The role of inertia in extensional fall of a viscous drop

In flows of very viscous fluids, it is often justifiable to neglect inertia and solve the resulting creeping-flow or Stokes equations. For drops hanging beneath a fixed wall and extending under gravity from an initial rest state, an inevitable consequence of neglect of inertia and surface tension is that the drop formally becomes infinite in length at a finite crisis time, at which time the acceleration of the drop, which has been assumed small relative to gravity g, formally also becomes infinite. This is a physical impossibility, and the acceleration must in fact approach the (finite) free-fall value g. However, we verify here, by a full Navier–Stokes computation and also with a slender-drop approximation, that the crisis time is a good estimate of the time at which the bulk of the drop goes into free fall. We also show that the drop shape at the crisis time is a good approximation to the final shape of the freely falling drop, prior to smoothing by surface tension. Additionally, we verify that the drop has an initial acceleration of g, which quickly decreases as viscous forces in the drop become dominant during the early stages of fall.Y. M. Stokes and E. O. Tuc

Numerical simulations of viscoelastic interfacial flows

While several experimental and numerical studies for Newtonian sprays have been conducted, the exploration of their non-Newtonian counterparts has received comparatively little attention. Achieving a fundamental understanding of the physical phenomena governing spray formation of this type of flow remains a challenge. The numerical simulations of the spray formation of a non-Newtonian fluid still offer substantial challenges, but it is reflective of industrial applications (i.e. spray-drying) and can lead to the optimisation of spray processes containing complex fluids. This thesis aims to provide the basis for the numerical examination of non-Newtonian atomisation and spray systems. We begin with axisymmetric simulations of an impulsively-started viscoelastic jet exiting a nozzle and entering a stagnant gas phase using the open-source code Basilisk. This code allows for efficient computations through an adaptively-refined volume-of-fluid technique that can accurately capture the interface. We use the FENE-P constitutive equation to describe the viscoelasticity of the fluid and employ the log-conformation transformation, which provides stable solutions for the conformation tensor. For the first time, the entire jetting and breakup process of a viscoelastic fluid is simulated, including the flow through the nozzle, which results in an inhomogeneous initial radial stress distribution that affects the subsequent breakup dynamics. The evolution of the velocity field and the elastic stresses in the nozzle are validated against analytical solutions, and the early-stage dynamics of the jet are compared favourably to the predictions of linear stability theory. We explore the effect of flow inside the nozzle on the thinning dynamics of the viscoelastic jet, which develops distinctive "beads-on-a-stringstructures", via analysis of the spatiotemporal evolution of the polymeric stresses. We also systematically investigate the dependence of the filament thinning and breakup characteristics on the axial momentum of the jet and the extensibility of the dissolved polymer chains. We also probe how the secondary droplet formation can be controlled by the finite extensibility of the polymeric chains, as well as the wavenumber of the forced oscillation of the injected liquid at the nozzle inlet. In addition, we study numerically the thinning and breakup in a Dripping-onto-Substrate (DoS) rheometry. The DoS is a conceptually-simple, but dynamically-complex, probe of the extensional rheology of low-viscosity non-Newtonian fluids. It exploits the capillary-driven thinning of a liquid bridge, produced by a single drop as it is dispensed from a syringe pump and spreads laterally onto a solid substrate. By following the filament thinning process, the extensional viscosity and relaxation time of the sample can be determined. Importantly, DoS rheometry allows experimentalists to measure the extensional properties of solutions with lower viscosity than is possible with commercially-available capillary break-up extensional rheometers. Understanding the fluid mechanics underlying the operation of DoS is essential for optimising and extending the performance of this protocol. To achieve this, we employ a computational rheology approach using adaptively-refined axisymmetric numerical simulations with the Basilisk code. The volume-of-fluid technique is used to resolve the moving interface, and the log-conformation transformation provides a stable and accurate solution of the viscoelastic constitutive equation that describes the rheology of the thinning liquid filament. Here, we focus on understanding the role of elasticity and finite chain extensibility in controlling the elasto-capillary (EC) regime, as well as the perturbative effects that gravity and the substrate wettability play in establishing the evolution of the self-similar thinning and pinch-off dynamics. To illustrate the interplay of these different forces, we construct a simple one-dimensional model that captures the initial rate of thinning when the interplay of inertia and capillarity dominates; the model also captures the structure of the transition region to the nonlinear EC regime where the rapidly growing elastic tensile stresses in the thread balance the capillary pressure as the filament thins towards breakup. Finally, we develop and test a rheological model for avoiding the numerical challenges associated with the commonly-used constitutive equations for viscoelastic extensional flows, which accounts for the changes in the fluid viscosity based on the principal invariants of the deviatoric stress tensor. We validate the predictions of the model against a free-filament thinning and a jetting flow configuration of a FENE-P fluid, highlighting its capability to account for a substantial increase in viscosity under elongation. The model, however, fails to exhibit all of the characteristic viscoelastic flow regimes observed in our FENE-P-based simulation results. This highlights the need for further model improvement incorporating the flow kinematics history, a distinctive characteristic of viscoelasticity, which will be the subject of future work.Open Acces

