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maxr - Third-order integrator for the Maxey-Riley equations, in Python

author: @jesserobertson

What happen?!

This uses a third-order (in time) scheme from Daitsche (2013). The equations solved are a rewritten version of the Maxey-Riley equations, sans the Faxen correction terms.

$$ w(t + \delta t) = w(t) + \int_t^{t+\delta t}G(\tau)d\tau + H(t + \delta t) - H(t) $$

and

$$ \frac{dr}{dt} = w + u $$

where $r$ is the particle location, $w$ is the difference between the particle velocity $v$ and the fluid velocity $u$, $G(t)$ is the 'velocity term' (the part of the Maxey-Riley equation corresponding to the instantaneous effects of the local fluid flow), and $H(t)$ is the 'history term'. The velocity term is given by

$$ G(t) = (R - 1)\frac{du}{dt} - Rw\cdot\nabla u - \frac{R}{S}w $$

and the history term by

$$ H(t) = -R\sqrt{\frac{3}{\pi S}}\int_{0}^{t+\delta t}\frac{w(\tau)}{\sqrt{t - \tau}}d\tau $$

and $R=3\rho_f/(\rho_f + 2\rho_p)$ is a density ratio, and $S=a^2/3\nu T$ is the relaxation timescale.

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Third-order integrator for the Maxey-Riley equations, in Python

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