5.2
Brownian motion in particulate suspensions.
Brownian motion is the random motion exhibited by micron and sub-micron particles im- mersed in liquid. It is a fundamental mechanism for material and chemical transport in micron-scale physical and biological systems (Grima et al.(2010)), and can play a key role in determining the mechanical response of colloidal suspensions to applied stresses (Batch- elor(1977);Bossis and Brady(1989);Foss and Brady(2000);Banchio and Brady(2003)). Brownian motion is also known to affect the aggregation and self-assembly of interacting particles (Anderson and Lekkerkerker(2002);Zaccarelli(2007);Lu et al.(2008)), a funda- mental process important in many engineering applications that utilize colloidal particles to tune rheological properties of fluids (Mabille et al. (2000); ten Brinke et al. (2007)) and construct new materials and devices (Whitesides and Boncheva(2002);Glotzer et al.
(2004);Promislow et al. (1995)). The rotational diffusivity, and therefore the motility, of micron-sized bacteria is strongly influenced by thermal agitation (Drescher et al.(2011)). Brownian motion also plays an important role in the transport and mixing properties of active systems (Leptos et al. (2009);Kurtuldu et al.(2011); Mi˜no et al.(2013)). The dif- fusion of a colloid or a molecule in an active suspension depends on the intricate coupling between its Brownian diffusivity and the suspension dynamics (Kasyap et al.(2014)). To address the issue of enhanced Brownian particle diffusion in active suspensions one needs to include Brownian motion. More importantly, in the semi-dilute and concen- trated regimes, the effect of hydrodynamic interactions is significant and particle rigidity constraints (i.e. stresslets) must be enforced. Therefore we need a tool which efficiently couples thermal fluctuations with an accurate description of hydrodynamic interactions within the same framework.
In Chapter7we detail all the technical difficulties arising when incorporating Brownian motion in particulate suspension and develop a new scheme, the Drifter-Corrector, which efficiently address these issues in the framework of the fluctuating force-coupling method (Keaveny (2014)).
Chapter 6
Modeling active suspensions with
the force-coupling method.
Contents
6.1 Solving hydrodynamic interactions with the force-coupling method. . . 84 6.1.1 FCM for passive suspensions . . . 84 6.1.2 Squirmer interactions and motion. . . 85 6.1.3 Notations . . . 86 6.2 Numerical tools for High Performance Computing . . . 86 6.2.1 Fluid solver . . . 86 6.2.2 Computational work . . . 86 6.2.3 Steric interactions . . . 87 6.2.4 Algorithm . . . 89 6.2.5 Including additional features . . . 90 6.3 Validations . . . 90 6.3.1 Interactions between a squirmer and an inert sphere . . . 90 6.3.2 Trajectories of two interacting squirmers . . . 90 This chapter presents a development of the force-coupling method (FCM) to address the accurate simulation of a large number of interacting micro-swimmers. Our approach is based on the squirmer model described in Chapter3, which we adapt to the FCM framework in the context of High Performance Computing. First, we detail the parallel implementa- tion of the method. Then the scalability of the code is evaluated and results for pairwise interactions are compared with the literature.
Chapter 6 : Modeling active suspensions with the force-coupling method.
6.1
Solving hydrodynamic interactions with the force-
coupling method
The FCM framework for particle suspensions is a straightforward extension of the isolated case presented in Section 3.1.1.
6.1.1
FCM for passive suspensions
Consider a suspension of Np rigid spherical particles, each having radius a. Each particle
n, (n = 1, . . . , Np), is centered at Ynand subject to force Fnand torque τn. To determine
their motion through the surrounding fluid, we first represent each particle by a low order, finite-force multipole expansion in the Stokes equations
∇p − η∇2u = X n Fn∆n(x) + 1 2τ n× ∇Θ n(x) + Sn· ∇Θn(x) ∇ · u = 0. (6.1)
In Eq. (6.1), Sn are the particle stresslets determined through a constraint on the local
rate-of-strain as described below. Also in Eq. (6.1) are the two Gaussian envelopes, ∆n(x) = (2πσ∆2) −3/2 e−|x−Yn|2/2σ∆2 Θn(x) = (2πσΘ2) −3/2 e−|x−Yn|2/2σ2Θ, (6.2)
used to project the particle forces onto the fluid.