Capillary breakup of a liquid bridge : identifying regimes and transitions

Computations of the breakup of a liquid bridge are used to establish the limits of applicability of similarity solutions derived for different breakup regimes. These regimes are based on particular viscous-inertial balances, that is different limits of the Ohnesorge number Oh. To accurately establish the transitions between regimes, the minimum bridge radius is resolved through four orders of magnitude using a purpose-built multiscale finite element method. This allows us to construct a quantitative phase diagram for the breakup phenomenon which includes the appearance of a recently discovered low-Oh viscous regime. The method used to quantify the accuracy of the similarity solutions allows us to identify a number of previously unobserved features of the breakup, most notably an oscillatory convergence towards the viscous-inertial similarity solution. Finally, we discuss how the new findings open up a number of challenges for both theoretical and experimental analysis

A discrete geometric approach for simulating the dynamics of thin viscous threads

We present a numerical model for the dynamics of thin viscous threads based on a discrete, Lagrangian formulation of the smooth equations. The model makes use of a condensed set of coordinates, called the centerline/spin representation: the kinematical constraints linking the centerline's tangent to the orientation of the material frame is used to eliminate two out of three degrees of freedom associated with rotations. Based on a description of twist inspired from discrete differential geometry and from variational principles, we build a full-fledged discrete viscous thread model, which includes in particular a discrete representation of the internal viscous stress. Consistency of the discrete model with the classical, smooth equations is established formally in the limit of a vanishing discretization length. The discrete models lends itself naturally to numerical implementation. Our numerical method is validated against reference solutions for steady coiling. The method makes it possible to simulate the unsteady behavior of thin viscous jets in a robust and efficient way, including the combined effects of inertia, stretching, bending, twisting, large rotations and surface tension

Scaling laws and dynamics of bubble coalescence

The coalescence of bubbles and drops plays a central role in nature and industry. During coalescence, two bubbles or drops touch and merge into one as the neck connecting them grows from microscopic to macroscopic scales. The hydrodynamic singularity that arises when two bubbles or drops have just touched and the flows that ensue have been studied thoroughly when two drops coalesce in a dynamically passive outer fluid. In this paper, the coalescence of two identical and initially spherical bubbles, which are idealized as voids, that are surrounded by an incompressible Newtonian liquid is analyzed by numerical simulation. This problem has recently been studied (a) experimentally using high-speed imaging and (b) by asymptotic analysis in which the dynamics is analyzed by determining the growth of a hole in the thin liquid sheet separating the two bubbles. In the latter, advantage is taken of the fact that the flow in the thin sheet of non-constant thickness is governed by a set of one-dimensional, radial extensional flow equations. While these studies agree on the power law scaling of the variation of the minimum neck radius with time, they disagree with respect to the numerical value of the prefactors in the scaling laws. In order to reconcile these differences and also provide insights into the dynamics that are difficult to probe by either of the aforementioned approaches, simulations are used to access both earlier times than it has been possible in the experiments and also later times when asymptotic analysis is no longer applicable. Early times and extremely small length scales are attained in the new simulations through the use of a truncated domain approach. Furthermore, it is shown by direct numerical simulations in which the flow within the bubbles is also determined along with the flow exterior to them that idealizing the bubbles as passive voids has virtually no effect on the scaling laws relating minimum neck radius and time.This research was sponsored by the Petroleum Research Fund of the American Chemical Society, the Basic Energy Sciences Program of the United States Department of Energy, and the Engineering and Physical Sciences Research Council

Stratospheric constituent measurements using UV solar occultation technique

The photochemistry of the stratospheric ozone layer was studied as the result of predictions that trace amounts of pollutants can significantly affect the layer. One of the key species in the determination of the effects of these pollutants is the OH radical. A balloon flight was made to determine whether data on atmospheric OH could be obtained from lower resolution solar spectra obtained from high altitude during sunset

Very viscous flows driven by gravity with particular application to slumping of molten glass

