arXiv subjects

Michael Wilkinson

Publications and source records attributed to Michael Wilkinson.

At least 55 records · Page 3Linked to original sources

arXiv2014

A test-tube model for rainfall

If the temperature of a cell containing two partially miscible liquids is changed very slowly, so that the miscibility is decreased, microscopic droplets nucleate, grow and migrate to the interface due to their buoyancy. The system may show an approximately periodic variation of the turbidity of the mixture, as the mean droplet size fluctuates. These precipitation events are analogous to rainfall from warm clouds. This paper considers a theoretical model for these experiments. After nucleation the initial growth is by Ostwald ripening, followed by a finite-time runaway growth of droplet sizes due to larger droplets sweeping up smaller ones. The model predicts that the period $Δt$ and the temperature sweep rate $ξ$ are related by $Δt\sim C ξ^{-3/7}$, and is in good agreement with experiments. The coefficient $C$ has a power-law divergence approaching the critical point of the miscibility transition: $C\sim (T-T_{\rm c})^{-η}$, and the critical exponent $η$ is determined.

physics.ao-ph↗

arXiv2013

Multiple collisions in turbulent flows

In turbulent suspensions, collision rates determine how rapidly particles coalesce or react with each other. To determine the collision rate, many numerical studies rely on the 'Ghost Collision Approximation' (GCA), which simply records how often pairs of point particles come within a threshold distance. In many applications, the suspended particles stick (or in the case of liquid droplets, coalesce) upon collision, and it is the frequency of first contact which is of interest. If a pair of 'ghost' particles undergoes multiple collisions, the GCA may overestimate the true collision rate. Here, using fully resolved Direct Numerical Simulations of turbulent flows at moderate Reynolds number (R_λ= 130), we investigate the prevalence and properties of multiple collisions. We demonstrate that the GCA leads to a systematic overestimate of the collision rate, which is of the order of 15% when the particle inertia is small, and slowly decreases when inertia increases. We investigate the probability P(N) for a given pair of ghost particles colliding N times. We find P(N)=βα^N for N_>1, where αand βare coefficients which depend upon the particle inertia. This result is used to explain the discrepancy between the GCA and the true collision rates. We also investigate the statistics of the times that ghost particles remain in contact. We show that the probability density function of the contact time is different for the first collision. The difference is explained by the effect of caustics in the phase space of the suspended particles.

physics.flu-dyn↗

arXiv2013

Prevalence of the sling effect for enhancing collision rates in turbulent suspensions

Turbulence facilitates collisions between particles suspended in a turbulent flow. Two effects have been proposed which can enhance the collision rate at high turbulence intensities: 'preferential concentration' (a clustering phenomenon) and the 'sling effect' (arising from the formation of caustic folds in the phase-space of the suspended particles). We have determined numerically the collision rate of small heavy particles as a function of their size and densities. The dependence on particle densities reveals that the enhancement by turbulence of the collision rate of particles with significant inertia is due almost entirely to the sling effect.

physics.flu-dyn↗

arXiv2013

A model for the shapes of advected triangles

Three particles floating on a fluid surface define a triangle. The aim of this paper is to characterise the shape of the triangle, defined by two of its angles, as the three vertices are subject to a complex or turbulent motion. We consider a simple class of models for this process, involving a combination of a random strain of the fluid and Brownian motion of the particles. Following D. G. Kendall, we map the space of triangles to a sphere, whose equator corresponds to degenerate triangles with colinear vertices, with equilaterals at the poles. We map our model to a diffusion process on the surface of the sphere and find an exact solution for the shape distribution. Whereas the action of the random strain tends to make the shape of the triangles infinitely elongated, in the presence of a Brownian diffusion of the vertices, the model has an equilibrium distribution of shapes. We determine here exactly this shape distribution in the simple case where the increments of the strain are diffusive.

nlin.CD↗

arXiv2012

Enhanced transport when Anderson localization is destroyed

We investigate the anomalous transport in optically-induced potentials that are random in both space and time. We find that the time variation destroys Anderson localization, replacing it by transport that is faster than diffusion, which in some cases can be even faster than ballistic. We relate this phenomenon to Chirikov's theory of overlapping resonances, and find radical differences between the anomalously-enhanced transport in one-dimensional and two-dimensional systems.

