A. Faxén laws and connection to Stokesian dynamics

Consider an inert (non-active) sphere of radius <sup>R</sup> in an ambient flow field <sup>u</sup><sup>∞</sup>ð Þ <sup>x</sup> . The sphere has translational velocity U and rotational velocity Ω. The flow field u can be written as u xð Þ¼ <sup>u</sup><sup>∞</sup>ð Þþ <sup>x</sup> <sup>u</sup><sup>D</sup>ð Þ <sup>x</sup> , where <sup>u</sup><sup>D</sup>ð Þ <sup>x</sup> is the velocity disturbance created by the presence of the sphere. The boundary condition for <sup>u</sup><sup>D</sup>ð Þ <sup>x</sup> on the sphere surface S is

$$\mathbf{u}^{D}(\mathbf{x}\_{\mathfrak{s}}) = \mathbf{U} + \mathfrak{Q} \times (\mathbf{x}\_{\mathfrak{s}} - \mathbf{x}\_{0}) - \mathbf{u}^{\infty}(\mathbf{x}\_{\mathfrak{s}}), \ \mathbf{x}\_{\mathfrak{s}} \in \mathcal{S}. \tag{69}$$

Additionally, <sup>u</sup><sup>D</sup>ð Þ! <sup>j</sup>xj ! <sup>∞</sup> 0. Taking the sphere position to be <sup>x</sup><sup>0</sup> <sup>¼</sup> 0, we can expand the ambient flow field around the sphere center as:

$$u\_i^{\infty}(\mathbf{x}) = u\_i^{\infty}(\mathbf{0}) + \left. \frac{\partial u\_i^{\infty}}{\partial \mathbf{x}\_j} \right|\_{\mathbf{x} = \mathbf{0}} \propto\_j + \frac{1}{2} \frac{\partial^2 u\_i^{\infty}}{\partial \mathbf{x}\_j \partial \mathbf{x}\_k} \bigg|\_{\mathbf{x} = \mathbf{0}} \propto\_j \mathbf{x}\_k + \dots \tag{70}$$

Now we recall the definitions of the (symmetric) rate of strain tensor eij,

$$\mathfrak{e}\_{ij} = \frac{1}{2} \left( \frac{\partial u\_i}{\partial \mathbf{x}\_j} + \frac{\partial u\_j}{\partial \mathbf{x}\_i} \right), \tag{71}$$

with <sup>u</sup><sup>∞</sup>ð Þ¼� <sup>x</sup><sup>s</sup> <sup>v</sup>sð Þ <sup>x</sup><sup>s</sup> on <sup>S</sup>? Constructing an effective flow field would allow us to obtain Fdrag, τdrag, and S by Faxén laws, without having to solve the complete hydrodynamic problem posed in Section II. Moreover, an understanding of how <sup>v</sup>sð Þ <sup>x</sup> determines <sup>F</sup>drag, <sup>τ</sup>drag, and <sup>S</sup> would pave the way towards development of a hybrid BEM-Stokesian Dynamics scheme, since these quantities are central to SD.

> <sup>i</sup> ð Þþ <sup>0</sup> <sup>∂</sup>u<sup>∞</sup> i ∂xj � � � � x¼0

<sup>i</sup> ð Þ 0 dS þ

4πR<sup>2</sup>

4πR<sup>2</sup>

ð vs

This equation is one of the major results obtained in Ref. 37 by use of the Lorentz reciprocal theorem. However, our rederivation and interpretation in terms of an effective ambient flow field u<sup>∞</sup> is (to our knowledge) novel. To obtain the vorticity associated with u<sup>∞</sup>, we multiply Eq. 80 by εlmixm and integrate over the sphere

> ð ∂u<sup>∞</sup> i ∂xj � � � � x¼0

<sup>3</sup> <sup>ε</sup>lji

� � � � x¼0

∂u<sup>∞</sup> i ∂xj � � � � x¼0

> <sup>¼</sup> <sup>8</sup>πR<sup>4</sup> 3

4πR<sup>2</sup>

<sup>i</sup> ð Þ 0 εlmi xm dS þ

The first integral on the right hand side of Eq. 85 vanishes. For the second

xm xj dS <sup>¼</sup> <sup>4</sup>πR<sup>4</sup>

<sup>i</sup> ð Þ <sup>x</sup> dS <sup>¼</sup> <sup>4</sup>πR<sup>4</sup>

<sup>∇</sup> � <sup>u</sup><sup>∞</sup>

ð vs

ð vs

<sup>i</sup> ð Þ 0 , we integrate both sides of Eq. 80 over the surface of the sphere:

ð ∂u<sup>∞</sup> i ∂xj � � � � x¼0

<sup>i</sup> ð Þ x on the surface of the sphere as �vsð Þ x<sup>s</sup> . The second integral

ð Þ x<sup>s</sup> dS � U

� �: (83)

xj: (80)

xj dS: (81)

ð Þ x<sup>s</sup> dS: (82)

ð Þ x<sup>s</sup> dS: (84)

