2 Overview

Prasath, Vasan & Govindarajan (Reference Prasath, Vasan and Govindarajan2019) give a fundamentally new take on the MR equation by reformulating (1.1) as a boundary condition for a simple diffusion equation $\boldsymbol{q}_{t}=\boldsymbol{q}_{xx}$ for a vectorial quantity $\boldsymbol{q}(x,t)$ , defined over the half-line $x\geqslant 0$ for times $t>0$ . Specifically, they select the initial condition $\boldsymbol{q}(x,0)=\mathbf{0}$ for $x>0$ , and the Dirichlet boundary condition $\boldsymbol{q}(0,t)=\boldsymbol{g}_{0}(t)$ . They then ask what equivalent Neumann boundary condition would lead to the exact same solution to this problem.

The result is that the equivalent Neumann condition, $\boldsymbol{q}_{x}(0,t),$ for this diffusion problem will be exactly of the form of the bracketed last two terms in the MR equation if one introduces the notation $\boldsymbol{q}(0,t)=\dot{\boldsymbol{y}}(t)-\boldsymbol{u}(\boldsymbol{y}(t),t)$ in these terms. The remaining terms in (1.1) can be expressed as functions of $\boldsymbol{q}(0,t)$ and $\boldsymbol{q}_{t}(0,t)$ , and the $\boldsymbol{y}(t)$ position appearing in the arguments of $\boldsymbol{u}$ and $\text{D}\boldsymbol{u}/\text{D}t$ in the equation can be computed on the run as the solution of the ordinary differential equation $\dot{\boldsymbol{y}}(t)=\boldsymbol{q}(0,t)+\boldsymbol{u}(\boldsymbol{y}(t),t)$ .

This insightful observation enables one to solve the classic diffusion equation for $\boldsymbol{q}(x,t)$ , with the MR equation imposed as a Robin-type boundary condition $\boldsymbol{q}_{t}(0,t)+\unicode[STIX]{x1D6FC}\boldsymbol{q}(0,t)-\unicode[STIX]{x1D6FE}\boldsymbol{q}_{x}(0,t)=\boldsymbol{f}(\boldsymbol{q}(0,t),\boldsymbol{y},t)$ for appropriate constants $\unicode[STIX]{x1D6FC}$ , $\unicode[STIX]{x1D6FE}>0$ and a smooth function  $\boldsymbol{f}$ . In this representation, the original non-local problem, along with all complications of the singular kernel and the integral over the past, disappears. This creates an opportunity to use analytical and numerical methods available for classical diffusion problems to study the dynamics of inertial particles.

As a first illustration of the power of this approach, Prasath et al. (Reference Prasath, Vasan and Govindarajan2019) show how particle motion in spatially homogeneous (but still unsteady) flows can be analytically deduced. They derive a formula for $\boldsymbol{q}(0,t)$ which is explicit up to the evaluation of an integral involving the forcing function $\boldsymbol{f}(\boldsymbol{q}(0,t),t)$ . In concrete problems, this integral form can be analysed using Watson’s lemma (Miller Reference Miller2006) to obtain asymptotic expressions for the particle velocity for increasing times.

Specifically, the velocity of a particle relaxing in a non-moving background flow ( $\boldsymbol{f}\equiv \mathbf{0}$ ) is found explicitly to decay to zero at a rate of $O(t^{-3/2})$ , confirming the only previously available analytic solution derived for the MR equation by Provencher-Langlois et al. (Reference Provencher-Langlois, Farazmand and Haller2015). Notably, ignoring the memory term in the MR equation leads to exponential decay. Sedimenting particles, such as aerosols and droplets in the atmosphere, are characterized by $\boldsymbol{f}\equiv (0,\text{const.})$ . A similar analysis yields that they approach a vertical terminal velocity at a rate of $O(t^{-1/2})$ , extending the earlier findings of Clift, Grace & Weber (Reference Clift, Grace and Weber2013) to the case of non-zero initial velocities. Finally, for a particle in an oscillating background flow ( $\boldsymbol{f}\equiv (0,\sin \unicode[STIX]{x1D706}t)$ ), the velocity decay to a time-periodic steady state is surprisingly found to be faster in the presence of the Boussinesq–Basset memory term than without it. This shows that the memory term cannot generally be considered as an additional drag on the particle.

For spatially dependent velocity fields, closed-form solutions are generally not available from this approach. An exception is particle migration in planar Couette flow, for which Prasath et al. (Reference Prasath, Vasan and Govindarajan2019) still derive a new, explicit solution, exploiting the linearity of the velocity in the spatial variable. This solution may serve as an important benchmark for testing numerical schemes for more general, spatially dependent solutions. In all such schemes, the complete absence of a singular kernel in the diffusive reformulation of the MR equation eliminates any need for approximating such a kernel, keeping the memory requirements low relative to other methods. The numerical scheme proposed by the authors reaches spectral accuracy by employing Chebyshev polynomials in the evaluation of the integrals arising from the nonlinear boundary conditions for the underlying diffusion problem. They illustrate the efficacy of this approach on the example of particle motion in a steady, infinite vortex.

3 Future

As many-particle simulations in various media are increasingly popular, the fresh approach of Prasath et al. (Reference Prasath, Vasan and Govindarajan2019) will undoubtedly generate further activity in this field. As they have already demonstrated, their reformulation yields new closed-form solutions, provides rigorous asymptotic estimates for the particle velocities and inspires new numerical methods.

Further testing and refinement of the available numerical scheme is highly desirable for problems involving a large number of particles in spatially complex flows with complicated time dependence. In such a challenging setting, a systematic comparison with prior numerical approaches will also be necessary to carry out, factoring in memory requirements, computational speed and coding complexity. Viewing inertial particle motion with memory as a classic diffusion problem may well have presently unforeseen further advantages, as it opens a new connection to a well-developed area of applied mathematics and computational science.