cond-mat.dis-nn↗

arXiv2012

Ostwald ripening and the kinetics of rain initiation

A central problem in cloud physics is understanding the kinetics of the growth of water droplets. It is believed that there exists a 'condensation-coalescence bottleneck' in the growth of intermediate size droplets. Here the Lifshitz-Slezov theory of Ostwald ripening is applied to the kinetics of the growth of rain drops. The analysis shows that kinetic barriers to rain initiation are not significant. It is also shown that Ostwald ripening can greatly enhance the apparent collision efficiency of droplets falling under gravity.

physics.ao-ph↗

arXiv2012

Universal anomalous diffusion of weakly damped particles

We show that anomalous diffusion arises in two different models for the motion of randomly forced and weakly damped particles: one is a generalisation of the Ornstein-Uhlenbeck process with a random force which depends on position as well as time, the other is a generalisation of the Chandrasekhar-Rosenbluth model of stellar dynamics, encompassing non-Coulombic potentials. We show that both models exhibit anomalous diffusion of position $x$ and momentum $p$ with the same exponents: $ \sim C_x t^2$ and $ \sim C_p t^{2/5}$. We are able to determine the prefactors $C_x$, $C_p$ analytically.

nlin.CD↗

arXiv2011

Spherical Ornstein-Uhlenbeck processes

The paper considers random motion of a point on the surface of a sphere, in the case where the angular velocity is determined by an Ornstein-Uhlenbeck process. The solution is fully characterized by only one dimensionless number, the persistence angle, which is the typical angle of rotation of the object during the correlation time of the angular velocity. We first show that the two-dimensional case is exactly solvable. When the persistence angle is large, a series for the correlation function has the surprising property that its sum varies much more slowly than any of its individual terms. In three dimensions we obtain asymptotic forms for the correlation function, in the limits where the persistence angle is very small and very large. The latter case exhibits a complicated transient, followed by a much slower exponential decay. The decay rate is determined by the solution of a radial Schrodinger equation in which the angular momentum quantum number takes an irrational value, namely j=(sqrt{17}-1)/2. Possible applications of the model to objects tumbling in a turbulent environment are discussed.

physics.flu-dyn↗

arXiv2010

Hot-Carrier Model for an Anomalous Exponent of Photoconduction

Experiments often show that the photoconductance $σ$ of a semiconductor system and the light intensity $I$ are related by $σ\sim I^γ$. Conventional theories give a satisfactory explanation for $γ=1$ or $γ=1/2$, but anamalous exponents close to $γ=3/4$ are often observed. This paper argues that there is a universal anomalous regime for which $γ=3/4$ (or $γ=2/3$ in two-dimensional samples), resulting from the kinetics of electron-hole recombination being controlled by Coulombic attraction. Because the local electric fields are extremely high, the theory appeals to the 'hot-carrier' model for transport in semiconductors.

cond-mat.other↗

arXiv2009

Lyapunov exponent for small particles in smooth one-dimensional flows

This paper discusses the Lyapunov exponent for small particles in a spatially and temporally smooth flow in one dimension. Using a plausible model for the statistics of the velocity gradient in the vicinity of a particle, the Lyapunov exponent is obtained as a series expansion in the Stokes number, St, which is a dimensionless measure of the importance of inertial effects. The approach described here can be extended to calculations of the Lyapunov exponents and of the correlation dimension for inertial particles in higher dimensions. It is also shown that there is correction to this theory which arises because the particles do not sample the velocity field ergodically. Using this non-ergodic correction, it is found that (contrary to expectations) the first order term in the expansion does not vanish.

physics.flu-dyn↗

arXiv2009

Perturbation theory for a stochastic process with Ornstein-Uhlenbeck noise

The Ornstein-Uhlenbeck process may be used to generate a noise signal with a finite correlation time. If a one-dimensional stochastic process is driven by such a noise source, it may be analysed by solving a Fokker-Planck equation in two dimensions. In the case of motion in the vicinity of an attractive fixed point, it is shown how the solution of this equation can be developed as a power series. The coefficients are determined exactly by using algebraic properties of a system of annihilation and creation operators.

