Solving the inertial particle equation with memory

Abstract

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.

The waltz of tiny droplets and the flow they live in

“…We then complicate our own argument by showing that the omission of the Basset-Boussinesq history, while expedient, may not be justifiable [36]. We have seen that droplet growth by gravitational settling is seriously overestimated if BBH is neglected, since the manner in which terminal velocity is attained by a falling droplet is algebraic rather than exponential.…”

“…We notice that the next term in the power series of St is the Basset-Boussinesq history (BBH) force, being a half-power of Stokes smaller than the Stokes drag term. This term is usually neglected, not because we are convinced about its smallness of magnitude, but simply because it is too hard to compute, if one were to follow the naive approach of performing the integral at every instant of time for every particle [36]. It is clear that this approach is not feasible for any large flow.…”

A Python toolbox for the numerical solution of the Maxey‐Riley equation

“…The reformulation of the MRE into an initial-boundary-value problem allows to use analytical methods for classical diffusion problems [4] instead of dealing with the memory term in direct integration of (1) as is done for example by Daitche [8]. Using Fokas' method [13], Prasath et al obtain the solution…”

“…The motion of inertial particles in a fluid is a fundamental problem of fluid dynamics that appears in many areas, for example spread of COVID-19 virus in the air [1], rain drop formation in clouds [2], settlement of plankton in the ocean [3] and more [4]. The Maxey-Riley equation (MRE) [5] has become an accepted model, at least for particles up to a certain size [6].…”

Nonlinear dynamics of inertial particles in the ocean: From drifters and floats to marine debris and Sargassum

“…It has been also noted [22] that it mainly tends to slow down the inertial particle motion without changing its qualitative dynamics fundamentally. However, the effects of the memory terms remain the subject of active research [34,66,70]. We have also ignored so-called Faxen corrections (terms of the form a 2 ∇ 2 v f ) in the added mass and drag forces; this is much easier to justify.…”