# Fluid Mechanics

### Brachistochronous motion of a flat plate parallel to its surface immersed in a fluid

Fluid mechanics Optimization Boundary LayersWe determine the globally minimum time đ needed to translate a thin submerged flat plate a given distance parallel to its surface within a work budget. The Reynolds number for the flow is assumed to be large so that the drag on the plate arises from skin friction in a thin viscous boundary layer. The minimum is determined computationally using a steepest descent, where an adjoint formulation is used to compute the gradients.

### Work-minimizing kinematics for small displacement of an infinitely long cylinder

Fluid mechanics Optimization Boundary LayersWe consider the time-dependent speed of an infinitely long cylinder that minimizes the net work done on the surrounding fluid to travel a given distance perpendicular to its axis in a fixed amount of time. The flow that develops is two-dimensional. An analytical solution is possible using calculus of variations for the case that the distance travelled and the viscous boundary layer thickness that develops are much smaller than the circle radius.

### Dynamics of a camphoric acid boat at the airâwater interface

Fluid Mechanics SurfactantWe report experiments on an agarose gel tablet loaded with camphoric acid (c-boat) spontaneously set into motion by surface tension gradients on the water surface. We observe three distinct modes of c-boat motion: harmonic mode where the c-boat speed oscillates sinusoidally in time, a steady mode where the c-boat maintains constant speed, and an intermittent mode where the c-boat maintains near-zero speed between sudden jumps in speed. Whereas all three modes have been separately reported before in different systems, controlled release of Camphoric Acid (CA) from the agarose gel matrix allowed the observation of all the three modes in the same system.

### Surface tension of flowing soap films

Fluid Mechanics SurfactantsThe surface tension of flowing soap films is measured with respect to the film thickness and the concentration of soap solution. We perform this measurement by measuring the curvature of the nylon wires that bound the soap film channel and use the measured curvature to parametrize the relation between the surface tension and the tension of the wire. We find that the surface tension of our soap films increases when the film is relatively thin or is made of soap solution of low concentration; otherwise, it approaches an asymptotic value of 30 mN/m.

### Hydrodynamic signatures of stationary Marangoni-driven surfactant transport

Fluid Mechanics SurfactantsWe experimentally study steady Marangoni-driven surfactant transport on the interface of a deep water layer. Using hydrodynamic measurements, and without using any knowledge of the surfactant physicochemical properties, we show that sodium dodecyl sulphate and Tergitol 15-S-9 introduced in low concentrations result in a flow driven by adsorbed surfactant. At higher surfactant concentration, the flow is dominated by the dissolved surfactant. Using camphoric acid, whose properties are a priori unknown, we demonstrate this methodâs efficacy by showing its spreading is adsorption dominated.

### Axisymmetric spreading of surfactant from a point source

Fluid Mechanics SurfactantsGuided by computation, we theoretically calculate the steady flow driven by the Marangoni stress due to a surfactant introduced on a fluid interface at a constant rate. Two separate extreme cases, where the surfactant dynamics is dominated by the adsorbed phase or the dissolved phase, are considered. We focus on the case where the size of the surfactant source is much smaller than the size of the fluid domain, and the resulting Marangoni stress overwhelms the viscous forces so that the flow is strongest in a boundary layer close to the interface.

### Marangoni elasticity of flowing soap films

Fluid Mechanics SurfactantsWe measure the Marangoni elasticity of a flowing soap film to be 22 mN/m irrespective of its width, thickness, flow speed, or the bulk soap concentration. We perform this measurement by generating an oblique shock in the soap film and measuring the shock angle, flow speed, and thickness. We postulate that the elasticity is constant because the film surface is crowded with soap molecules. Our method allows nondestructive measurement of flowing soap film elasticity and the value 22 mN/m is likely applicable to other similarly constructed flowing soap films.

### Energy harvesting performance and flow structure of an oscillating hydrofoil with finite span

Fluid Mechanics Renewable energy Oscillating foils Hydrokinetic energyThe energy harvesting performance and resulting flow structures of a hydrofoil oscillating in pitch and heave are studied experimentally in a water flume. The shape of a hydrofoil cross-section is shown to have negligible influence on the power generation for the geometries tested. It is found that contribution to efficiency from heaving motion increases with reduced frequency at optimal pitching amplitude. However, contribution to efficiency from pitching motion decreases with reduced frequency because the development of a leading-edge vortex during the stroke is delayed at the high reduced frequency.