physics.data-an↗

arXiv2009

Emergent order in rheoscopic swirls

We discuss the reflection of light by a rheoscopic fluid (a suspension of microscopic rod-like crystals) in a steady two-dimensional flow. This is determined by an order parameter which is a non-oriented vector, obtained by averaging solutions of a nonlinear equation containing the strain rate of the fluid flow. Exact solutions of this equation are obtained from solutions of a linear equation which are analogous to Bloch bands for a one-dimensional Schrodinger equation with a periodic potential. On some contours of the stream function, the order parameter approaches a limit, and on others it depends increasingly sensitively upon position. However, in the long-time limit a local average of the order parameter is a smooth function of position in both cases. We analyse the topology of the order parameter and the structure of the generic zeros of the order parameter field.

physics.flu-dyn↗

arXiv2009

Correlation Dimension of Inertial Particles in Random Flows

We obtain an implicit equation for the correlation dimension of dynamical systems in terms of an integral over a propagator. We illustrate the utility of this approach by evaluating the correlation dimension for inertial particles suspended in a random flow. In the limit where the correlation time of the flow field approaches zero, taking the short-time limit of the propagator enables the correlation dimension to be determined from the solution of a partial differential equation. We develop the solution as a power series in a dimensionless parameter which represents the strength of inertial effects.

physics.flu-dyn↗

arXiv2009

Poincare Indices of Rheoscopic Visualisations

Suspensions of small anisotropic particles, termed 'rheoscopic fluids', are used for flow visualisation. By illuminating the fluid with light of three different colours, it is possible to determine Poincare indices for vector fields formed by the longest axis of the particles. Because this vector field is non-oriented, half-integer Poincare indices are possible, and are observed experimentally. An exact solution for the direction vector appears to preclude the existence of topological singularities. However, we show that upon averaging over the random initial orientations of particles, singularities with half-integer Poincare index appear. We describe their normal forms.

physics.flu-dyn↗

arXiv2008

Planet Formation by Concurrent Collapse

After reviewing the difficulties faced by the conventional theory of planet formation (based upon the aggregation of microscopic dust particles), we describe an alternative hypothesis. We propose that planets form by gravitational collapse at the same time as the star about which they orbit. This 'concurrent collapse' hypothesis avoids theoretical difficulties associated with the conventional model and suggests satisfying explanations for various poorly understood phenomena. We introduce new explanations for FU Orionis outbursts seen in young stars, the discovery of exoplanets with eccentric orbits and the existence of small rocky objects such as chondrules in the solar system.

astro-ph↗

arXiv2008

Mutually Unbiased Bases for Continuous Variables

The concept of mutually unbiased bases is studied for N pairs of continuous variables. To find mutually unbiased bases reduces, for specific states related to the Heisenberg-Weyl group, to a problem of symplectic geometry. Given a single pair of continuous variables, three mutually unbiased bases are identified while five such bases are exhibited for two pairs of continuous variables. For N = 2, the golden ratio occurs in the definition of these mutually unbiased bases suggesting the relevance of number theory not only in the finite-dimensional setting.

quant-ph↗

arXiv2008

Fingerprints of Random Flows?

We consider the patterns formed by small rod-like objects advected by a random flow in two dimensions. An exact solution indicates that their direction field is non-singular. However, we find from simulations that the direction field of the rods does appear to exhibit singularities. First, ` scar lines' emerge where the rods abruptly change direction by $π$. Later, these scar lines become so narrow that they ` heal over' and disappear, but their ends remain as point singularities, which are of the same type as those seen in fingerprints. We give a theoretical explanation for these observations.

nlin.PS↗

arXiv2006

Colliding Particles in Highly Turbulent Flows

We discuss relative velocities and the collision rate of small particles suspended in a highly turbulent fluid. In the limit where the viscous damping is very weak, we estimate the relative velocities using the Kolmogorov cascade principle.

nlin.CD↗