# 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

- Lauga, E.; Powers, T.R. The hydrodynamics of swimming microorganisms. Rep. Prog. Phys.
**2009**, 72, 096601. [Google Scholar] [CrossRef] - Elgeti, J.; Winkler, R.G.; Gompper, G. Physics of microswimmers—Single particle motion and collective behavior: A review. Rep. Prog. Phys.
**2015**, 78, 056601. [Google Scholar] [CrossRef] [PubMed] - Bechinger, C.; Leonardo, R.D.; Löwen, H.; Reichhardt, C.; Volpe, G.; Volpe, G. Active particles in complex and crowded environments. Rev. Mod. Phys.
**2016**, 88, 045006. [Google Scholar] [CrossRef] - Purcell, E.M. Life at low Reynolds number. Am. J. Phys.
**1977**, 45, 3–11. [Google Scholar] [CrossRef] - Zöttl, A.; Stark, H. Emergent behavior in active colloids. J. Phys. Condens. Matter
**2016**, 28, 253001. [Google Scholar] [CrossRef] - Sokolov, A.; Apodaca, M.M.; Grzybowski, B.A.; Aranson, I.S. Swimming bacteria power microscopic gears. Proc. Natl. Acad. Sci. USA
**2010**, 107, 969–974. [Google Scholar] [CrossRef] [PubMed] - Thampi, S.P.; Doostmohammadi, A.; Shendruk, T.N.; Golestanian, R.; Yeomans, J.M. Active micromachines: Microfluidics powered by mesoscale turbulence. Sci. Adv.
**2016**, 2, e1501854. [Google Scholar] [CrossRef] [PubMed] - Kim, M.J.; Breuer, K.S. Enhanced diffusion due to motile bacteria. Phys. Fluids
**2004**, 16, L78–L81. [Google Scholar] [CrossRef] - Pushkin, D.O.; Shum, H.; Yeomans, J.M. Fluid transport by individual microswimmers. J. Fluid Mech.
**2013**, 726, 5–25. [Google Scholar] [CrossRef] - Hatwalne, Y.; Ramaswamy, S.; Rao, M.; Simha, R. Rheology of active-particle suspensions. Phys. Rev. Lett.
**2004**, 92, 118101. [Google Scholar] [CrossRef] [PubMed] - López, H.M.; Gachelin, J.; Douarche, C.; Auradou, H.; Clément, E. Turning bacteria suspensions into superfluids. Phys. Rev. Lett.
**2015**, 115, 028301. [Google Scholar] [CrossRef] [PubMed] - Fung, Y.C. Biomechanics: Mechanical Properties of Living Tissues; Springer: Heidelberg, Germany, 1981. [Google Scholar]
- Luby-Phelps, K. Cytoarchitecture and physical properties of pytoplasm: volume, viscosity, diffusion, intracellular surface area. Int. Rev. Cytol.
**1999**, 192, 189–221. [Google Scholar] - Rey, A.D. Liquid crystal models of biological materials and processes. Soft Matter
**2010**, 6, 3402–3429. [Google Scholar] [CrossRef] - Kumar, A.; Galstian, T.; Pattanayek, S.K.; Rainville, S. The Motility of bacteria in an anisotropic liquid environment. Mol. Cryst. Liq. Cryst.
**2013**, 574, 33–39. [Google Scholar] [CrossRef] - Toner, J.; Löwen, H.; Wensink, H.H. Following fluctuating signs: Anomalous active superdiffusion of swimmers in anisotropic media. Phys. Rev. E
**2016**, 93, 062610. [Google Scholar] [CrossRef] [PubMed] - Mushenheim, P.C.; Trivedi, R.R.; Roy, S.S.; Arnold, M.S.; Weibel, D.B.; Abbott, N.L. Effects of confinement, surface-induced orientations and strain on dynamical behaviors of bacteria in thin liquid crystalline films. Soft Matter
**2015**, 11, 6821–6831. [Google Scholar] [CrossRef] [PubMed] - Sokolov, A.; Zhou, S.; Lavrentovich, O.D.; Aranson, I.S. Individual behavior and pairwise interactions between microswimmers in anisotropic liquid. Phys. Rev. E
**2015**, 91, 013009. [Google Scholar] [CrossRef] [PubMed] - Mushenheim, P.C.; Trivedi, R.R.; Tuson, H.H.; Weibel, D.B.; Abbott, N.L. Dynamic self-assembly of motile bacteria in liquid crystals. Soft Matter
**2013**, 10, 88–95. [Google Scholar] [CrossRef] [PubMed] - Zhou, S.; Sokolov, A.; Lavrentovich, O.D.; Aranson, I.S. Living liquid crystals. Proc. Natl. Acad. Sci. USA
**2014**, 111, 1265–1270. [Google Scholar] [CrossRef] [PubMed] - Trivedi, R.R.; Maeda, R.; Abbott, N.L.; Spagnolie, S.E.; Weibel, D.B. Bacterial transport of colloids in liquid crystalline environments. Soft Matter
**2015**, 11, 8404–8408. [Google Scholar] [CrossRef] [PubMed] - Lavrentovich, O.D. Active colloids in liquid crystals. Curr. Opin. Colloid Interface Sci.
