# Elementary Flow Field Profiles of Micro-Swimmers in Weakly Anisotropic Nematic Fluids: Stokeslet, Stresslet, Rotlet and Source Flows

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## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Nematic Green Function for the Stokes Equation

#### 2.2. Flow Fields of Point Force in Nematic Fluid

#### 2.3. Flow Field of Force Dipole

#### 2.3.1. Stresslet Flow Field in Nematics

#### 2.3.2. Rotlet Flow

#### 2.4. Flow Fields of Sources and Sinks in Homogeneous Nematics

#### 2.5. Source Dipole Flow

## 3. Discussion

#### 3.1. Assumption of Weakly Anisotropic Nematic Fluid

#### 3.2. Deformations in the Director Profile

#### 3.3. Possible Application to Experiments

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) an outline of the problem—a microswimmer exerts forces upon the surrounding nematic fluid with homogeneous director $\overrightarrow{n}$, driving a flow field $\overrightarrow{v}(\overrightarrow{r})$. A set of elementary flow profile solutions for isotropic fluids is shown as reference for later comparison: (

**b**) Stokeslet flow due to a point force; (

**c**) stresslet flow due to a pair of opposite forces (a force dipole), and (

**d**) rotlet flow due to a point torque. The magnitude of the flow field as noted in the colorbar is given in basic quantities (see text for more).

**Figure 2.**Flow field of point force in nematic fluid oriented (

**a**,

**d**) parallel, (

**b**,

**e**) perpendicular, or (

**c**,

**f**) at angle ${45}^{\circ}$ to the director. The flow field decays as $1/r$ with the distance from the point force. In the bottom row, the velocity field component parallel to the force ${\overrightarrow{v}}_{\parallel}$ (along ${\overrightarrow{e}}_{\parallel}$) and perpendicular to the force ${\overrightarrow{v}}_{\perp}$ (along ${\overrightarrow{e}}_{\perp}$) at distance ${d}_{0}$ from the centre is shown as a function of azimuthal and polar angle. The value of ${d}_{0}$ can be chosen arbitrarily as long it is much larger than a swimmer size, since it only rescales the velocity magnitude. The values are compared to the isotropic case (dashed lines, see also Figure 1). Flow field is drawn for arbitrary values of the length ${d}_{0}$ and force F. Results show that for the given nematic anisotropy of the viscosity tensor, spreading of the momentum in the direction perpendicular to the director is suppressed, while the direction along the director offers much less resistance to the fluid flow. Note that graphs (

**d**,

**e**) are symmetrical with respect to $\theta =0$ case (or to $\varphi =0$ case). In (

**f**), which is no longer the case—the velocity field is tilted with respect to the direction of the applied force.

**Figure 3.**Flow field of a force dipole at different orientations with respect to the director field. The bottom row shows radial and azimuthal (or polar) component of the velocity at distance ${d}_{0}$ from the centre as a function of azimuthal (or polar) angle compared to the result for isotropic fluids. (

**a**,

**e**) flow field of force dipole aligned with the director has only radial component. Note that radial flow field is characteristic for dipolar flow in isotropic fluids; however, in nematic fluids, there is still a difference between the magnitudes of the flow field between the isotropic case and a dipole aligned along the director due to anisotropic viscosity (see,

**e**). In (

**a**,

**e**), the magnitude of the velocity field falls to zero at angle $\theta =48.{3}^{\circ}$, whereas, in the isotropic case, the transition from inward to outward flow occurs at $\theta =54.{7}^{\circ}$. At (

**b**,

**c**,

**f**,

**g**) ${90}^{\circ}$ or (

**d**,

**h**) ${45}^{\circ}$, angles between force dipole and the director before radial flow configuration gains additional terms in the azimuthal and polar directions.

**Figure 4.**Flow field of a point torque in nematic. (

**a**,

**b**) flow field due to torque that is perpendicular to the director. Compared to the isotropic case (

**b**, see also Figure 1), the vortex stretches in the direction of the director; (

**c**,

**d**) same flow field in the $xy$ cross-section. In (

**c**), flow lines point in (out) of the plane.

**Figure 5.**Flow field of point source in (

**a**) isotropic and (

**b**) nematic fluid; and (

**c**) radial flow velocity for a source flow (Equation (40)) shown for an isotropic and for a nematic fluid as a function of the azimuthal angle. In nematic fluid, the velocity field retains the radial direction; however, its amplitude as a function of azimuthal angle $\theta $ shows an increase along the z-axis—the director—and a decrease perpendicular to the director. Note that the flow field of a point sink is obtained directly by taking the opposite sign of the flow field of the source; (

**d**) flow source and sink can be combined in a source dipole, here shown for isotropic medium.

**Figure 6.**Flow field of source dipole in nematic (Equation (48)) for the dipole aligned (

**a**) parallel to the director; (

**b**,

**c**) perpendicular to the director, and (

**d**) at angle ${45}^{\circ}$ to the director. The bottom row shows radial and azimuthal (or polar) components of the velocity field compared to the solution for isotropic fluid (${\alpha}_{1}={\alpha}_{6}=0$) for each of the cases above. Spreading of the velocity magnitude along the director is observed.

**Figure 7.**Evaluation of small ${\alpha}_{i}/{\alpha}_{4}$ expansion assumption. Results for the point force along the director (first row) and perpendicular to the director (second row) obtained through the nematic Green function in Equations (24)–(29) are compared to the results obtained by numerical inverse Fourier transform of Equations (10)–(13) at distance ${d}_{0}$ from the point of force origin. The comparison is performed for the values of Leslie coefficients used throughout this article (middle column), values twice as small (first column) and twice as large (third column).

**Table 1.**A list of inverse Fourier transforms used in this article, obtained by the procedure, described in [51]. Note that the expressions form self-consistent pairs bound by the relation $\frac{\partial}{\partial {r}_{i}}{\mathcal{F}}^{-1}\left(\right)open="("\; close=")">\widehat{f}(\overrightarrow{k}).$

$\widehat{\mathit{f}}\left(\right)open="("\; close=")">\overrightarrow{\mathit{k}}$ | ${\mathcal{F}}^{-1}\left(\right)open="("\; close=")">\widehat{\mathit{f}}$ | $\widehat{\mathit{g}}\left(\right)open="("\; close=")">\overrightarrow{\mathit{k}}={\mathit{k}}_{\mathit{z}}\times \widehat{\mathit{f}}\left(\right)open="("\; close=")">\overrightarrow{\mathit{k}}$ | ${\mathcal{F}}^{-1}\left(\right)open="("\; close=")">\widehat{\mathit{g}}$ |
---|---|---|---|

$\frac{1}{{k}^{2}}$ | $\frac{1}{4\pi r}$ | $\frac{{k}_{z}}{{k}^{2}}$ | $\frac{iz}{4\pi {r}^{3}}$ |

$\frac{{k}_{i}{k}_{j}}{{k}^{4}}$ | $\frac{{\delta}_{ij}}{8\pi r}}-{\displaystyle \frac{{r}_{i}{r}_{j}}{8\pi {r}^{3}}$ | $\frac{{k}_{z}^{3}}{{k}^{4}}$ | $-{\displaystyle \frac{3i}{8\pi {r}^{2}}}\left(\right)open="("\; close=")">{\displaystyle \frac{{z}^{3}}{{r}^{3}}}-{\displaystyle \frac{z}{r}}$ |

$\frac{{k}_{x}{k}_{z}^{2}}{{k}^{4}}$ | $\frac{i}{8\pi {r}^{2}}}\left(\right)open="("\; close=")">-{\displaystyle \frac{3x{z}^{2}}{{r}^{3}}}+{\displaystyle \frac{x}{r}$ | ||

$\frac{{k}_{z}^{4}}{{k}^{6}}$ | $\frac{1}{4\pi r}}\left(\right)open="("\; close=")">{\displaystyle \frac{3}{8}}{\displaystyle \frac{{z}^{4}}{{r}^{4}}}-{\displaystyle \frac{3}{4}}{\displaystyle \frac{{z}^{2}}{{r}^{2}}}+{\displaystyle \frac{3}{8}$ | $\frac{{k}_{z}^{5}}{{k}^{6}}$ | $\frac{5i}{4\pi {r}^{2}}}\left(\right)open="("\; close=")">{\displaystyle \frac{3}{8}}{\displaystyle \frac{{z}^{5}}{{r}^{5}}}-{\displaystyle \frac{3}{4}}{\displaystyle \frac{{z}^{3}}{{r}^{3}}}+{\displaystyle \frac{3}{8}}{\displaystyle \frac{z}{r}$ |

$\frac{{k}_{x}{k}_{z}^{3}}{{k}^{6}}$ | $\frac{3}{32\pi r}}\left(\right)open="("\; close=")">{\displaystyle \frac{x{z}^{3}}{{r}^{4}}}-{\displaystyle \frac{xz}{{r}^{2}}$ | $\frac{{k}_{x}{k}_{z}^{4}}{{k}^{6}}$ | $\frac{i}{4\pi {r}^{2}}}\left(\right)open="("\; close=")">{\displaystyle \frac{15}{8}}{\displaystyle \frac{x{z}^{4}}{{r}^{5}}}-{\displaystyle \frac{9}{4}}{\displaystyle \frac{x{z}^{2}}{{r}^{3}}}+{\displaystyle \frac{3}{8}}{\displaystyle \frac{x}{r}$ |

$\frac{{k}_{x}^{2}{k}_{z}^{2}}{{k}^{6}}$ | $\frac{1}{32\pi r}}\left(\right)open="("\; close=")">3{\displaystyle \frac{{x}^{2}{z}^{2}}{{r}^{4}}}-{\displaystyle \frac{{x}^{2}}{{r}^{2}}}-{\displaystyle \frac{{z}^{2}}{{r}^{2}}}+1$ | $\frac{{k}_{x}^{2}{k}_{z}^{3}}{{k}^{6}}$ | $\frac{i}{32\pi {r}^{2}}}\left(\right)open="("\; close=")">15{\displaystyle \frac{{x}^{2}{z}^{3}}{{r}^{5}}}-3{\displaystyle \frac{{z}^{3}}{{r}^{3}}}-9{\displaystyle \frac{{x}^{2}z}{{r}^{3}}}+3{\displaystyle \frac{z}{r}$ |

