The Boundary Element Method for Fluctuating Active Colloids DOI: http://dx.doi.org/10.5772/intechopen.86738

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.

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

$$u\_i^{\infty}(\mathbf{x}) = u\_i^{\infty}(\mathbf{0}) + \left. \frac{\partial u\_i^{\infty}}{\partial \mathbf{x}\_j} \right|\_{\mathbf{x} = \mathbf{0}} \propto\_j. \tag{80}$$

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

$$\int u\_i^{\infty}(\mathbf{x})d\mathbf{S} = \int u\_i^{\infty}(\mathbf{0})d\mathbf{S} + \left. \int \frac{\partial u\_i^{\infty}}{\partial \mathbf{x}\_j} \right|\_{\mathbf{x}=\mathbf{0}} \mathbf{x}\_j d\mathbf{S}.\tag{81}$$

We identify u<sup>∞</sup> <sup>i</sup> ð Þ x on the surface of the sphere as �vsð Þ x<sup>s</sup> . The second integral on the right hand side of Eq. 81 vanishes, giving

$$\mathbf{u}^{\infty}(\mathbf{0}) = -\frac{1}{4\pi R^2} \left[ \mathbf{v}'(\mathbf{x}\_i) \, \text{d}\mathbf{S}.\tag{82}$$

Using Eq. 75, we obtain

u<sup>∞</sup>

Non-Equilibrium Particle Dynamics

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

and the (anti-symmetric) vorticity tensor

force on the sphere (see Ref. [59] for details):

where the angular velocity of the fluid <sup>ω</sup> � <sup>1</sup>

<sup>2</sup> xjσiknk <sup>þ</sup> xiσjknk � � � <sup>1</sup>

<sup>S</sup> <sup>¼</sup> <sup>20</sup>

where S is defined as an integral over the particle surface

the stresslet [59, 83, 84]

Sij ¼

flow field

68

ð 1

<sup>i</sup> ð Þþ <sup>0</sup> <sup>∂</sup>u<sup>∞</sup> i ∂xj � � � � x¼0 xj þ 1 2 ∂2 u<sup>∞</sup> i ∂xj∂xk

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

∂ui ∂xj þ ∂uj ∂xi � �

> ∂ui ∂xj

Using the Lorentz reciprocal theorem, one can obtain Faxén's law for the drag

R2 <sup>6</sup> <sup>∇</sup><sup>2</sup> � �

(In our shorthand notation, the Laplacian is first applied to <sup>u</sup><sup>∞</sup>ð Þ <sup>x</sup> and then

R2 <sup>10</sup> <sup>∇</sup><sup>2</sup> � �

� �

<sup>u</sup><sup>∞</sup>ð Þ� <sup>x</sup><sup>0</sup> <sup>U</sup>

<sup>τ</sup>drag <sup>¼</sup> <sup>8</sup>πμR<sup>3</sup> <sup>ω</sup><sup>∞</sup> ð Þ ð Þ� <sup>x</sup><sup>0</sup> <sup>Ω</sup> , (76)

ð Þ xkσklnl δij � μ uinj þ ujni

<sup>u</sup><sup>∞</sup>ð Þ¼ <sup>x</sup> <sup>u</sup><sup>∞</sup>ð Þþ <sup>0</sup> <sup>e</sup><sup>∞</sup> � <sup>x</sup> <sup>þ</sup> <sup>ω</sup><sup>∞</sup> � <sup>x</sup> (79)

� ∂uj ∂xi � �

eij <sup>¼</sup> <sup>1</sup> 2

Wij <sup>¼</sup> <sup>1</sup> 2

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

<sup>W</sup> <sup>¼</sup> <sup>1</sup> 2

Here, ε is the Levi-Civita tensor. The first derivative in Eq. 70 can be

∂u<sup>∞</sup> i ∂xj ¼ e ∞ ij <sup>þ</sup> <sup>W</sup><sup>∞</sup>

decomposed into symmetric and anti-symmetric contributions:

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

evaluated at x0.) One can also obtain Faxén law for the drag torque:

<sup>3</sup> πμR<sup>3</sup> <sup>1</sup> <sup>þ</sup>

3

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

� � � �

So far we have only presented standard results, but now we raise the following

� � � � x¼0

xjxk þ … (70)

, (71)

, (72)

ε � w: (73)

ij : (74)

<sup>2</sup> w. Finally, there is a Faxén law for

<sup>e</sup><sup>∞</sup>ð Þ <sup>x</sup><sup>0</sup> , (77)

dS: (78)

: (75)

$$\mathbf{F}^{\text{drag}} = 6\pi\mu R \left[ -\frac{1}{4\pi R^2} \int \mathbf{v}'(\mathbf{x}\_i) \, dS - \mathbf{U} \right]. \tag{83}$$

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

$$\mathbf{U} = -\frac{1}{4\pi R^2} \int \mathbf{v}^\prime(\mathbf{x}\_\prime) \, dS. \tag{84}$$

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 surface:

$$\int \varepsilon\_{lmi} \boldsymbol{\mathcal{x}}\_{m} \boldsymbol{u}\_{i}^{\infty}(\mathbf{x}) \, d\mathbf{S} = \int \boldsymbol{u}\_{i}^{\infty}(\mathbf{0}) \boldsymbol{e}\_{lmi} \boldsymbol{\mathcal{x}}\_{m} \, d\mathbf{S} + \left. \int \frac{\partial \boldsymbol{u}\_{i}^{\infty}}{\partial \boldsymbol{\mathcal{x}}\_{j}} \right|\_{\mathbf{x}=\mathbf{0}} \varepsilon\_{lmi} \boldsymbol{\mathcal{x}}\_{m} \, d\mathbf{S}. \tag{85}$$

The first integral on the right hand side of Eq. 85 vanishes. For the second integral on the right hand side, we use the identity

$$
\int \mathbf{x}\_m \mathbf{x}\_j \, d\mathbf{S} = \frac{4\pi R^4}{3} \delta\_{jm} \,. \tag{86}
$$

We obtain:

$$\int \varepsilon\_{lmi} \mathbf{x}\_m \, u\_i^{\infty}(\mathbf{x}) \, d\mathbf{S} = \frac{4\pi R^4}{3} \varepsilon\_{lji} \frac{\partial u\_i^{\infty}}{\partial \mathbf{x}\_j} \bigg|\_{\mathbf{x} = \mathbf{0}} \tag{87}$$

$$-\int \mathbf{x} \times \mathbf{v}'(\mathbf{x}\_t) \, d\mathbf{S} = \frac{4\pi R^4}{3} \nabla \times \mathbf{u}^\infty \Big|\_{\mathbf{x}=0} = \frac{8\pi R^4}{3} \boldsymbol{\alpha}^\infty(\mathbf{0}),\tag{88}$$

so that

$$\boldsymbol{\phi}^{\infty}(\mathbf{0}) = -\frac{3}{8\pi R^4} \left[ \mathbf{x} \times \mathbf{v}'(\mathbf{x}\_t) \, d\mathbf{S}.\tag{89}$$

Using Eq. 76, we obtain:

$$\boldsymbol{\sigma}^{\text{drag}} = 8\pi\mu R^3 \left( -\frac{3}{8\pi R^4} \int \mathbf{x} \times \mathbf{v}^\epsilon(\mathbf{x}\_s) d\mathbf{S} - \boldsymbol{\mathfrak{Q}} \right). \tag{90}$$

For a torque-free swimmer, <sup>τ</sup>drag <sup>¼</sup> 0, and we obtain a second major result from Ref. [37]:

$$\mathbf{\dot{\Omega}} = -\frac{3}{8\pi R^4} \left[ \mathbf{x} \times \mathbf{v}'(\mathbf{x}\_t) \, d\mathbf{S}.\tag{91}$$

Finally, we consider how to obtain the stresslet S. We multiply Eq. 80 by xm and integrate over the surface of the sphere:

$$\int \boldsymbol{u}\_{i}^{\infty}(\mathbf{x}) \boldsymbol{\varkappa}\_{m} d\mathbf{S} = \int \boldsymbol{u}\_{i}^{\infty}(\mathbf{0}) \boldsymbol{\varkappa}\_{m} d\mathbf{S} + \left. \int \frac{\partial \boldsymbol{u}\_{i}^{\infty}}{\partial \boldsymbol{\varkappa}\_{j}} \right|\_{\mathbf{x}=\mathbf{0}} \boldsymbol{\varkappa}\_{m} \boldsymbol{\varkappa}\_{j} d\mathbf{S}.\tag{92}$$

The first integral on the right hand vanishes, giving

$$\int u\_i^{\infty}(\mathbf{x}) \varkappa\_m \, dS = \frac{4\pi R^4}{3} \frac{\partial u\_i^{\infty}}{\partial \varkappa\_m} \Big|\_{\mathbf{x}=0} \,. \tag{93}$$

Swapping the indices i and m, we can also write:

$$\int u\_m^{\infty}(\mathbf{x}) \propto\_i d\mathbf{S} = \frac{4\pi R^4}{\mathfrak{Z}} \frac{\partial u\_m^{\infty}}{\partial \mathbf{x}\_i} \bigg|\_{\mathbf{x}=\mathbf{0}}.\tag{94}$$

Author details

William E. Uspal

HI, United States

71

\*Address all correspondence to: uspal@hawaii.edu

The Boundary Element Method for Fluctuating Active Colloids

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

provided the original work is properly cited.

