arXiv subjects

Roberto Camassa

Publications and source records attributed to Roberto Camassa.

17 recordsLinked to original sources

arXiv2024

Diffusion-limited settling of highly porous particles in density-stratified fluids

The vertical transport of solid material in a stratified medium is fundamental to a number of environmental applications, with implications for the carbon cycle and nutrient transport in marine ecosystems. In this work, we study the diffusion-limited settling of highly porous particles in a density-stratified fluid through a combination of experiment, analysis, and numerical simulation. By delineating and appealing to the diffusion-limited regime wherein buoyancy effects due to mass adaptation dominate hydrodynamic drag, we derive a simple expression for the steady settling velocity of a sphere as a function of the density, size, and diffusivity of the solid, as well as the density gradient of the background fluid. In this regime, smaller particles settle faster, in contrast with most conventional hydrodynamic drag mechanisms. Furthermore, we outline a general mathematical framework for computing the steady settling speed of a body of arbitrary shape in this regime and compute exact results for the case of general ellipsoids. Using hydrogels as a highly porous model system, we validate the predictions with laboratory experiments in linear stratification for a wide range of parameters. Lastly, we show how the predictions can be applied to arbitrary slowly varying background density profiles and demonstrate how a measured particle position over time can be used to reconstruct the background density profile.

physics.flu-dyn↗

arXiv2023

Tauberian identities and the connection to Wile E. Coyote physics

The application of the motion of a vertically suspended mass-spring system released under tension is studied focusing upon the delay timescale for the bottom mass as a function of the spring constants and masses. This ``hang-time", reminiscent of the Coyote and Road Runner cartoons, is quantified using the far-field asymptotic expansion of the bottom mass' Laplace transform. These asymptotics are connected to the short time mass dynamics through Tauberian identities and explicit residue calculations. It is shown, perhaps paradoxically, that this delay timescale is maximized in the large mass limit of the top ``boulder". Experiments are presented and compared with the theoretical predictions. This system is an exciting example for the teaching of mass-spring dynamics in classes on Ordinary Differential Equations, and does not require any normal mode calculations for these predictions.

math.DS↗

arXiv2023

Critical density triplets for the arrestment of a sphere falling in a sharply stratified fluid

We study the motion of a rigid sphere falling in a two-layer stratified fluid under the action of gravity in the potential flow regime. Experiments at a moderate Reynolds number of approximately 20 to 450 indicate that a sphere with the precise critical density, higher than the bottom layer density, can display behaviors such as bounce or arrestment after crossing the interface. We experimentally demonstrate that such a critical sphere density increases linearly as the bottom fluid density increases with a fixed top fluid density. Additionally, the critical density approaches the bottom layer fluid density as the thickness of density transition layer increases. We propose an estimation of the critical density based on the potential energy. With assuming the zero layer thickness, the estimation constitutes an upper bound of the critical density with less than 0.043 relative difference within the experimental density regime 0.997 $g/cm^{3}$ $\sim $ 1.11 $g/cm^{3}$ under the zero layer thickness assumption. By matching the experimental layer thickness, we obtain a critical density estimation with less than 0.01 relative difference within the same parameter regime.

physics.flu-dyn↗

arXiv2022

Numerical Algorithms for Water Waves with Background Flow over Obstacles and Topography

We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply-connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).

physics.flu-dyn↗

arXiv2019

On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations

Self-similar solutions of the so called Airy equations, equivalent to the dispersionless nonlinear Schrödinger equation written in Madelung coordinates, are found and studied from the point of view of complete integrability and of their role in the recurrence relation from a bi-Hamiltonian structure for the equations. This class of solutions reduces the PDEs to a finite ODE system which admits several conserved quantities, which allow to construct explicit solutions by quadratures and provide the bi-Hamiltonian formulation for the reduced ODEs.

math-ph↗

arXiv2019

Cluster formation and self-assembly in stratified fluids: a novel mechanism for particulate aggregation

An extremely broad and important class of phenomena in nature involves the settling and aggregation of matter under gravitation in fluid systems. Some examples include: sedimenting marine snow particles in lakes and oceans (central to carbon sequestration), dense microplastics in the oceans (which impact ocean ecology and the food chain), and even iron snow on Mercury (conjectured as its magnetic field source). These fluid systems all have stable density stratification, which is known to trap particulates through upper lightweight fluid coating the sinking particles, thus providing transient buoyancy. The current understanding of aggregation of such trapped matter involves collisions (due to Brownian motion, shear, and differential settling) and adhesion. Here, we observe and rationalize a new fundamental effective attractive mechanism by which particles suspended within stratification may self-assemble and form large aggregates without need for short range binding effects. This phenomenon arises through a complex interplay involving solute diffusion, impermeable boundaries, and aggregate geometry, which produces toroidal flows. We show that these toroidal flows yield attractive horizontal forces between particles. We observe that many particles demonstrate a collective motion revealing a system which self-assembles, appearing to solve jigsaw-like puzzles on its way to organizing into a disc-like shape, with the effective force increasing as the collective disc radius grows. Control experiments with two objects isolate the individual dynamics, which are quantitatively predicted through numerical integration of the underlying equations of motion. This new mechanism may be an important process in formation of marine snow aggregates and distribution of phytoplankton in lakes and oceans. Further, it potentially provides a new mechanism for general sorting and packing of layered material.