**2016**, 21, 97–109. [Google Scholar] [CrossRef] - Krieger, M.S.; Spagnolie, S.E.; Powers, T.R. Locomotion and transport in a hexatic liquid crystal. Phys. Rev. E
**2014**, 90, 052503. [Google Scholar] [CrossRef] [PubMed] - Krieger, M.S.; Spagnolie, S.E.; Powers, T. Microscale locomotion in a nematic liquid crystal. Soft Matter
**2015**, 11, 9115–9125. [Google Scholar] [CrossRef] [PubMed] - Heuer, H.; Kneppe, H.; Schneider, F. Flow of a nematic liquid crystal around a sphere. Mol. Cryst. Liq. Cryst.
**1992**, 214, 43–61. [Google Scholar] [CrossRef] - Kneppe, H.; Schneider, F.; Schwesinger, B. Axisymmetrical flow of a nematic liquid crystal around a sphere. Mol. Cryst. Liq. Cryst.
**1991**, 205, 9–28. [Google Scholar] [CrossRef] - Kozachok, M.V.; Lev, B.I. Analytical calculation of the Stokes drag of the spherical particle in a nematic liquid crystal. Condens. Matter Phys.
**2014**, 17, 79–86. [Google Scholar] [CrossRef] - Stark, H.; Ventzki, D.; Reichert, M. Recent developments in the field of colloidal dispersions in nematic liquid crystals: the Stokes drag. J. Phys. Condens. Matter
**2003**, 15, S191. [Google Scholar] [CrossRef] - Stark, H.; Ventzki, D. Stokes drag of spherical particles in a nematic environment at low Ericksen numbers. Phys. Rev. E
**2001**, 64, 031711. [Google Scholar] [CrossRef] [PubMed] - Pokrovskii, V.; Tskhai, A. Slow motion of a particle in a weakly anisotropic viscous fluid. J. Appl. Math. Mech.
**1986**, 50, 391–394. [Google Scholar] [CrossRef] - Gómez-González, M.; del Álamo, J.C. Flow of a viscous nematic fluid around a sphere. J. Fluid Mech.
**2013**, 725, 299–331. [Google Scholar] [CrossRef] - Drescher, K.; Goldstein, R.E.; Michel, N.; Polin, M.; Tuval, I. Direct measurement of the flow field around swimming microorganisms. Phys. Rev. Lett.
**2010**, 105, 168101. [Google Scholar] [CrossRef] [PubMed] - Drescher, K.; Dunkel, J.; Cisneros, L.H.; Ganguly, S.; Goldstein, R.E. Fluid dynamics and noise in bacterial cell-cell and cell-surface scattering. Proc. Natl. Acad. Sci. USA
**2011**, 108, 10940–10945. [Google Scholar] [CrossRef] [PubMed] - Pushkin, D.O.; Yeomans, J.M. Fluid mixing by curved trajectories of microswimmers. Phys. Rev. Lett.
**2013**, 111, 188101. [Google Scholar] [CrossRef] [PubMed] - Alexander, G.P.; Yeomans, J.M. Dumb-bell swimmers. EPL
**2008**, 83, 34006. [Google Scholar] [CrossRef] - Alexander, G.P.; Pooley, C.M.; Yeomans, J.M. Hydrodynamics of linked sphere model swimmers. J. Phys. Condens. Matter
**2009**, 21, 204108. [Google Scholar] [CrossRef] [PubMed] - Baskaran, A.; Marchetti, M.C. Statistical mechanics and hydrodynamics of bacterial suspensions. Proc. Natl. Acad. Sci. USA
**2009**, 106, 15567–15572. [Google Scholar] [CrossRef] [PubMed] - Saintillan, D.; Shelley, M. Instabilities and pattern formation in active particle suspensions: Kinetic theory and continuum simulations. Phys. Rev. Lett.
**2008**, 100, 178103. [Google Scholar] [CrossRef] [PubMed] - Mathijssen, A.J.T.M.; Doostmohammadi, A.; Yeomans, J.M.; Shendruk, T.N. Hydrodynamics of micro-swimmers in films. J. Fluid Mech.
