arXiv subjects

Jack S. Keeler

Publications and source records attributed to Jack S. Keeler.

5 recordsLinked to original sources

arXiv2024

On the stability of fully nonlinear hydraulic-fall solutions to the forced water-wave problem

Two-dimensional free-surface flow over localised topography is examined with the emphasis on the stability of hydraulic-fall solutions. A Gaussian topography profile is assumed with a positive or negative amplitude modelling a bump or a dip, respectively. Steady hydraulic-fall solutions to the full incompressible, irrotational Euler equations are computed, and their linear and nonlinear stability is analysed by computing eigenspectra of the pertinent linearised operator and by solving an initial value problem. The computations are carried out numerically using a specially developed computational framework based on the finite element method. The Hamiltonian structure of the problem is demonstrated and stability is determined by computing eigenspectra of the pertinent linearised operator. It is found that a hydraulic-fall flow over a bump is spectrally stable. The corresponding flow over a dip is found to be linearly unstable. In the latter case, time-dependent simulations show that the flow ultimately settles into a time-periodic motion that corresponds to an invariant solution in an appropriately defined phase space. Physically, the solution consists of a localised large amplitude wave that pulsates above the dip while simultaneously emitting nonlinear cnoidal waves in the upstream direction and multi-harmonic linear waves in the downstream direction.

physics.flu-dyn↗

arXiv2024

Stable bubble formations in a Hele-Shaw channel

Deformable bubbles propagated by the flow of a viscous liquid in a planar Hele-Shaw channel of uniform depth tend to travel steadily along the channel's streamwise axis and pairs of neighbouring bubbles will either separate or coalesce because an individual bubble's propagation speed increases monotonically with its size. Thus, any group of bubbles will eventually rearrange itself in order of decreasing size and all of the bubbles will separate. We show that, by introducing a small geometric perturbation to the channel in the form of an axially-uniform depth reduction along its centreline, the system supports a multitude of stable bubble formations and this can disrupt the usual reordering by bubble size. The constituent bubbles of a stable formation lie in alternation on opposite sides of the depth-perturbation, retain fixed shapes and propagate steadily at the same speed. A stable formation is always led by the smallest of its constituent bubbles, which would be the slowest bubble in isolation. The leading bubble propagates as if it were isolated and the trailing bubbles reduce their speeds by adjusting their shapes and overlaps of the depth-perturbation in the perturbation fields of their preceding nearest-neighbours in order to match that of the leading bubble. The trailing bubbles can be arranged in any order and, hence, the number of stable formations increases factorially as the number of bubbles is increased.

physics.flu-dyn↗

arXiv2022

Putting the micro into the macro: A molecularly-augmented hydrodynamic model of dynamic wetting applied to flow instabilities during forced dewetting

We report a molecularly-augmented continuum-based computational model of dynamic wetting and apply it to the displacement of an externally-driven liquid plug between two partially-wetted parallel plates. The results closely follow those obtained in a recent molecular-dynamics (MD) study of the same problem Toledano (2021) which we use as a benchmark. We are able to interpret the maximum speed of dewetting $U^*_{\mathrm{crit}}$ as a fold bifurcation in the steady phase diagram and show that its dependence on the true contact angle $θ_{\mathrm{cl}}$ is quantitatively similar to that found using MD. A key feature of the model is that the contact angle is dependent on the speed of the contact line, with $θ_{\mathrm{cl}}$ emerging as part of the solution. The model enables us to study the formation of a thin film at dewetting speeds $U^*>U^*_{\mathrm{crit}}$ across a range of length scales, including those that are computationally prohibitive to MD simulations. We show that the thickness of the film scales linearly with the channel width and is only weakly dependent on the capillary number. This work provides a link between matched asymptotic techniques (valid for larger geometries) and MD simulations (valid for smaller geometries). In addition, we find that the apparent angle, the experimentally visible contact angle at the fold bifurcation, is not zero. This is in contrast to the prediction of conventional treatments based on the lubrication model of flow near the contact line, but consistent with experiment.

physics.flu-dyn↗

arXiv2020

The life and fate of a bubble in a geometrically-perturbed Hele-Shaw channel

Motivated by the desire to understand complex transient behaviour in fluid flows, we study the dynamics of an air bubble driven by the steady motion of a suspending viscous fluid within a Hele-Shaw channel with a centred depth perturbation. Using both experiments and numerical simulations of a depth-averaged model, we investigate the evolution of an initially centred bubble of prescribed volume as a function of flow rate and initial shape. The experiments exhibit a rich variety of organised transient dynamics, involving bubble break up as well as aggregation and coalescence of interacting neighbouring bubbles. The long-term outcome is either a single bubble or multiple separating bubbles, positioned along the channel in order of increasing velocity. Up to moderate flow rates, the life and fate of the bubble are reproducible and can be categorised by a small number of characteristic behaviours that occur in simply-connected regions of the parameter plane. Increasing the flow rate leads to less reproducible time evolutions with increasing sensitivity to initial conditions and perturbations in the channel. Time-dependent numerical simulations that allow for break up and coalescence are found to reproduce most of the dynamical behaviour observed experimentally including enhanced sensitivity at high flow rate. An unusual feature of this system is that the set of steady and periodic solutions can change during temporal evolution because both the number of bubbles and their size distribution evolve due to break up and coalescence events. Calculation of stable and unstable solutions in the single- and two-bubble cases reveals that the transient dynamics are orchestrated by weakly-unstable solutions of the system termed edge states that can appear and disappear as the number of bubbles changes.

physics.flu-dyn↗

arXiv2019

Termination points and homoclinic glueing for a class of inhomogeneous nonlinear ordinary differential equations

Solutions $u(x)$ to the class of inhomogeneous nonlinear ordinary differential equations taking the form \[u'' + u^2 = αf(x) \] for parameter $α$ are studied. The problem is defined on the $x$ line with decay of both the solution $u(x)$ and the imposed forcing $f(x)$ as $|x| \to \infty $. The rate of decay of $f(x)$ is important and has a strong influence on the structure of the solution space. Three particular forcings are examined primarily: a rectilinear top-hat, a Gaussian, and a Lorentzian, the latter two exhibiting exponential and algebraic decay, respectively, for large $x$. The problem for the top hat can be solved exactly, but for the Gaussian and the Lorentzian it must be computed numerically in general. Calculations suggest that an infinite number of solution branches exist in each case. For the top-hat and the Gaussian the solution branches terminate at a discrete set of $α$ values starting from zero. A general asymptotic description of the solutions near to a termination point is constructed that also provides information on the existence of local fold behaviour. The solution branches for the Lorentzian forcing do not terminate in general. For large $α$ the asymptotic analysis of Keeler, Binder $\&$ Blyth (2018 "On the critical free-surface flow over localised topography", J. Fluid Mech., 832, 73-96) is extended to describe the behaviour on any given solution branch using a method for glueing homoclinic connections.

math.CA↗