Skip to main content Accessibility help
Hostname: page-component-cf9d5c678-dkwk2 Total loading time: 0.549 Render date: 2021-08-02T12:21:53.820Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Solving the inertial particle equation with memory

Published online by Cambridge University Press:  03 July 2019

Rights & Permissions[Opens in a new window]


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.

Focus on Fluids
© 2019 Cambridge University Press 

1 Introduction

The motion of finite-size (or inertial) particles immersed in a fluid flow can differ significantly from that of fluid elements. This leads to clustering or scattering effects that cannot be explained using the fluid velocity field alone. The resulting dynamical phenomena are important in a number of areas, including rain drop formation in clouds (Pinsky & Khain Reference Pinsky and Khain1997), sinking of biogenic particles in the ocean (Monroy et al. Reference Monroy, Hernández-García, Rossi and López2017), transport of volcanic ash in the atmosphere (Espinosa Arenal, Avila & Raza Reference Espinosa Arenal, Avila and Raza2018), ocean-drifter motion (Beron-Vera et al. Reference Beron-Vera, Olascoaga, Haller, Farazmand, Triñanes and Wang2015) and cell collection in oncological studies (Haddadi, Naghsh-Nilchi & Di Carlo Reference Haddadi, Naghsh-Nilchi and Di Carlo2018), just to name a few.

Most related studies assume that the particles are spherical, do not interact with each other, their associated Stokes and Reynolds numbers are small, the effect of gravity is negligible and the particles do not modify the ambient flow whose velocity field is $\boldsymbol{u}(\boldsymbol{x},t)$ . Under these assumptions, the equation of motion for inertial particles simplifies from its general form derived by Maxey & Riley (Reference Maxey and Riley1983) to

(1.1) $$\begin{eqnarray}\displaystyle R\ddot{\boldsymbol{y}} & = & \displaystyle \frac{\text{D}\boldsymbol{u}(\boldsymbol{y}(t),t)}{\text{D}t}-\frac{1}{S}(\dot{\boldsymbol{y}}(t)-\boldsymbol{u}(\boldsymbol{y}(t),t))\nonumber\\ \displaystyle & & \displaystyle -\,\sqrt{\frac{3}{S}}\left\{\frac{\dot{\boldsymbol{y}}(0)-\boldsymbol{u}(\boldsymbol{y}(0),0)}{\sqrt{\unicode[STIX]{x03C0}t}}+\frac{1}{\sqrt{\unicode[STIX]{x03C0}}}\int _{0}^{t}\frac{\dot{\boldsymbol{y}}(s)-\boldsymbol{u}(\boldsymbol{y}(s),s)}{\sqrt{t-s}}\,\text{d}s\right\},\end{eqnarray}$$

where $\text{D}/\text{D}t\doteq \unicode[STIX]{x2202}_{t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}$ is the material derivative along fluid trajectories; $R$ is a dimensionless number depending on the ratio of the particle and fluid densities and $S$ is the Stokes number. The simplified Maxey–Riley (MR) equation given above is a second-order, nonlinear system of implicit integro-differential equations with a singular kernel and with an initial-condition term that blows up as $t\rightarrow 0$ . The implicit nature of the equation arises from its last term, the Boussinesq–Basset memory term, which accounts for boundary layer development around the moving particle and renders the equation highly non-trivial to solve. Classical solutions to the MR equation turn out to exist only under the unphysical assumption that the initial particle velocity is exactly equal to the local fluid velocity. In all other cases, only mild solutions can be shown to exist (Farazmand & Haller Reference Farazmand and Haller2014; Provencher-Langlois, Farazmand & Haller Reference Provencher-Langlois, Farazmand and Haller2015). More broadly speaking, the MR equation is not a dynamical system, as the present state does not uniquely determine the future evolution of the inertial particle. Rather, the full history of particle positions and velocities is required for the solution.

Because of these subtleties, the memory term in the MR equation is often omitted in applications (Michaelides Reference Michaelides1997). At the same time, growing numerical and experimental evidence for its significance (Candelier, Angilella & Souhar Reference Candelier, Angilella and Souhar2004; Toegel, Luther & Lohse Reference Toegel, Luther and Lohse2006; Garbin et al. Reference Garbin, Dollet, Overvelde, Cojoc, Di Fabrizio, van Wijngaarden, Prosperetti, de Jong, Lohse and Versluis2009; Daitche & Tél Reference Daitche and Tél2011; Guseva, Feudel & Tél Reference Guseva, Feudel and Tél2013; Daitche & Tél Reference Daitche and Tél2014) has inspired an ongoing effort to handle this term appropriately in numerical schemes. The main challenge to overcome is the increasing memory cost in many-particle computations, given that the full history of each particle needs to be stored to advance the numerical scheme. To address this challenge, Van Hinsberg, Ten Thije Boonkkamp & Clercx (Reference Van Hinsberg, Ten Thije Boonkkamp and Clercx2011), Klinkenberg, de Lange & Brandt (Reference Klinkenberg, de Lange and Brandt2014), Elghannay & Tafti (Reference Elghannay and Tafti2016) and Parmar et al. (Reference Parmar, Annamalai, Balachandar and Prosperetti2018) have proposed approximations or modifications to the memory kernel, whereas Daitche (Reference Daitche2013) developed higher-order schemes to handle the singularity of the kernel. All these advances, however, involve modifications to the original problem, and assume the initial particle velocity to be equal the flow velocity (i.e. $\dot{\boldsymbol{y}}(0)=\boldsymbol{u}(\boldsymbol{y}(0),0)$ ), which is unlikely in practical applications.