Contrary to the isolated case, no analytical solution can be derived for the fluid flow u. A numerical solver is thus necessary. It could be of any type. In Section 6.2.1 we provide more details about the fluid solver we use throughout the thesis.
After solving Eq. (6.1), the velocity, Vn, angular velocity, Ωn, and local rate-of-strain,
En, of each particle n are found by volume averaging of the resulting fluid flow,
Vn = ˆ u∆n(x)d3x (6.3) Ωn = 1 2 ˆ [∇ × u] Θn(x)d3x, (6.4) En = 1 2 ˆ ∇u + (∇u)T Θ n(x)d3x, (6.5)
where the integration is performed over R3. In order for Eqs. (6.3) – (6.5) to recover the
correct mobility relations for a single, isolated sphere, namely that V = F/(6πaη) and Ω = τ /(8πa3η), the envelope length scales need to be σ
∆= a/
√
π and σΘ = a/ (6
√ π)1/3. As the particles are rigid, the stresslets are found by enforcing the constraint that En = 0
6.1 Solving hydrodynamic interactions with the force-coupling method
6.1.2
Squirmer interactions and motion
Using FCM, the task of computing the interactions between squirmers is relatively straight- forward. We now consider Np independent squirmers with positions Yn and orientations
pn. Each squirmer has swimming dipole
Gn= 4 3πηa
2(3pnpn− I) B
2, (6.6)
and degenerate quadrupole Hn= −4
3πηa
3B
1pn. (6.7)
The squirmers may also be subject to external forces Fn and torques τn. Using the
linearity of the Stokes equations we obtain ∇p − η∇2u = X n Fn∆n(x) + 1 2τ n× ∇Θ n(x) + Sn· ∇Θn(x) +Gn· ∇∆n(x) + Hn∇2Θn(x) (6.8) ∇ · u = 0 (6.9)
for the flow field generated by the suspension.
After finding the flow field, we determine the motion of the squirmers using Eqs. (6.3) – (6.5) with two modifications. First, we need to add the swimming velocity, U pn, to
Eq. (6.3). Second, we must subtract the artificial, self-induced velocity and the local rate-of-strain due to the squirming modes. The self-induced velocity is given by
Wn= ˆ
A · Hn∆(x)d3x (6.10)
where the tensor A is given in Eq. (3.23). The self-induced rate-of-strain is given by
Kn= ˆ
1
2 ∇R · G
n+ (∇R · Gn)T Θ(x)d3x (6.11)
where the expression for the third rank tensor R can be found in Eq. (3.24).
Taking these self-induced effects into account, the motion of a squirmer n is given by Vn = U pn− Wn+ ˆ u∆n(x)d3x (6.12) Ωn = 1 2 ˆ [∇ × u] Θn(x)d3x (6.13) En = −Kn+1 2 ˆ ∇u + (∇u)T Θ n(x)d3x. (6.14)
As for passive particles, the stresslets Sn due to squirmer rigidity are obtained from the
usual constraint on the local rate-of-strain, namely En= 0 for all n. Squirmer positions
Yn and orientations pn are then updated with the Lagrangian equations dYn
dt = V
Chapter 6 : Modeling active suspensions with the force-coupling method.
dpn
dt = Ω
n× pn. (6.16)
In Section 6.3, we show through a comparison with the boundary element simulations fromIshikawa et al.(2006) that our FCM squirmer model recovers the velocities, angular velocities, stresslets (Sn) and trajectories for two interacting squirmers for a wide range of separations.
6.1.3
Notations
In the following, we denote Y, P, V and W the 3Np vectors containing the particle
positions, orientations, translational and rotational velocities. E and S are 6Np and
contain the particle rate-of-strains and stresslets respectively. F , T are the 3Np forces
and torques. H is a 3Npvector containing the degenerate quadrupoles and G is 6Npvector
containing the swimming stresslets.