Shreyas Mandre

Dynamics of roll waves

Shallow-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. Second, the nonlinear dynamics of the roll waves is explored, with numerical solutions of the shallow-water equations complementing an asymptotic theory relevant near onset. We find that trains of roll waves undergo coarsening due to waves overtaking one another and merging, lengthening the scale of the pattern. Unlike previous investigations, we find that coarsening does not always continue to its ultimate conclusion (a single roll wave with the largest spatial scale). Instead, coarsening becomes interrupted at intermediate scales, creating patterns with preferred wavelengths. We quantify the coarsening dynamics in terms of linear stability of steady roll-wave trains.

Events before droplet splashing on a solid surface

A high-velocity (≈1 m s ―1 ) impact between a liquid droplet (≈1 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 μs over which the coupled dynamics between the gas and the droplet becomes important. The droplet initially tries to contact the substrate by either draining gas out of a thin layer or compressing it, with the local behaviour described by a self-similar solution of the governing equations. This similarity solution is not asymptotically consistent: forces that were initially negligible become relevant and dramatically change the behaviour. Depending on the radius and impact velocity of the droplet, we show that the solution is overtaken by initially subdominant physical effects such as the surface tension of the liquid―gas interface or viscous forces in the liquid. At low impact velocities surface tension stops the droplet from impacting the surface, whereas at higher velocities viscous forces become important before surface tension. The ultimate dynamics of the interface once droplet viscosity cannot be neglected is not yet known.

SKATING ON A FILM OF AIR: DROPS IMPACTING ON A SURFACE

The 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 thin film of air serves to lubricate the drop enabling the fluid to skate on the air film laterally outward at surprisingly high velocities, consistent with theoretical predictions. Eventually this thin film of air breaks down as the fluid wets the surface via a spinodal-like mechanism. Our results show that the dynamics of impacting drops are much more complex than previously thought, with a rich array of unexpected phenomena that require rethinking classic paradigms.

Partial coalescence between a drop and a liquid-liquid interface

This 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. For drops that are too large or too small, partial coalescence is arrested by gravity or viscosity, respectively.

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

Abstract Oscillations 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. For each of the mechanisms considered, oscillations are generated if flow speeds exceed a critical value. Roll waves are waves of channel thickness variation that propagate in the direction of flow and are equivalent to traveling crack waves. The convective instability criterion is that the flow is faster than those travelling waves. Similarly, wall modes and clarinet modes require that the flow speed exceeds a critical value related to a wave speed (e.g. elastic or acoustic wave) multiplied by a geometrical factor. Flow destabilized modes offer a plausible explanation for low-frequency volcano seismicity, but there are limitations on what kind of standing waves comprises them.

Algorithm for a Microfluidic Assembly Line

Microfluidic 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.

A generalized theory of viscous and inviscid flutter

We present a unified theory of flutter in inviscid and viscous flows interacting with flexible structures based on the phenomenon of 1 : 1 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. This separation of time scales slaves the hydrodynamic force to the instantaneous elastic displacement and allows us to calculate explicitly the dependence of the critical flow speed for flutter on the various problem parameters. We show that the origin of the instability lies in the coincidence of the real frequencies of the first two modes at a critical flow speed beyond which the frequencies become complex, thus making the system unstable to oscillations. This critical flow speed depends on the difference between the frequencies of the first few modes and the nature of the hydrodynamic coupling between them. Our generalized framework applies to a range of elastohydrodynamic systems and further extends the Benjamin–Landahl classification of fluid–elastic instabilities.

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

When 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. The results are compared with lubrication models for film drainage. If the surrounding matrix is a dilute polymer solution, the rest time is identical to that for a matrix of the solvent alone. Further investigation indicates that the polymer molecules have been cleared from the film by surface adsorption. Depending on the fluid properties and drop size, the drop-interface merging may be completed in one shot or through a cascade of partial coalescence. Partial coalescence occurs for an intermediate range of drop sizes; it is arrested by vis...

Capillary interactions between nearby interfacial objects

We 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 and the prefactor depends on the mean radius of curvature R at the closest points. After contact the objects roll and/or slide with respect to each other to locally maximize R and therefore the force. For smaller objects (, the force scales as , and the prefactor depends only weakly on the shape and relative orientation of the objects.

Bounds on double-diffusive convection

We 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 Rayleigh-Benard 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. Our bound extends Josephs result beyond his energy stability boundary. At large Rayleigh number, the bound is found to behave like R 1/2 T for fixed ratio R S /R T , where R T and R S are the Rayleigh numbers of the unstably and stably stratified components, respectively.