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.


Beron-Vera, F. J., Olascoaga, M. J., Haller, G., Farazmand, M., Triñanes, J. & Wang, Y. 2015 Dissipative inertial transport patterns near coherent Lagrangian eddies in the ocean. Chaos 25, 087412.CrossRefGoogle ScholarPubMed
Candelier, F., Angilella, J. R. & Souhar, M. 2004 On the effect of the Boussinesq Basset force on the radial migration of a Stokes particle in a vortex. Phys. Fluids 16 (5), 17651776.CrossRefGoogle Scholar
Clift, R., Grace, J. R. & Weber, M. E. 2013 Bubbles, Drops, and Particles. Dover.Google Scholar
Daitche, A. 2013 Advection of inertial particles in the presence of the history force: higher order numerical schemes. J. Comput. Phys. 54, 93106.CrossRefGoogle Scholar
Daitche, A. & Tél, T. 2011 Memory effects are relevant for chaotic advection of inertial particles. Phys. Rev. Lett. 107, 244501.CrossRefGoogle ScholarPubMed
Daitche, A. & Tél, T. 2014 Memory effects in chaotic advection of inertial particles. New J. Phys. 16 (7), 073008.Google Scholar
Elghannay, H. A. & Tafti, D. K. 2016 Development and validation of a reduced order history force model. Intl J. Multiphase Flow 85, 284297.CrossRefGoogle Scholar
Espinosa Arenal, F., Avila, R. & Raza, S. S. 2018 A new method of simulating volcanic eruption column formation and dispersion of ejected ash clouds. Meteorol. Appl. 25, 151160.CrossRefGoogle Scholar
Farazmand, M. & Haller, G. 2014 The Maxey–Riley equation: existence, uniqueness and regularity of solutions. Nonlinear Anal. 22, 98106.CrossRefGoogle Scholar
Garbin, V., Dollet, B., Overvelde, M., Cojoc, D., Di Fabrizio, E., van Wijngaarden, L., Prosperetti, A., de Jong, N., Lohse, D. & Versluis, M. 2009 History force on coated microbubbles propelled by ultrasound. Phys. Fluids 21 (9), 092003.CrossRefGoogle Scholar
Guseva, K., Feudel, U. & Tél, T. 2013 Influence of the history force on inertial particle advection: gravitational effects and horizontal diffusion. Phys. Rev. E 88 (4), 042909.Google ScholarPubMed
Haddadi, H., Naghsh-Nilchi, H. & Di Carlo, D. 2018 Separation of cancer cells using vortical microfluidic flows. Biomicrofluidics 12, 014112.CrossRefGoogle ScholarPubMed
Klinkenberg, J., de Lange, H. C. & Brandt, L. 2014 Linear stability of particle laden flows: the influence of added mass, fluid acceleration and Basset history force. Meccanica 49 (4), 811827.CrossRefGoogle Scholar
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26, 883889.CrossRefGoogle Scholar
Michaelides, E. E. 1997 The transient equation of motion for particles, bubbles, and droplets. Trans. ASME J. Fluids Engng 119, 233247.CrossRefGoogle Scholar
Miller, P. D. 2006 Applied Asymptotic Analysis. AMS.CrossRefGoogle Scholar
Monroy, P., Hernández-García, E., Rossi, V. & López, C. 2017 Modeling the dynamical sinking of biogenic particles in oceanic flow. Nonlinear Process. Geophys. 24, 293305.CrossRefGoogle Scholar
Parmar, M., Annamalai, S., Balachandar, S. & Prosperetti, A. 2018 Differential formulation of the viscous history force on a particle for efficient and accurate computation. J. Fluid Mech. 844, 970993.CrossRefGoogle Scholar
Pinsky, M. & Khain, A. 1997 Turbulence effects on droplets growth and size distribution in clouds a review. J. Aero. Sci. 28, 11771214.CrossRefGoogle Scholar
Prasath, S. G., Vasan, V. & Govindarajan, R. 2019 Accurate solution method for the Maxey–Riley equation, and the effects of Basset history. J. Fluid. Mech. 868, 428460.CrossRefGoogle Scholar
Provencher-Langlois, G., Farazmand, M. & Haller, G. 2015 Asymptotic dynamics of inertial particles with memory. J. Nonlinear Sci. 25, 12251255.CrossRefGoogle Scholar
Toegel, R., Luther, S. & Lohse, D. 2006 Viscosity destabilizes sonoluminescing bubbles. Phys. Rev. Lett. 96 (11), 114301.CrossRefGoogle ScholarPubMed
Van Hinsberg, M. A. T., Ten Thije Boonkkamp, J. H. M. & Clercx, H. J. 2011 An efficient, second order method for the approximation of the Basset history force. J. Comput. Phys. 230 (4), 14651478.CrossRefGoogle Scholar
You have Access
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Solving the inertial particle equation with memory
Available formats

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Solving the inertial particle equation with memory
Available formats

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Solving the inertial particle equation with memory
Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *