Continuum model for the transport of active Brownian particles in arbitrary flow field
Date:
Understanding the transport of Stokesian swimmers such as motile microorganisms and artificial swimmers in complex flow is vital to many ecological and industrial applications. In particular, it is important to capture the two-way interactions between the flow and the swimmers. While individual-based modelling is of great merit in capturing these interactions and remains a vital avenue to test theories, continuum modelling is also as important in enabling a deep theoretical un- derstanding of the suspension dynamics. There were numerous examples, from the instability of pusher suspension [1] to the modelling of bioconvection [2] and gyrotactic focusing [3], where a continuum model has made deep insight into suspension dynamics possible. This is partly attributed to the wide range of classical fluid mechanical tools made available by the continuum modelling of these suspensions.
However, a key challenge in the continuum modelling for active suspension is that the transport of swimmers is coupled with the orientation, while the orientation is determined by the local flow field. Therefore, accurately modelling the swimmer phase as a continuum requires solving a variable in both positional and orientational space (see eq. 1). When coupled with the flow equation, the direct numerical simulation of the equations would be too expensive unless some simplification is made. This talk will discuss how we can reduce such a high-dimensional model into a lower-dimension transport equation while maintaining the coupling between the orientation-dependent transport and the local flow field. In many applications, the suspension is dilute, and the flow is at a larger lengthscale than the swimmers. Therefore, we will employ the framework of active Brownian particles under a dilute assumption, where we assume the swimmers are small enough such that they only experience the flow as a linear flow field and far apart enough that they do not interact directly. To model a suspension as a continuum, one can employ the Fokker-Planck equation
\[\partial_t \Psi + \nabla_x \cdot [\dot{\mathbf{x}} \Psi] + \nabla_p \cdot [\dot{\mathbf{p}} \Psi] = D_r \nabla^2_p \Psi + D_T \nabla_x^2 \Psi\]based on the spatial \(\dot{\mathbf{x}}\) and orientational \(\dot{\mathbf{p}}\) trajectories of swimmers. Here, \(\Psi(\mathbf{x},\mathbf{p},t)\) is the probability density function of finding a swimmer in position \({\mathbf{x}}\) and orientation \({\mathbf{p}}\) at time \(t\), \(D_T\) the translational diffusivity and \(D_r\) the rotational diffusivity. Then, one can couple the mean-field hydrodynamic contribution from the swimmers with the Navier-Stokes equation governing the flow field to study the coupling between the swimmers and the flow.
However, as mentioned, the high-dimensionality of the Fokker-Planck equation may deem the DNS of these equations almost impossible. While reducing the equation is possible, past macro-transport models are either inaccurate [4] or not generalised enough [5] to be applied in any arbitrary flow fields or turbulent flow. We will present a new local approximation model that gives the effective transport as a function of the local flow field and swimming. It overcomes the limitation of the past models while giving effective transport properties as a function of the local flow field more accurately. This framework can then be coupled with the Navier-Stokes equation to study the interactions between the swimmers and the flow field. It can also be extended to estimate the effective sedimentation and dispersion of long non-motile planktons, whose orientation can affect their sedimentation rate.
References
[1] Saintillan,D. & Shelley, M.J. Instabilities and pattern formation inactive particle suspensions, Phys.Rev.Lett. 100 (17), 178103 (2008).
[2] Kessler, J. & Pedley, T. Hydrodynamic Phenomena in Suspensions Of Swimming Microorganisms, Annu. Rev. Fluid Mech. 24, 313–358 (1992).
[3] Fung, L., Bearon, R., & Hwang, Y. Bifurcation and stability of downflowing gyrotactic micro-organism suspensions in a vertical pipe, J. Fluid Mech. 902, A26 (2020).
[4] Pedley, T. & Kessler J. A new continuum model for suspensions of gyrotactic micro-organisms, J. Fluid Mech. 212, 155–182 (1990).
[5] Hill, N. & Bees, M. Taylor dispersion of gyrotactic swimming micro-organisms in a linear flow, Phys. Fluids. 14 (8), 2598—2605 (2002).