physics.flu-dyn↗

arXiv2018

Experimental investigation of nonlinear internal waves in deep water with miscible fluids

Laboratory experimental results are presented for nonlinear Internal Solitary Waves (ISW) propagation in deep water configuration with miscible fluids. The results are validated against direct numerical simulations and traveling wave exact solutions where the effect of the diffused interface is taken into account. The waves are generated by means of a dam break and their evolution is recorded with Laser Induced Fluorescence (LIF) and Particle Image Velocimetry (PIV). In particular, data collected in a frame moving with the waves are presented here for the first time. Our results are representative of geophysical applications in the deep ocean where weakly nonlinear theories fail to capture the characteristics of large amplitude ISWs from field observations.

physics.flu-dyn↗

arXiv2018

On the symmetry properties of a random passive scalar with and without boundaries, and their connection between hot and cold states

We consider the evolution of a decaying passive scalar in the presence of a gaussian white noise fluctuating linear shear flow known as the Majda Model. We focus on deterministic initial data and establish the symmetry properties of the evolving point wise probability measure for the random scalar. We identify, for both point line source initial data, regions in the x-y plane outside of which the PDF skewness is sign definite for all time, while inside these regions we observe multiple sign changes corresponding to exchanges in symmetry between hot and cold leaning states using exact representation formula for the PDF at the origin, and away from the origin, using numerical evaluation of the exact available Mehler kernels for the scalars statistical moments. A new, rapidly convergent Monte-Carlo method is developed, dubbed Direct Monte-Carlo (DMC), using the random Green's functions allowing for the fast construction of the PDF for single point statistics, and multi-point statistics natural for full Monte-Carlo simulations of the underlying stochastic differential equations (FMC). This new method demonstrates the full evolution of the PDF from short times, to its long time, limiting and collapsing universal distribution at arbitrary points in the plane. Further, this method provides a strong benchmark for FMC and we document numbers of field realization criteria for the FMC to faithfully compute this complete dynamics. Armed with this benchmark, we apply the FMC to a channel with a no-flux boundary condition enforced on a channel and observe a dramatically different long time state resulting from the wall. In particular, the channel case collapsing invariant measure has negative skewness, with random states heavily leaning heavily towards the hot state, in stark contrast to free space, where the limiting skewness is positive, with its states leaning heavily towards the cold state.

physics.flu-dyn↗

arXiv2018

Mass distribution and skewness for passive scalar transport in pipes with polygonal and smooth cross-sections

We extend our previous results characterizing the loading properties of a diffusing passive scalar advected by a laminar shear flow in ducts and channels to more general cross-sectional shapes, including regular polygons and smoothed corner ducts originating from deformations of ellipses. For the case of the triangle, short time skewness is calculated exactly to be positive, while long-time asymptotics shows it to be negative. Monte-Carlo simulations confirm these predictions, and document the time scale for sign change. Interestingly, the equilateral triangle is the only regular polygon with this property, all other polygons possess positive skewness at all times, although this cannot cannot be proved on finite times due to the lack of closed form flow solutions for such geometries. Alternatively, closed form flow solutions can be constructed for smooth deformations of ellipses, and illustrate how the possibility of multiple sign switching in time is unrelated to domain corners. Exact conditions relating the median and the skewness to the mean are developed which guarantee when the sign for the skewness implies front (back) loading properties of the evolving tracer distribution along the pipe. Short and long time asymptotics confirm this condition, and Monte-Carlo simulations verify this at all times.

physics.flu-dyn↗

arXiv2015

A class of fast geodesic shooting algorithms for template matching and its applications via the $N$-particle system of the Euler-Poincar\'e equations

The Euler-Poincar\'e (EP) equations describe the geodesic motion on the diffeomorphism group. For template matching (template deformation), the Euler-Lagrangian equation, arising from minimizing an energy function, falls into the Euler-Poincar\'e theory and can be recast into the EP equations. By casting the EP equations in the Lagrangian (or characteristics) form, we formulate the equations as a finite dimensional particle system. The evolution of this particle system describes the geodesic motion of landmark points on a Riemann manifold. In this paper we present a class of novel algorithms that take advantage of the structure of the particle system to achieve a fast matching process between the reference and the target templates. The strong suit of the proposed algorithms includes (1) the efficient feedback control iteration, which allows one to find the initial velocity field for driving the deformation from the reference template to the target one, (2) the use of the conical kernel in the particle system, which limits the interaction between particles and thus accelerates the convergence, and (3) the availability of the implementation of fast-multipole method for solving the particle system, which could reduce the computational cost from $O(N^2)$ to $O(N\log N)$, where $N$ is the number of particles. The convergence properties of the proposed algorithms are analyzed. Finally, we present several examples for both exact and inexact matchings, and numerically analyze the iterative process to illustrate the efficiency and the robustness of the proposed algorithms.

math.NA↗

arXiv2014

Solitary waves and $N$-particle algorithms for a class of Euler-Poincar\'e equations

