Academic literature on the topic 'Maxey Riley equation'

The dynamics of spherical particles in a fluid flow is governed by the well-accepted Maxey–Riley equation. This equation of motion simply represents Newton’s second law, equating the rate of change of the linear momentum with all forces acting on the particle. One of these forces, the Basset–Boussinesq memory term, however, is notoriously difficult to handle, which prompts most studies to ignore this term despite ample numerical and experimental evidence of its significance. This practice may well change now due to a clever reformulation of the particle equation of motion by Prasath, Vasan & Govindarajan (J. Fluid Mech., vol. 868, 2019, pp. 428–460), who convert the Maxey–Riley equation into a one-dimensional heat equation with non-trivial boundary conditions. Remarkably, this reformulation confirms earlier estimates on particle asymptotics, yields previously unknown analytic solutions and leads to an efficient numerical scheme for more complex flow fields.

AbstractLinearized viscous compressible Navier–Stokes equations are solved for the transient force on a spherical particle undergoing unsteady motion in an inhomogeneous unsteady ambient flow. The problem is formulated in a reference frame attached to the particle and the force contributions from the undisturbed ambient flow and the perturbation flow are separated. Using a density-weighted velocity transformation and reciprocal relation, the total force is first obtained in the Laplace domain and then transformed to the time domain. The total force is separated into the quasi-steady, inviscid unsteady, and viscous unsteady contributions. The above rigorously derived particle equation of motion can be considered as the compressible extension of the Maxey–Riley–Gatignol equation of motion and it incorporates interesting physics that arises from the combined effects of inhomogeneity and compressibility.

The Maxey–Riley equation has been extensively used by the fluid dynamics community to study the dynamics of small inertial particles in fluid flow. However, most often, the Basset history force in this equation is neglected. Analytical solutions have almost never been attempted because of the difficulty in handling an integro-differential equation of this type. Including the Basset force in numerical solutions of particulate flows involves storage requirements which rapidly increase in time. Thus the significance of the Basset history force in the dynamics has not been understood. In this paper, we show that the Maxey–Riley equation in its entirety can be exactly mapped as a forced, time-dependent Robin boundary condition of the one-dimensional diffusion equation, and solved using the unified transform method. We obtain the exact solution for a general homogeneous time-dependent flow field, and apply it to a range of physically relevant situations. In a particle coming to a halt in a quiescent environment, the Basset history force speeds up the decay as a stretched exponential at short time while slowing it down to a power-law relaxation, ${\sim}t^{-3/2}$, at long time. A particle settling under gravity is shown to relax even more slowly to its terminal velocity (${\sim}t^{-1/2}$), whereas this relaxation would be expected to take place exponentially fast if the history term were to be neglected. An important mechanism for the growth of raindrops is by the gravitational settling of larger drops through an environment of smaller droplets, and repeatedly colliding and coalescing with them. Using our solution we estimate that the rate of growth rate of a raindrop can be grossly overestimated when history effects are not accounted for. We solve exactly for particle motion in a plane Couette flow and show that the location (and final velocity) to which a particle relaxes is different from that due to Stokes drag alone. For a general flow, our approach makes possible a numerical scheme for arbitrary but smooth flows without increasing memory demands and with spectral accuracy. We use our numerical scheme to solve an example spatially varying flow of inertial particles in the vicinity of a point vortex. We show that the critical radius for caustics formation shrinks slightly due to history effects. Our scheme opens up a method for future studies to include the Basset history term in their calculations to spectral accuracy, without astronomical storage costs. Moreover, our results indicate that the Basset history can affect dynamics significantly.

We use a dynamical systems approach to identify coherent structures from often chaotic motions of inertial particles in open flows. We show that particle Lagrangian coherent structures (pLCS) act as boundaries between regions in which particles have different kinematics. They provide direct geometric information about the motion of ensembles of inertial particles, which is helpful to understand their transport. As an application, we apply the methodology to a planktonic predator–prey system in which moon jellyfish Aurelia aurita uses its body motion to generate a flow that transports small plankton such as copepods to its vicinity for feeding. With the flow field generated by the jellyfish measured experimentally and the dynamics of plankton described by a modified Maxey–Riley equation, we use the pLCS to identify a capture region in which prey can be captured by the jellyfish. The properties of the pLCS and the capture region enable analysis of the effect of several physiological and mechanical parameters on the predator–prey interaction, such as prey size, escape force, predator perception, etc. The methods developed here are equally applicable to multiphase and granular flows, and can be generalized to any other particle equation of motion, e.g. equations governing the motion of reacting particles or charged particles.

The paper describes the calibration of a numerical model to simulate the 2D motion of floating rigid bodies. The proposed model follows a one-way coupling Eulerian-Lagrangian approach, in which the solution of the Shallow Water Equations (SWE) is combined with the Discrete Element Method (DEM) to compute the displacement of rigid bodies. Floating bodies motion is computed by adapting the Maxey-Riley equation to the case of semi-submerged bodies at high Reynolds number. In order to account for the flow velocity distribution along the body axis, the elements are divided into shorter subsections. A specific formulation is proposed to calculate the rotation of wooden cylinders, by computing the angular momentum. The model includes also a term of added inertia, which accounts for the resistance to rotation and requires the calibration of a specific inertia coefficient. A series of flume experiments is performed to calibrate the model. The 2D trajectories of floating spheres and the linear and angular displacement of cylinders are recorded in stationary conditions. The comparison between the experimental data and the simulation shows that the numerical results are in agreement with the experimental ones, although less accuracy is observed in the reproduction of the angular displacement.

