Accurate solution method for the Maxey–Riley equation, and the effects of Basset history

Abstract

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 stretchedexponential at short time while slowing it down to a power-law relaxation, ∼ t −3/2 , at long time. A particle settling under gravity is shown to relax even more slowly to its terminal velocity (∼ 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 rain drops 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 rain drop can be gross overestimated when history effects are not accounted. 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.

Drops in the wind: their dispersion and COVID-19 implications

“…This is a rather surprising result, since the orderof-magnitude of the B-B term is really small. This t −1/2 decay of the sedimenting velocity has been also recently reported 17 . Further consequences of the B-B term on the motion of particles at low Reynolds numbers may be search for in future investigations.…”

Ultrasonic propagation in concentrated colloidal dispersions: improvements in a hydrodynamic model

“…Richardson and Zaki [25] determined the Stokes correction factor to be 1 − .,-, as we will demonstrate, this will serve our purpose to model attenuation effectively. Although the history/Basset force for a single sphere has been investigated by a number of authors, [24,[26][27][28] an expression to account for the concentration dependency of the force is not known and most probably does not exist in the literature. Since the effect of Basset force with respect to that of the Stokes' force or inertial force is smaller, one can safely use the isolated-sphere form of the history force.…”

Solving the inertial particle equation with memory

“…As a first illustration of the power of this approach, Prasath et al (2019) show how particle motion in spatially homogeneous (but still unsteady) flows can be analytically deduced. They derive a formula for q(0, t) which is explicit up to the evaluation of an integral involving the forcing function f (q(0, t), t).…”

“…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 (2019) 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.…”

“…As many-particle simulations in various media are increasingly popular, the fresh approach of Prasath et al (2019) 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.…”