### Linear stability analysis for monami in a submerged seagrass bed

Fluid Mechanics Biomechanics EnvironmentThe onset of monami â the synchronous waving of seagrass beds driven by a steady flow â is modelled as a linear instability of the flow. Unlike previous works, our model considers the drag exerted by the grass in establishing the steady flow profile, and in damping out perturbations to it. We find two distinct modes of instability, which we label modes 1 and 2. Mode 1 is closely related to KelvinâHelmholtz instability modified by vegetation drag, whereas mode 2 is unrelated to KelvinâHelmholtz instability and arises from an interaction between the flow in the vegetated and unvegetated layers.

### Dynamics of evaporative colloidal patterning

Fluid Mechanics Materials ScienceDrying suspensions often leave behind complex patterns of particulates, as might be seen in the coffee stains on a table. Here, we consider the dynamics of periodic band or uniform solid film formation on a vertical plate suspended partially in a drying colloidal solution. Direct observations allow us to visualize the dynamics of band and film deposition, where both are made of multiple layers of close packed particles. We further see that there is a transition between banding and filming when the colloidal concentration is varied.

### Capillary interactions between nearby interfacial objects

Fluid Mechanics Capillary Cheerios EffectWe develop a general method to study the capillary interactions between objects of arbitrary shape which float close to each other on an interface, a regime in which multipole expansion is not useful. The force is represented as a power series in the small distance between the objects, of which the leading order is finite. For objects with size a much larger than the capillary length lc, the force scales as (a/lc)1/2 and the prefactor depends on the mean radius of curvature R at the closest points.

### Skating on a film of air: Drops impacting a surface

Fluid Mechanics Droplet impactThe commonly accepted description of drops impacting on a surface typically ignores the essential role of the air that is trapped between the impacting drop and the surface. Here we describe a new imaging modality that is sensitive to the behavior right at the surface. We show that a very thin film of air, only a few tens of nanometers thick, remains trapped between the falling drop and the surface as the drop spreads.

### The mechanism of a splash on a dry solid surface

Fluid Mechanics Droplet impactFrom rain storms to ink jet printing, it is ubiquitous that a high-speed liquid droplet creates a splash when it impacts on a dry solid surface. Yet, the fluid mechanical mechanism causing this splash is unknown. About fifty years ago it was discovered that corona splashes are preceded by the ejection of a thin fluid sheet very near the vicinity of the contact point. Here we present a first-principles description of the mechanism for sheet formation, the initial stages of which occur before the droplet physically contacts the surface.

### Algorithm for microfluidic assembly line

Fluid Mechanics Materials Science MicrofluidicsMicrofluidic technology has revolutionized the control of flows at small scales giving rise to new possibilities for assembling complex structures on the microscale. We analyze different possible algorithms for assembling arbitrary structures, and demonstrate that a sequential assembly algorithm can manufacture arbitrary 3D structures from identical constituents. We illustrate the algorithm by showing that a modified Hele-Shaw cell with 7 controlled flow rates can be designed to construct the entire English alphabet from particles that irreversibly stick to each other.

### Events before droplet splashing on a solid surface

Fluid Mechanics Droplet impactA high-velocity (â1 m/s) impact between a liquid droplet (â1 mm) and a solid surface produces a splash. Classical observations traced the origin of this splash to a thin sheet of fluid ejected near the impact point, though the fluid mechanical mechanism leading to the sheet is not known. Mechanisms of sheet formation have heretofore relied on initial contact of the droplet and the surface. In this paper, we theoretically and numerically study the events within the time scale of about 1 ÎŒs over which the coupled dynamics between the gas and the droplet becomes important.

### A generalized theory of viscous and inviscid flutter

Fluid Mechanics Biomechanics Geophysics Fluid-structure interaction FlutterWe present a unified theory of flutter in inviscid and viscous flows interacting with flexible structures based on the phenomenon of 1â:â1 resonance. We show this by treating four extreme cases corresponding to viscous and inviscid flows in confined and unconfined flows. To see the common mechanism clearly, we consider the limit when the frequencies of the first few elastic modes are closely clustered and small relative to the convective fluid time scale.

### Precursors to splashing of a liquid droplet on a solid surface

Fluid Mechanics Droplet impactA high velocity impact between a liquid droplet and a solid surface produces a splash. Classical work traced the origin of the splash to a thin sheet of fluid ejected near the impact point. Mechanisms of sheet formation have heretofore relied on initial contact of the droplet and the surface. We demonstrate that, neglecting intermolecular forces between the liquid and the solid, the liquid does not contact the solid, and instead spreads on a very thin air film.

### Short-time dynamics of partial wetting

Fluid Mechanics Capillarity Wetting Materials ScienceWhen a liquid drop contacts a wettable surface, the liquid spreads over the solid to minimize the total surface energy. The first moments of spreading tend to be rapid. For example, a millimeter-sized water droplet will wet an area having the same diameter as the drop within a millisecond. For perfectly wetting systems, this spreading is inertially dominated. Here we identify that even in the presence of a contact line, the initial wetting is dominated by inertia rather than viscosity.

### The feasibility of generating low-frequency volcano seismicity by flow through a deformable channel

Fluid Mechanics Fluid-structure interaction GeophysicsOscillations generated by flow of magmatic or hydrothermal fluids through tabular channels in elastic rocks are a possible source of low-frequency seismicity. We assess the conditions required to generate oscillations of approximately 1 Hz via hydrodynamic flow instabilities (roll waves), flow-destabilized standing waves set up on the elastic channel walls (wall modes), and unstable normal modes ringing in an adjacent fluid reservoir (clarinet modes). Stability criteria are based on physical and dimensional arguments, and discussion of destabilized elastic modes is supplemented with laboratory experiments of gas flow through a channel in a block of gelatine, and between a rigid plate and a rubber membrane.

### Bounds on double-diffusive convection

Fluid Mechanics Geophysics Analysis Double diffusionWe consider double-diffusive convection between two parallel plates and compute bounds on the flux of the unstably stratified species using the background method. The bound on the heat flux for RayleighâBĂ©nard convection also serves as a bound on the double-diffusive problem (with the thermal Rayleigh number equal to that of the unstably stratified component). In order to incorporate a dependence of the bound on the stably stratified component, an additional constraint must be included, like that used by Joseph (Stability of Fluid Motion, 1976, Springer) to improve the energy stability analysis of this system.

### An experimental study of the coalescence between a drop and an interface in Newtonian and polymeric liquids

Fluid Mechanics Droplet coalescence CapillarityWhen a water drop falls onto an oil-water interface, the drop usually rests for some time before merging with the water underneath the interface. We report experiments on this process using water- and oil-based Newtonian liquids and polymer solutions, with an emphasis on the non-Newtonian effects. We deduce that the drop surface is immobilized by contaminants pre-existing in the fluids, and find that the rest time scales with the matrix viscosity for Newtonian fluids.

### Partial coalescence between a drop and a liquid-liquid interface

Fluid Mechanics Capillarity Droplet coalescenceThis Letter reports experimental results for partial coalescence when a drop merges with an interface. We find an intermediate range of drop sizes in which the merger is not complete but a daughter drop is left behind. This phenomenon is governed primarily by inertia and interfacial tension, and three regimes can be further delineated depending on the roles of viscosity and gravity. Scaling relationships are developed for the drop size ratio and the coalescence time.

### Dynamics of roll waves

Fluid Mechanics Instability WavesShallow-water equations with bottom drag and viscosity are used to study the dynamics of roll waves. First, we explore the effect of bottom topography on linear stability of turbulent flow over uneven surfaces. Low-amplitude topography is found to destabilize turbulent roll waves and lower the critical Froude number required for instability. At higher amplitude, the trend reverses and topography stabilizes roll waves. At intermediate topographic amplitude, instability can be created at much lower Froude numbers due to the development of hydraulic jumps in the equilibrium flow.

### Mechanisms of liquid slip and solid surfaces

Fluid Mechanics Molecular Dynamics Materials Science Liquid slipOne of the oldest unresolved problems in fluid mechanics is the nature of liquid flow along solid surfaces. It is traditionally assumed that across the liquid-solid interface, liquid and solid speeds exactly match. However, recent observations document that on the molecular scale, liquids can slip relative to solids. We formulate a model in which the liquid dynamics are described by a stochastic differential-difference equation, related to the Frenkel-Kontorova equation. The model, in agreement with molecular dynamics simulations, reveals that slip occurs via two mechanisms: localized defect propagation and concurrent slip of large domains.

### A simple model illustrating the role of turbulence on phytoplankton blooms

Biomechanics Fluid MechanicsThe problem of the vertical distribution of phytoplankton is considered in the presence of gravitational settling, turbulent mixing, population growth due to cell division and a constant rate of loss due to predation and natural death. Nutrients are assumed to be plentiful so that the production rate depends only on the light available for photosynthesis. The non-linear saturation of plankton growth is modeled by allowing the attenuation rate of light to be a linear function of the plankton density.