Dr Ravi Pethiyagoda

Hypothesis: Evaporation of surfactant droplets on leaves is complicated due to the complex physical and chemical properties of the leaf surfaces. However, for certain leaf surfaces for which the evaporation process appears to follow the standard constant-contact-radius or constant-contact-angle modes, it should be possible to mimic the droplet evaporation with both a well-chosen synthetic surface and a relatively simple mathematical model. Experiments: Surfactant droplet evaporation experiments were performed on two commercial crop species, wheat and capsicum, along with two synthetic surfaces, up to a 90° incline. The time-dependence of the droplets' contact angles, height, volume and contact radius was measured throughout the evaporation experiments. Mathematical models were developed to simulate the experiments. Findings: With one clear exception, for all combinations of surfaces, surfactant concentrations and angles, the experiments appear to follow the standard evaporation modes and are well described by the mathematical models (modified Popov and Young¿Laplace-Popov). The exception is wheat with a high surfactant concentration, for which droplet evaporation appears nonstandard and deviates from the diffusion limited models, perhaps due to additional mechanisms such as the adsorption of surfactant, stomatal density or an elongated shape in the direction of the grooves in the wheat surface.

The surface gravity wave pattern that forms behind a steadily moving disturbance is well known to comprise divergent waves and transverse waves, contained within a distinctive-shaped wake. In this paper, we are concerned with a theoretical study of the limit of a slow-moving disturbance (small Froude numbers) in the absence of surface tension, for which the wake is dominated by transverse waves. Three configurations are considered: flow past a submerged source singularity, a submerged doublet and a pressure distribution applied to the surface. We treat the linearised version of these problems and use the method of stationary phase and exponential asymptotics to demonstrate that the apparent wake angle is less than the classical Kelvin angle and to quantify the decrease in apparent wake angle as the Froude number decreases. These results complement a number of recent studies for sufficiently fast-moving disturbances (large Froude numbers) where the apparent wake angle has been also shown to be less than the classical Kelvin angle. As well as shedding light on the issue of apparent wake angle, we also study the fully nonlinear problems for our three configurations under various limits to demonstrate the unique and interesting features of Kelvin wake patterns at small Froude numbers.

Thin surfaces, such as the leaves of a plant, pose a significant challenge for implicit surface reconstruction techniques, which typically assume a closed, orientable surface. We show that by approximately interpolating a point cloud of the surface (augmented with off-surface points) and restricting the evaluation of the interpolant to a tight domain around the point cloud, we need only require an orientable surface for the reconstruction. We use polyharmonic smoothing splines to fit approximate interpolants to noisy data, and a partition of unity method with an octree-like strategy for choosing subdomains. This method enables us to interpolate an N-point dataset in O(N) operations. We present results for point clouds of capsicum and tomato plants, scanned with a handheld device. An important outcome of the work is that sufficiently smooth leaf surfaces are generated that are amenable for droplet spreading simulations.

Spectrograms provide an efficient way to analyze surface elevation signals of ship waves taken from a sensor fixed at a single point in space. Recent work based on a simplified model for the ship's disturbance suggests that matching the spectrogram heat-map patterns to a so-called dispersion curve has the potential for estimating of properties of a steadily moving ship, such as the ship's speed and closest distance to the sensor. Here we extend the theory behind the dispersion curve so that it can be applied to ships accelerating along arbitrary paths and demonstrate how acceleration affects the structure of the associated spectrograms. Examples are provided for a simple model of a ship accelerating/decelerating in a straight line or traveling in a circle with constant angular speed. We highlight a problem with nonuniqueness of the dispersion curve when comparing ships moving along different paths. Finally, we validate the new dispersion curve against experimental results of ship models accelerating in a finite depth basin. Our work will provide a basis for more comprehensive studies that extend the simplified model to take into account the shape of the hull in question.

The Stokes phenomenon is a class of asymptotic behaviour that was first discovered by Stokes in his study of the Airy function. It has since been shown that the Stokes phenomenon plays a significant role in the behaviour of surface waves on flows past submerged obstacles. A detailed review of recent research in this area is presented, which outlines the role that the Stokes phenomenon plays in a wide range of free surface flow geometries. The problem of inviscid, irrotational, incompressible flow past a submerged step under a thin elastic sheet is then considered. It is shown that the method for computing this wave behaviour is extremely similar to previous work on computing the behaviour of capillary waves. Exponential asymptotics are used to show that free-surface waves appear on the surface of the flow, caused by singular fluid behaviour in the neighbourhood of the base and top of the step. The amplitude of these waves is computed and compared to numerical simulations, showing excellent agreements between the asymptotic theory and computational solutions. This article is part of the theme issue 'Stokes at 200 (part 2)'.

BACKGROUND: A suite of plant retention spray models has been developed to simulate spray retention using virtual surfaces (either single leaves or whole plants) and their outputs compared with experimental data for the equivalent spray scenarios. RESULTS: The results for a single formulation (0.1% v/v lecithin mixture in water) and difficult to wet plant species Chenopodium album L (common lambsquarters) are presented. They include experimental observations with single leaves, as well as simulations of virtual impaction events, conducted to provide for the first time estimates of f (the proportion of theoretical impact drop diameter at shatter). With this factor prescribed, multi-plant simulations using a range of nozzle types and droplet sizes (volume mean diameter (VMD) range 241 to 530 µm) are compared with equivalent experimentally determined spray retention by real plants. The simulations demonstrated that impaction resulted predominantly in shatter with the production of daughter droplets, and that retention is mainly due to re-capture of these droplets. Overall the simulations show the same trends as experimental retention results from different nozzle applications, but at best predicted retention results were 68% to 79% of experimental percentage retention, depending on plant spacing. CONCLUSIONS: Retention is the result of some primary drop capture but predominantly by recapture of shatter droplets as the modelling illustrates. The value of f affects the droplet shatter outcome and can result in fewer, more energetic daughter droplets, or more droplets but with lower energies. However, this effect alone cannot explain the discrepancy between actual and simulated results. Possible operational influences are discussed. © 2020 Society of Chemical Industry.