This thesis examines the flow of very viscous Newtonian fluids driven by gravity. It is written with concern for specific applications in the optics industry, with emphasis on the slumping of molten glass into a mould, as in the manufacture of optical components, which are in turn used to manufacture ophthalmic lenses. This process is known as thermal replication. However, the work has more general applicability, and disc viscometry, used to determine the viscosity of very viscous fluids, is also considered. In addition, one chapter of the thesis is devoted to the flow of dripping honey, as another example of a very viscous flow to which the model can be applied. The Stokes creeping-flow equations are used to model the very viscous flows of interest. The main solution method is finite elements, and a purpose-written computer program has been developed to solve the creeping-flow equations by this method. The present program is restricted to solving for either two-dimensional or axisymmetric flows but is extendible to three dimensions. In addition, semi-analytic series and asymptotic methods are used for some small portions of the work. The optical applications of this work demand consideration of the topic of computing surface curvature, and therefore second derivatives, from inexact and discrete numerical and experimental data. For this purpose, fitting of B-splines by a least-squares method to coordinate data defining the surface has been used. Much of the work assumes isothermal conditions, but in the context of the accuracy required in optical component manufacture it is also possible that non-isothermal effects will be important. Consequently, this restriction is eventually relaxed and some consideration given to non-isothermal conditions. In order to validate the creeping-flow model and finite-element program, comparisons of numerical simulations with experimental results are performed. A preliminary assessment of the importance of non-isothermal conditions to the thermal-replication process is also made by comparing isothermal and non-isothermal simulations with experimental results. The isothermal model is found to best match the experimental data.Thesis (Ph.D.)--School of Applied Mathematics, 1998

Gravitational extension of a fluid cylinder with internal structure

First published online 3 February 2016Motivated by the fabrication of microstructured optical fibres, a model is presented for the extension under gravity of a slender fluid cylinder with internal structure. It is shown that the general problem decouples into a two-dimensional surface-tension-driven Stokes flow that governs the transverse shape and an axial problem that depends upon the transverse flow. The problem and its solution differ from those obtained for fibre drawing, because the problem is unsteady and the fibre tension depends on axial position. Solutions both with and without surface tension are developed and compared, which show that the relative importance of surface tension depends upon both the parameter values and the geometry under consideration. The model is compared with experimental data and is shown to be in good agreement. These results also show that surface-tension effects are essential to accurately describing the cross-sectional shape.Hayden Tronnolone, Yvonne M. Stokes, Herbert Tze Cheung Foo and Heike Ebendorff-Heideprie

Numerical simulations of thin viscoelastic films

This dissertation is developed in the field of Computational Fluid Dynamics (CFD) and it focuses on numerical simulations of the dynamics of thin viscoelastic films in different settings. The first part of this dissertation presents a novel computational investigation of thin viscoelastic films and drops, that are subject to the van der Waals interaction force, in two spatial dimensions. The liquid films are deposited on a flat solid substrate, that can have a zero or nonzero inclination with respect to the base. The equation that governs the interfacial dynamics of the thin films and drops is obtained within the long-wave approximation of the Navier-Stokes equations, with the Jeffreys model for viscoelastic stresses. The effects of viscoelasticity and the substrate slippage on the dynamics of thin viscoelastic films are investigated. Moreover, the effects of viscoelasticity on drops that spread or recede on a prewetted flat substrate are analyzed. For dewetting films, the numerical results show the presence of multiple secondary droplets for higher values of the relaxation time, consistently with experimental findings. These secondary length scales are found to be suppressed by gravitational effects when the case of dewetting films on inverted planes is analyzed. For spreading and receding drops on flat, prewetted substrates, viscoelastic effects are found to lead to deviations from the Cox-Voinov law for partially wetting fluids. In general, viscoelasticity enhances the spreading and suppresses the retraction of viscoelastic drops, compared to Newtonian ones. The second part of this dissertation presents a novel numerical investigation of the dynamics of free-boundary flows of viscoelastic liquid membranes, not necessarily deposited on solid substrates. The governing equation describes the balance of linear momentum, in which the stresses include the viscoelastic response to deformations of Maxwell type. A penalty method is utilized to enforce near incompressibility of the viscoelastic media, in which the penalty constant is proportional to the viscosity of the fluid. A finite element method is used, in which the slender geometry representing the liquid membrane is discretized by linear three-node triangular elements under plane stress conditions. Two applications of interest are considered for the numerical framework provided: shear flow, and extensional flow in drawing processes. Finally, the last part of this dissertation considers the expansion of the study of the dynamics of viscoelastic membranes by applying the general theory of shells, in which any application of loading or external forces causes both bending and stretching, so that buckling or wrinkling phenomena can be investigated as future work