We study a class of partial differential equations (PDEs) in the family of the so-called Euler-Poincar\'e differential systems, with the aim of developing a foundation for numerical algorithms of their solutions. This requires particular attention to the mathematical properties of this system when the associated class of elliptic operators possesses non-smooth kernels. By casting the system in its Lagrangian (or characteristics) form, we first formulate a particles system algorithm in free space with homogeneous Dirichlet boundary conditions for the evolving fields. We next examine the deformation of the system when non-homogeneous "constant stream" boundary conditions are assumed. We show how this simple change at the boundary deeply affects the nature of the evolution, from hyperbolic-like to dispersive with a non-trivial dispersion relation, and examine the potentially regularizing properties of singular kernels offered by this deformation. From the particle algorithm viewpoint, kernel singularities affect the existence and uniqueness of solutions to the corresponding ordinary differential equations systems. We illustrate this with the case when the operator kernel assumes a conical shape over the spatial variables, and examine in detail two-particle dynamics under the resulting lack of Lipschitz-continuity. Curiously, we find that for the conically-shaped kernels the motion of the related two-dimensional waves can become completely integrable under appropriate initial data. This reduction projects the two-dimensional system to the one-dimensional completely integrable Shallow-Water equation [Camassa, R. and Holm, D. D., Phys. Rev. Lett., 71, 1961-1964, 1993], while retaining the full dependence on two spatial dimensions for the single channel solutions.

math.NA↗

arXiv2014

A connection between the shallow-water equations and the Euler-Poincaré equations

The Euler-Poincaré differential (EPDiff) equations and the shallow water (SW) equations share similar wave characteristics. Using the Hamiltonian structure of the SW equations with flat bottom topography, we establish a connection between the EPDiff equations and the SW equations in one and multi-dimensions. Additionally, we show that the EPDiff equations can be recast in a curl formulation.

math-ph↗

arXiv2012

An inertia 'paradox' for incompressible stratified Euler fluids

The interplay between incompressibility and stratification can lead to non-conservation of horizontal momentum in the dynamics of a stably stratified incompressible Euler fluid filling an infinite horizontal channel between rigid upper and lower plates. Lack of conservation occurs even though in this configuration only vertical external forces act on the system. This apparent paradox was seemingly first noticed by Benjamin (J. Fluid Mech., vol. 165, 1986, pp. 445-474) in his classification of the invariants by symmetry groups with the Hamiltonian structure of the Euler equations in two dimensional settings, but it appears to have been largely ignored since. By working directly with the motion equations, the paradox is shown here to be a consequence of the rigid lid constraint coupling through incompressibility with the infinite inertia of the far ends of the channel, assumed to be at rest in hydrostatic equilibrium. Accordingly, when inertia is removed by eliminating the stratification, or, remarkably, by using the Boussinesq approximation of uniform density for the inertia terms, horizontal momentum conservation is recovered. This interplay between constraints,action at a distance by incompressibility, and inertia is illustrated by layer-averaged exact results, two-layer long-wave models, and direct numerical simulations of the incompressible Euler equations with smooth stratification.

physics.flu-dyn↗

arXiv2008

Vortex Induced Oscillations of Cylinders

This article submitted to the APS-DFD 2008 conference, accompanies the fluid dynamics video depicting the various orientational dynamics of a hinged cylinder suspended in a flow tank. The different behaviors displayed by the cylinder range from steady orientation to periodic oscillation and even autorotation. We illustrate these features using a phase diagram which captures the observed phenomena as a function of Reynolds number and reduced inertia. A hydrogen bubble flow visualization technique is also used to show vortex shedding structure in the cylinder's wake which results in these oscillations.

physics.flu-dyn↗

arXiv2001

The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's of Shallow Water and Dym Type

An extension of the algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new piecewise smooth weak solutions of a class of $N$-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional solutions of these nonlinear PDE's, which are associated to nonlinear subvarieties of hyperelliptic Jacobians. The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDE's that admit piecewise smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDE's to gain further insight into, and explicit formulas for, piecewise-smooth finite-gap solutions.

nlin.CD↗

arXiv2000

On billiard weak solutions of nonlinear PDE's and Toda flows

A certain class of partial differential equations possesses singular solutions having discontinuous first derivatives ("peakons"). The time evolution of peaks of such solutions is governed by a finite dimensional completely integrable system. Explicit solutions of this system are constructed by using algebraic-geometric method which casts it as a flow on an appropriate Riemann surface and reduces it to a classical Jacobi inversion problem. The algebraic structure of the finite dimensional flow is also examined in the context of the Toda flow hierarchy. Generalized peakon systems are obtained for any simple Lie algebra and their complete integrability is demonstrated.

nlin.SI↗

arXiv1993

An integrable shallow water equation with peaked solitons

We derive a new completely integrable dispersive shallow water equation that is biHamiltonian and thus possesses an infinite number of conservation laws in involution. The equation is obtained by using an asymptotic expansion directly in the Hamiltonian for Euler's equations in the shallow water regime. The soliton solution for this equation has a limiting form that has a discontinuity in the first derivative at its peak.

patt-sol↗