Department of Mechanical Engineering, University of Hawai'i at Manoa, Honolulu,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Adding these two equations and dividing by two, we obtain

$$\frac{1}{2}\int \left[u\_i^{\infty}(\mathbf{x})\mathbf{x}\_m + u\_m^{\infty}(\mathbf{x})\mathbf{x}\_i\right]d\mathbf{S} = \frac{4\pi R^4}{3}e\_{im}^{\infty}(\mathbf{0}).\tag{95}$$

Accordingly,

$$e\_{im}^{\infty}(\mathbf{0}) = -\frac{3}{8\pi R^4} \int [v\_{\boldsymbol{\epsilon},i}(\mathbf{x})\boldsymbol{\epsilon}\_m + v\_{\boldsymbol{\epsilon},m}(\mathbf{x})\boldsymbol{\epsilon}\_i] d\mathbf{S}.\tag{96}$$

Using the Faxén Law in Eq. 77, we obtain:

$$\mathbf{S} = -\frac{5\mu}{2R} \int [\mathbf{v}\_{\boldsymbol{\epsilon}}(\mathbf{x}\_{\boldsymbol{\epsilon}})\,\mathbf{x} + \mathbf{x}\,\mathbf{v}\_{\boldsymbol{\epsilon}}(\mathbf{x}\_{\boldsymbol{\epsilon}})]d\mathbf{S}.\tag{97}$$

This is the major result obtained in Ref. 84 via the Lorentz reciprocal theorem. As before, this manuscript provides a novel alternative derivation and interpretation of Eq. 97 in terms of an effective ambient flow field. (Note that, due to the linearity of the Stokes equation, our approach is easily extended to model active particles in a real ambient flow field.)

The Boundary Element Method for Fluctuating Active Colloids DOI: http://dx.doi.org/10.5772/intechopen.86738

so that

Ref. [37]:

Using Eq. 76, we obtain:

Non-Equilibrium Particle Dynamics

integrate over the surface of the sphere:

<sup>i</sup> ð Þ x xm dS ¼

The first integral on the right hand vanishes, giving

ð u<sup>∞</sup>

Swapping the indices i and m, we can also write:

ð u∞

imð Þ¼� <sup>0</sup> <sup>3</sup>

<sup>S</sup> ¼ � <sup>5</sup><sup>μ</sup> 2R ð

Using the Faxén Law in Eq. 77, we obtain:

1 2 ð u∞

e ∞

particles in a real ambient flow field.)

Accordingly,

70

ð u<sup>∞</sup> <sup>ω</sup><sup>∞</sup>ð Þ¼� <sup>0</sup> <sup>3</sup>

<sup>Ω</sup> ¼ � <sup>3</sup>

ð u<sup>∞</sup>

8πR<sup>4</sup>

<sup>τ</sup>drag <sup>¼</sup> <sup>8</sup>πμR<sup>3</sup> � <sup>3</sup>

8πR<sup>4</sup>

8πR<sup>4</sup>

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

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

For a torque-free swimmer, <sup>τ</sup>drag <sup>¼</sup> 0, and we obtain a second major result from

Finally, we consider how to obtain the stresslet S. We multiply Eq. 80 by xm and

ð ∂u<sup>∞</sup> i ∂xj � � � � x¼0

∂u<sup>∞</sup> i ∂xm � � � � x¼0

∂u<sup>∞</sup> m ∂xi � � � � x¼0

> <sup>3</sup> <sup>e</sup> ∞

vs,ið Þ x xm þ vs,mð Þ x xi ½ �dS: (96)

vsð Þ x<sup>s</sup> x þ xvsð Þ x<sup>s</sup> ½ �dS: (97)

<sup>i</sup> ð Þ 0 xm dS þ

3

3

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

<sup>m</sup> ð Þ <sup>x</sup> xi dS <sup>¼</sup> <sup>4</sup>πR<sup>4</sup>

<sup>m</sup> ð Þ x xi � �dS <sup>¼</sup> <sup>4</sup>πR<sup>4</sup>

This is the major result obtained in Ref. 84 via the Lorentz reciprocal theorem. As before, this manuscript provides a novel alternative derivation and interpretation of Eq. 97 in terms of an effective ambient flow field. (Note that, due to the linearity of the Stokes equation, our approach is easily extended to model active

Adding these two equations and dividing by two, we obtain

<sup>i</sup> ð Þ <sup>x</sup> xm <sup>þ</sup> <sup>u</sup><sup>∞</sup>

8πR<sup>4</sup>

ð

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

� �

ð Þ x<sup>s</sup> dS: (89)

ð Þ x<sup>s</sup> dS: (91)

: (90)

xm xj dS: (92)

: (93)

: (94)

imð Þ 0 : (95)

ð Þ x<sup>s</sup> dS � Ω
