick Gourdon, for hospitality during an extended visit supported by the Schlumberger Visiting Professorship (REG)

A Hybrid Variational-Level Set Approach to Handle Topological Changes

We present a method for allowing explicit, Lagrangian meshes to undergo topological changes in an automatic way. We employ a method for detecting when topological changes are imminent. When a change occurs, we use a level set method to guide the change of topology of the domain mesh. This is then followed by an optimization step, using a variational formulation, that seeks to improve boundary mesh conformity to the zero level contour of the level set function. The advantage of this method is that it directly allows for using a variational formulation of the physics being modeled and simulated, including the ability to account for important geometric information in the model (such as for surface tension driven flow). Furthermore, the level set update and optimization step are only needed during a topological change. Hence, our method does not significantly affect computational cost

Reduced-order modelling of fluid flows using analytical and numerical methods

Coating the exterior of a cylinder with a layer of fluid is a fundamental problem in fluid mechanics and occurs in numerous natural processes and industrial applications, such as heat and mass transfer and the production of orthopaedic implants. This thesis formulates and analyses novel models for two different coating flow problems which are not restricted by the common assumptions that the cylinder has circular cross-section and/or that the film is thin. The first problem involves the unsteady, two-dimensional flow on the exterior of a uniformly rotating horizontal cylinder with elliptical cross-section. By using a long-wave approximation we derive a thick-film model, and by using a thin-film approximation we derive a thin-film model. Both models incorporate the effects of cylinder eccentricity, rotation, gravity, centrifugation, viscosity, and surface tension. By studying the thin-film model, we demonstrate both analytically and numerically that the behaviour of the film coating the elliptical cylinder significantly differs from that in the circular case. In particular, it is shown that even a relatively mild departure from circularity produces significant qualitative and quantitative differences from the behaviour in the circular case. The second problem involves the unsteady, three-dimensional flow of a thick film on the exterior of a vertical fibre with circular cross-section. By using a longwave approximation and the method of weighted residuals, we derive a thick-film weighted-residual model, which incorporates the effects of gravity, viscosity, surface tension, and inertia. We study the thick-film weighted-residual model in the linear regime in order to elucidate the mechanics that determine both the stability and the axisymmetry of the flow. We demonstrate that these results in the linear regime, in general, correctly predict the results of the linear calculations of the Navier–Stokes equations and the results of numerical simulations of the thick-film weighted-residual model in the nonlinear regime.Coating the exterior of a cylinder with a layer of fluid is a fundamental problem in fluid mechanics and occurs in numerous natural processes and industrial applications, such as heat and mass transfer and the production of orthopaedic implants. This thesis formulates and analyses novel models for two different coating flow problems which are not restricted by the common assumptions that the cylinder has circular cross-section and/or that the film is thin. The first problem involves the unsteady, two-dimensional flow on the exterior of a uniformly rotating horizontal cylinder with elliptical cross-section. By using a long-wave approximation we derive a thick-film model, and by using a thin-film approximation we derive a thin-film model. Both models incorporate the effects of cylinder eccentricity, rotation, gravity, centrifugation, viscosity, and surface tension. By studying the thin-film model, we demonstrate both analytically and numerically that the behaviour of the film coating the elliptical cylinder significantly differs from that in the circular case. In particular, it is shown that even a relatively mild departure from circularity produces significant qualitative and quantitative differences from the behaviour in the circular case. The second problem involves the unsteady, three-dimensional flow of a thick film on the exterior of a vertical fibre with circular cross-section. By using a longwave approximation and the method of weighted residuals, we derive a thick-film weighted-residual model, which incorporates the effects of gravity, viscosity, surface tension, and inertia. We study the thick-film weighted-residual model in the linear regime in order to elucidate the mechanics that determine both the stability and the axisymmetry of the flow. We demonstrate that these results in the linear regime, in general, correctly predict the results of the linear calculations of the Navier–Stokes equations and the results of numerical simulations of the thick-film weighted-residual model in the nonlinear regime