$\frac{{k}_{z}^{6}}{{k}^{8}}$ | $\frac{5}{64\pi r}}\left(\right)open="("\; close=")">-{\displaystyle \frac{{z}^{6}}{{r}^{6}}}+3{\displaystyle \frac{{z}^{4}}{{r}^{4}}}-3{\displaystyle \frac{{z}^{2}}{{r}^{2}}}+1$ | $\frac{{k}_{z}^{7}}{{k}^{8}}$ | $\frac{35}{64\pi {r}^{2}}}\left(\right)open="("\; close=")">-{\displaystyle \frac{{z}^{7}}{{r}^{7}}}+3{\displaystyle \frac{{z}^{5}}{{r}^{5}}}-3{\displaystyle \frac{{z}^{3}}{{r}^{3}}}+{\displaystyle \frac{z}{r}$ |

$\frac{{k}_{x}{k}_{z}^{5}}{{k}^{8}}$ | $\frac{5}{32\pi r}}\left(\right)open="("\; close=")">-{\displaystyle \frac{1}{2}}{\displaystyle \frac{x{z}^{5}}{{r}^{6}}}+{\displaystyle \frac{x{z}^{3}}{{r}^{4}}}-{\displaystyle \frac{1}{2}}{\displaystyle \frac{xz}{{r}^{2}}$ | $\frac{{k}_{x}{k}_{z}^{6}}{{k}^{8}}$ | $\frac{5i}{64\pi {r}^{2}}}\left(\right)open="("\; close=")">-7{\displaystyle \frac{x{z}^{6}}{{r}^{7}}}+15{\displaystyle \frac{x{z}^{4}}{{r}^{5}}}-9{\displaystyle \frac{x{z}^{2}}{{r}^{3}}}+{\displaystyle \frac{x}{r}$ |

$\frac{{k}_{x}^{2}{k}_{z}^{4}}{{k}^{8}}$ | $\begin{array}{c}\hfill \left[l\right]{\displaystyle \frac{1}{32\pi r}}\left(\right)open="("\; close>-{\displaystyle \frac{15}{6}}{\displaystyle \frac{{x}^{2}{z}^{4}}{{r}^{6}}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{z}^{4}}{{r}^{4}}}\end{array}$ | $\frac{{k}_{x}^{2}{k}_{z}^{5}}{{k}^{8}}$ | $\begin{array}{c}\hfill \left[l\right]{\displaystyle \frac{i}{32\pi {r}^{2}}}\left(\right)open="("\; close>-{\displaystyle \frac{35}{2}}{\displaystyle \frac{{x}^{2}{z}^{5}}{{r}^{7}}}+{\displaystyle \frac{5}{2}}{\displaystyle \frac{{z}^{5}}{{r}^{5}}}\end{array}$ |

$\frac{{k}_{x}{k}_{y}{k}_{z}^{4}}{{k}^{8}}$ | $\frac{1}{32\pi r}}\left(\right)open="("\; close=")">-{\displaystyle \frac{15}{6}}{\displaystyle \frac{xy{z}^{4}}{{r}^{6}}}+3{\displaystyle \frac{xy{z}^{2}}{{r}^{4}}}-{\displaystyle \frac{1}{2}}{\displaystyle \frac{xy}{{r}^{2}}$ | $\frac{{k}_{x}{k}_{y}{k}_{z}^{5}}{{k}^{8}}$ | $\frac{5i}{32\pi {r}^{2}}}\left(\right)open="("\; close=")">-{\displaystyle \frac{7}{2}}{\displaystyle \frac{xy{z}^{5}}{{r}^{7}}}+5{\displaystyle \frac{xy{z}^{3}}{{r}^{5}}}-{\displaystyle \frac{3}{2}}{\displaystyle \frac{xyz}{{r}^{3}}$ |

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**MDPI and ACS Style**

Kos, Ž.; Ravnik, M.
Elementary Flow Field Profiles of Micro-Swimmers in Weakly Anisotropic Nematic Fluids: Stokeslet, Stresslet, Rotlet and Source Flows. *Fluids* **2018**, *3*, 15.
https://doi.org/10.3390/fluids3010015

**AMA Style**

Kos Ž, Ravnik M.
Elementary Flow Field Profiles of Micro-Swimmers in Weakly Anisotropic Nematic Fluids: Stokeslet, Stresslet, Rotlet and Source Flows. *Fluids*. 2018; 3(1):15.
https://doi.org/10.3390/fluids3010015

**Chicago/Turabian Style**

Kos, Žiga, and Miha Ravnik.
2018. "Elementary Flow Field Profiles of Micro-Swimmers in Weakly Anisotropic Nematic Fluids: Stokeslet, Stresslet, Rotlet and Source Flows" *Fluids* 3, no. 1: 15.
https://doi.org/10.3390/fluids3010015