<sup>3</sup> <sup>δ</sup>jm: (86)

εlmi xm xj dS: (85)

<sup>ω</sup><sup>∞</sup>ð Þ <sup>0</sup> , (88)

(87)

As our starting point, we write the Taylor expansion of <sup>u</sup><sup>∞</sup>ð Þ <sup>x</sup> :

<sup>i</sup> ð Þ¼ <sup>x</sup> <sup>u</sup><sup>∞</sup>

ð u<sup>∞</sup>

<sup>u</sup><sup>∞</sup>ð Þ¼� <sup>0</sup> <sup>1</sup>

If we consider a force-free swimmer, <sup>F</sup>drag <sup>¼</sup> 0, giving the result:

<sup>U</sup> ¼ � <sup>1</sup>

<sup>F</sup>drag <sup>¼</sup> <sup>6</sup>πμ<sup>R</sup> � <sup>1</sup>

u∞

The Boundary Element Method for Fluctuating Active Colloids

DOI: http://dx.doi.org/10.5772/intechopen.86738

<sup>i</sup> ð Þ x dS ¼

ð u∞

on the right hand side of Eq. 81 vanishes, giving

To obtain u<sup>∞</sup>

We identify u<sup>∞</sup>

surface:

ð

We obtain:

69

εlmixmu<sup>∞</sup>

<sup>i</sup> ð Þ x dS ¼

ð u<sup>∞</sup>

ð

εlmi xm u<sup>∞</sup>

ð Þ <sup>x</sup><sup>s</sup> dS <sup>¼</sup> <sup>4</sup>πR<sup>4</sup>

3

integral on the right hand side, we use the identity

ð

� ð <sup>x</sup> � <sup>v</sup><sup>s</sup>

Using Eq. 75, we obtain

and the (anti-symmetric) vorticity tensor

$$\mathcal{W}\_{\vec{\eta}} = \frac{1}{2} \left( \frac{\partial u\_i}{\partial \mathbf{x}\_j} - \frac{\partial u\_j}{\partial \mathbf{x}\_i} \right), \tag{72}$$

The vorticity tensor can be related to the vorticity vector w ¼ ∇ � u by

$$\mathbf{W} = \frac{1}{2}\mathbf{e} \cdot \mathbf{w}.\tag{73}$$

Here, ε is the Levi-Civita tensor. The first derivative in Eq. 70 can be decomposed into symmetric and anti-symmetric contributions:

$$\frac{\partial \boldsymbol{u}\_{i}^{\infty}}{\partial \boldsymbol{x}\_{j}} = \boldsymbol{e}\_{ij}^{\infty} + \boldsymbol{W}\_{ij}^{\infty}. \tag{74}$$

Using the Lorentz reciprocal theorem, one can obtain Faxén's law for the drag force on the sphere (see Ref. [59] for details):

$$\mathbf{F}^{\text{drag}} = 6\pi\mu R \left[ \left( \mathbf{1} + \frac{R^2}{6} \nabla^2 \right) \mathbf{u}^{\text{oc}}(\mathbf{x}\_0) - \mathbf{U} \right]. \tag{75}$$

(In our shorthand notation, the Laplacian is first applied to <sup>u</sup><sup>∞</sup>ð Þ <sup>x</sup> and then evaluated at x0.) One can also obtain Faxén law for the drag torque:

$$
\pi^{\text{drag}} = 8\pi\mu R^3 (\boldsymbol{\omega}^{\infty}(\mathbf{x}\_0) - \boldsymbol{\Omega}),
\tag{76}
$$

where the angular velocity of the fluid <sup>ω</sup> � <sup>1</sup> <sup>2</sup> w. Finally, there is a Faxén law for the stresslet [59, 83, 84]

$$\mathbf{S} = \frac{20}{3}\pi\mu R^3 \left(1 + \frac{R^2}{10}\nabla^2\right) \mathbf{e}^{\infty}(\mathbf{x}\_0),\tag{77}$$

where S is defined as an integral over the particle surface

$$\mathbf{S}\_{ij} = \int \left[ \frac{\mathbf{1}}{2} \left( \mathbf{x}\_{\vec{j}} \sigma\_{ik} n\_k + \mathbf{x}\_i \sigma\_{jk} n\_k \right) - \frac{\mathbf{1}}{3} (\mathbf{x}\_k \sigma\_{kl} n\_l) \delta\_{ij} - \mu \left( \mathbf{u}\_i n\_j + \mathbf{u}\_j n\_i \right) \right] d\mathbf{S}. \tag{78}$$

So far we have only presented standard results, but now we raise the following question. Consider an active sphere with a slip velocity vsð Þ x . Comparing the boundary conditions in Eq. 3 and Eq. 69, can we construct an ambient linear flow field

$$\mathbf{u}^{\infty}(\mathbf{x}) = \mathbf{u}^{\infty}(\mathbf{0}) + \mathbf{e}^{\infty} \cdot \mathbf{x} + \mathbf{o}^{\infty} \times \mathbf{x} \tag{79}$$