The dynamics of a non-neutrally buoyant particle moving in a rotating cylinder filled with a Newtonian fluid is examined analytically. The particle is set in motion from the centre of the cylinder due to the acceleration caused by the presence of a gravitational field. The problem is formulated in Cartesian coordinates and a relevant fractional Lagrangian equation is proposed. This equation is solved exactly by recognizing that the eigenfunctions of the problem are Mittag–Leffler functions. Virtual mass, gravity, pressure, and steady and history drag effects at low particle Reynolds numbers are considered and the balance of forces acting on the particle is studied for realistic cases. The presence of lift forces, both steady and unsteady, is taken into account. Results are compared to the exact solution of the Maxey–Riley equation for the same conditions. Substantial differences are found by including lift in the formulation when departing from the infinitesimal particle Reynolds number limit. For particles lighter than the fluid, an asymptotically stable equilibrium position is found to be at a distance from the origin characterized by X ≈ −Vτ/Ω and Y/X ≈ (CS/3π√2) Res1/2, where Vτ is the terminal velocity of the particle, Ω is the positive angular velocity of the cylinder, Res is the shear Reynolds number a2Ω/v, and CS is a constant lift coefficient. To the knowledge of the authors this work is the first to solve the particle Lagrangian equation of motion in its complete form, with or without lift, for a non-uniform flow using an exact method.

Droplet-laden flows with phase change are common. This study brings to light a mechanism by which droplet inertial dynamics and local phase change, taking place at sub-Kolmogorov scales, affect vortex dynamics in the inertial range of turbulence. To do this we consider vortices placed in a supersaturated ambient initially at constant temperature, homogeneous vapour concentration and uniformly distributed droplets. The droplets also act as sites of phase change. This allows the time scales associated with particle inertia and phase change, which could be significantly different from each other and from the time scale of the flow, to become coupled, and for their combined dynamics to govern the flow. The thermodynamics of condensation and evaporation have a characteristic time scale $\unicode[STIX]{x1D70F}_{s}$. The water droplets are treated as Stokesian inertial particles with a characteristic time scale $\unicode[STIX]{x1D70F}_{p}$, whose behaviour we approximate using an $O(\unicode[STIX]{x1D70F}_{p})$ truncation of the Maxey–Riley equation for heavy particles. This inertia leads the water droplets to vacate the vicinity of vortices, leaving no nuclei for the vapour to condense. The condensation process is thus spatially inhomogeneous, and leaves vortices in the flow colder than their surroundings. The combination of buoyancy and vorticity generates a lift force on the vortices perpendicular to their velocity relative to the fluid around them. In the case of a vortex dipole, this lift force can propel the vortices towards each other and undergo collapse, a phenomenon studied by Ravichandran et al. (Phys. Rev. Fluids, vol. 2, 2017, 034702). We find, spanning the space of the two time scales, $\unicode[STIX]{x1D70F}_{p}$ and $\unicode[STIX]{x1D70F}_{s}$, the region in which lift-induced dipole collapse can occur, and show numerically that the product of the time scales is the determining parameter. Our findings agree with our results from scaling arguments. We also study the influence of varying the initial supersaturation, and find that the strength of the lift-induced mechanism has a power-law dependence on the phase-change time scale $\unicode[STIX]{x1D70F}_{s}$. We then study systems of many vortices and show that the same coupling between the two time scales alters the dynamics of such systems, by energising the smaller scales. We show that this effect is significantly more pronounced at higher Reynolds numbers. Finally, we discuss how this effect could be relevant in conditions typical of clouds.

In many problems in dynamical systems one is interested in the identification of sets which have qualitatively different fates. The finite-time Lyapunov exponent (FTLE) method is a general and equation-free method that identifies codimension-one sets which have a locally high rate of stretching around which maximal exponential expansion of line elements occurs. These codimension-one sets thus act as transport barriers. This geometric framework of transport barriers is used to study various problems in phase space transport, specifically problems of separation in flows that can vary in scale from the micro to the geophysical. The first problem which we study is of the nontrivial motion of inertial particles in a two-dimensional fluid flow. We use the method of FTLE to identify transport barriers that produce segregation of inertial particles by size. The second problem we study is the long range advective transport of plant pathogen spores in the atmosphere. We compute the FTLE field for isobaric atmospheric flow and identify atmospheric transport barriers (ATBs). We find that rapid temporal changes in the spore concentrations at a sampling point occur due to the passage of these ATBs across the sampling point. We also investigate the theory behind the computation of the FTLE and devise a new method to compute the FTLE which does not rely on the tangent linearization. We do this using the 925 matrix of a probability density function. This method of computing the geometric quantities of stretching and FTLE also heuristically bridge the gap between the geometric and probabilistic methods of studying phase space transport. We show this with two examples.
Ph. D.