BACKGROUND: A key challenge for developing computer models of spray retention by plants is to accurately predict how spray drops behave when impacting leaf surfaces. One poorly understood outcome occurs when drops bounce or shatter on impact but leave behind a proportion of the liquid on the surface (designated as pinning). This process is studied via impaction experiments with two hard-to-wet leaf surfaces (fat-hen: Chenopodium album and barnyard grass: Echinochloa crus-galli L. P. Beauv) and one hydrophobic artificial surface (Teflon) using three liquid formulations. RESULTS: Drops that impact upon Teflon underwent pinning shatter events via a well-known mechanism referred to as receding breakup. Drops impacting on leaf surfaces did not undergo receding breakup because the liquid rim was not in direct contact with the leaf surface when it broke into secondary droplets. However, pinning did occur on plant surfaces via a different mechanism, especially when using formulations containing a surfactant. CONCLUSION: Newly developed image analysis and methodology has allowed quantification of the volume fraction pinned to surfaces when drops shatter. The addition of surfactant can increase both the probability of pinning and the pinned volume when drops shatter on fat-hen or Teflon. However, the surfactants studied did not substantially improve the probability of pinning on barnyard grass. The difference in behaviour between the two leaf surfaces and the underlying mechanism is worth further study. © 2020 Society of Chemical Industry.

Steady and unsteady linearised flow past a submerged source are studied in the small-surface-Tension limit, in the absence of gravitational effects. The free-surface capillary waves generated are exponentially small in the surface tension, and are determined using the theory of exponential asymptotics. In the steady problem, capillary waves are found to extend upstream from the source, switching on across curves on the free surface known as Stokes lines. Asymptotic predictions are compared with computational solutions for the position of the free surface. In the unsteady problem, transient effects cause the solution to display more complicated asymptotic behaviour, such as higher-order Stokes lines. The theory of exponential asymptotics is applied to show how the capillary waves evolve over time, and eventually tend to the steady solution.

We consider steady nonlinear free surface flow past an arbitrary bottom topography in three dimensions, concentrating on the shape of the wave pattern that forms on the surface of the fluid. Assuming ideal fluid flow, the problem is formulated using a boundary integral method and discretised to produce a nonlinear system of algebraic equations. The Jacobian of this system is dense due to integrals being evaluated over the entire free surface. To overcome the computational difficulty and large memory requirements, a Jacobian-free Newton-Krylov (JFNK) method is utilised. Using a block-banded approximation of the Jacobian from the linearised system as a preconditioner for the JFNK scheme, we find significant reductions in computational time and memory required for generating numerical solutions. These improvements also allow for a larger number of mesh points over the free surface and the bottom topography. We present a range of numerical solutions for both subcritical and supercritical regimes, and for a variety of bottom configurations. We discuss nonlinear features of the wave patterns as well as their relationship to ship wakes.

A spectrogram of a ship wake is a heat map that visualises the time-dependent frequency spectrum of surface height measurements taken at a single point as the ship travels by. Spectrograms are easy to compute and, if properly interpreted, have the potential to provide crucial information about various properties of the ship in question. Here we use geometrical arguments and analysis of an idealised mathematical model to identify features of spectrograms, concentrating on the effects of a finite-depth channel. Our results depend heavily on whether the flow regime is subcritical or supercritical. To support our theoretical predictions, we compare with data taken from experiments we conducted in a model test basin using a variety of realistic ship hulls. Finally, we note that vessels with a high aspect ratio appear to produce spectrogram data that contains periodic patterns. We can reproduce this behaviour in our mathematical model by using a so-called two-point wavemaker. These results highlight the role of wave interference effects in spectrograms of ship wakes.

We consider a family of steady free-surface flow problems in two dimensions, concentrating on the effect of nonlinearity on the train of gravity waves that appear downstream of a disturbance. By exploiting standard complex variable techniques, these problems are formulated in terms of a coupled system of Bernoulli equation and an integral equation. When applying a numerical collocation scheme, the Jacobian for the system is dense, as the integral equation forces each of the algebraic equations to depend on each of the unknowns. We present here a strategy for overcoming this challenge, which leads to a numerical scheme that is much more efficient than what is normally used for these types of problems, allowing for many more grid points over the free surface. In particular, we provide a simple recipe for constructing a sparse approximation to the Jacobian that is used as a preconditioner in a Jacobian-free Newton-Krylov method for solving the nonlinear system. We use this approach to compute numerical results for a variety of prototype problems including flows past pressure distributions, a surface-piercing object and bottom topographies.

A spectrogram is a useful way of using short-time discrete Fourier transforms to visualise surface height measurements taken of ship wakes in real-world conditions. For a steadily moving ship that leaves behind small-amplitude waves, the spectrogram is known to have two clear linear components, a sliding-frequency mode caused by the divergent waves and a constant-frequency mode for the transverse waves. However, recent observations of high-speed ferry data have identified additional components of the spectrograms that are not yet explained. We use computer simulations of linear and nonlinear ship wave patterns and apply time-frequency analysis to generate spectrograms for an idealised ship. We clarify the role of the linear dispersion relation and ship speed on the two linear components. We use a simple weakly nonlinear theory to identify higher-order effects in a spectrogram and, while the high-speed ferry data are very noisy, we propose that certain additional features in the experimental data are caused by nonlinearity. Finally, we provide a possible explanation for a further discrepancy between the high-speed ferry spectrograms and linear theory by accounting for ship acceleration.