**2016**, 806, 35–70. [Google Scholar] [CrossRef] - Reigh, S.Y.; Zhu, L.; Gallaire, F.; Lauga, E. Swimming with a cage: Low-Reynolds-number locomotion inside a droplet. Soft Matter
**2017**, 13, 3161–3173. [Google Scholar] [CrossRef] [PubMed] - Marchetti, M.C.; Joanny, J.F.; Ramaswamy, S.; Liverpool, T.B.; Prost, J.; Rao, M.; Simha, R.A. Hydrodynamics of soft active matter. Rev. Mod. Phys.
**2013**, 85, 1143–1189. [Google Scholar] [CrossRef] - Pozrikidis, C. Introduction to Theoretical and Computational Fluid Dynamics, 2nd ed.; Oxford University Press: New York, NY, USA, 2011. [Google Scholar]
- De Gennes, P.G.; Prost, J. Physics of Liquid Crystals; Oxford University Press: New York, NY, USA, 1993. [Google Scholar]
- Denniston, C.; Orlandini, E.; Yeomans, J. Lattice Boltzmann simulations of liquid crystal hydrodynamics. Phys. Rev. E
**2001**, 63, 056702. [Google Scholar] [CrossRef] [PubMed] - Giomi, L.; Kos, Ž.; Ravnik, M.; Sengupta, A. Cross-talk between topological defects in different fields revealed by nematic microfluidics. Proc. Natl. Acad. Sci. USA
**2017**, 114, E5771–E5777. [Google Scholar] [CrossRef] [PubMed] - Taratuta, V.G.; Hurd, A.J.; Meyer, R.B. Light-scattering study of a polymer nematic liquid crystal. Phys. Rev. Lett.
**1985**, 55, 246–249. [Google Scholar] [CrossRef] [PubMed] - Jamieson, A.M.; Gu, D.; Chen, F.; Smith, S. Viscoelastic behavior of nematic monodomains containing liquid crystal polymers. Prog. Polym. Sci.
**1996**, 21, 981–1033. [Google Scholar] [CrossRef] - Kuzuu, N.; Doi, M. Constitutive equation for nematic liquid crystals under weak velocity gradient derived from a molecular kinetic equation. J. Phys. Soc. Jpn.
**1983**, 52, 3486–3494. [Google Scholar] [CrossRef] - Herba, H.; Szymański, A.; Drzymala, A. Experimental test of hydrodynamic theories for nematic liquid crystals. Mol. Cryst. Liq. Cryst.
**1985**, 127, 153–158. [Google Scholar] [CrossRef] - Cui, M.; Kelly, J.R. Temperature dependence of visco-elastic properties of 5CB. Mol. Cryst. Liq. Cryst.
**1999**, 331, 49–57. [Google Scholar] [CrossRef] - Adkins, G. Three-dimensional Fourier transforms, integrals of spherical Bessel functions, and novel delta function identities. arXiv, 2013; arXiv:1302.1830. [Google Scholar]
- Doostmohammadi, A.; Stocker, R.; Ardekani, A.M. Low-Reynolds-number swimming at pycnoclines. Proc. Natl. Acad. Sci. USA
**2012**, 109, 3856–3861. [Google Scholar] [CrossRef] [PubMed] - Hu, J.; Yang, M.; Gompper, G.; Winkler, R.G. Modelling the mechanics and hydrodynamics of swimming E. coli. Soft Matter
**2015**, 11, 7867–7876. [Google Scholar] [CrossRef] [PubMed] - Lintuvuori, J.; Würger, A.; Stratford, K. Hydrodynamics defines the stable swimming direction of spherical squirmers in a nematic liquid crystal. Phys. Rev. Lett.
**2017**, 119, 068001. [Google Scholar] [CrossRef] [PubMed] - Lubensky, T.C.; Pettey, D.; Currier, N. Topological defects and interactions in nematic emulsions. Phys. Rev. E
**1998**, 57, 610–625. [Google Scholar] [CrossRef] - Peng, C.; Turiv, T.; Guo, Y.; Wei, Q.H.; Lavrentovich, O.D. Command of active matter by topological defects and patterns. Science
**2016**, 354, 882–885. [Google Scholar] [CrossRef] [PubMed] - Hernàndez-Navarro, S.; Tierno, P.; Farrera, J.A.; Ignés-Mullol, J.; Sagués, F. Reconfigurable swarms of nematic colloids controlled by photoactivated surface patterns. Angew. Chem. Int. Ed.
**2014**, 53, 10696–10700. [Google Scholar] [CrossRef] [PubMed] - Zhou, S.; Tovkach, O.; Golovaty, D.; Sokolov, A.; Aranson, I.S.; Lavrentovich, O.D. Dynamic states of swimming bacteria in a nematic liquid crystal cell with homeotropic alignment. New J. Phys.
**2017**, 19, 055006. [Google Scholar] [CrossRef]

**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 $\frac{\partial}{\partial {r}_{i}}{\mathcal{F}}^{-1}\left(\widehat{f}(\overrightarrow{k})\right)=i{\mathcal{F}}^{-1}\left(\widehat{f}(\overrightarrow{k})\times {k}_{i}\right).$

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

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

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

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

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

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

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

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

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

$\frac{{k}_{x}^{2}{k}_{z}^{4}}{{k}^{8}}$ | $\begin{array}{c}\hfill \left[l\right]{\displaystyle \frac{1}{32\pi r}}\left(-{\displaystyle \frac{15}{6}}{\displaystyle \frac{{x}^{2}{z}^{4}}{{r}^{6}}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{z}^{4}}{{r}^{4}}}\right.\\ \hfill +3\left.{\displaystyle \frac{{x}^{2}{z}^{2}}{{r}^{4}}}-{\displaystyle \frac{1}{2}}{\displaystyle \frac{{x}^{2}}{{r}^{2}}}-{\displaystyle \frac{{z}^{2}}{{r}^{2}}}+{\displaystyle \frac{1}{2}}\right)\end{array}$ | $\frac{{k}_{x}^{2}{k}_{z}^{5}}{{k}^{8}}$ | $\begin{array}{c}\hfill \left[l\right]{\displaystyle \frac{i}{32\pi {r}^{2}}}\left(-{\displaystyle \frac{35}{2}}{\displaystyle \frac{{x}^{2}{z}^{5}}{{r}^{7}}}+{\displaystyle \frac{5}{2}}{\displaystyle \frac{{z}^{5}}{{r}^{5}}}\right.\\ \hfill +25\left.{\displaystyle \frac{{x}^{2}{z}^{3}}{{r}^{5}}}-{\displaystyle \frac{15}{2}}{\displaystyle \frac{{x}^{2}z}{{r}^{3}}}-5{\displaystyle \frac{{z}^{3}}{{r}^{3}}}+{\displaystyle \frac{5}{2}}{\displaystyle \frac{z}{r}}\right)\end{array}$ |

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

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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