# Three-Dimensional Low Reynolds Number Flows near Biological Filtering and Protective Layers

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Relevant Dimensionless Numbers and the Leaky to Solid Transition

#### 1.2. Analytical Porous Models

#### 1.3. Fully Resolved Flow (Not Averaged) Past 3D Structures

- Flow around physical models of filtering layers is measured using particle image velocimetry.
- A 1D Brinkman model of flow through porous layers is compared to three-dimensional dynamically scaled physical models. The goal is to confirm that the 1D Brinkman model captures bulk flow outside of the porous layers.
- Three-dimensional flow through idealized filtering layers is numerically simulated using the immersed boundary method.
- A 1D Brinkman model of flow within the layer is compared to the numerical simulations. The goal is to confirm that the Brinkman model adequately captures average flow but does not capture movement in the third dimension (which would enhance exchange into and out of the layer).

## 2. Methods

#### 2.1. Immersed Boundary Method

#### 2.2. Description of the Numerical Setup, Example Output, and Validation

#### 2.3. 1D Analytical Model Using Brinkman Equations

## 2.4. Physical Models

^{TM}brand light corn syrup, with dynamic viscosity ($\mu $) of 1.229 kg·m${}^{-1}\xb7$s${}^{-1}$, and density ($\rho $) of 1340 kg·m${}^{-3}$ at an ambient room temperature of 20 ${}^{\circ}$C.

## 2.5. Experimental Diagnostics

## 3. Results

#### 3.1. Experimental Results

#### 3.1.1. Comparison of Experimental Results to 1D Theory

#### 3.2. 3D Simulations of Flow through Arrays of Cylinders

#### 3.2.1. Effect of Re

#### 3.3. Effect of the Number of Cylinders

#### 3.3.1. Effect of Height

#### 3.3.2. Effect of Spacing

#### 3.4. Brinkman vs. Explicit Treatment of Cylinders

## 4. Discussion and Conclusion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Jackson, G.; Winant, C. Effect of a kelp forest on coastal currents. Cont. Shelf Res.
**1983**, 2, 75–80. [Google Scholar] [CrossRef] - Gaylord, B.; Reed, D.; Washburn, L.; Raimondi, P. Physical-biological coupling in spore dispersal of kelp forest macroalgae. J. Mar. Syst.
**2004**, 49, 19–39. [Google Scholar] [CrossRef] - Koehl, M.A.R.; Reidenbach, M.A. Swimming by Microscopic Organisms in Ambient Water Flow; Animal Locomotion, Springer: Berlin, Germany, 2010; pp. 117–130. [Google Scholar]
- Sutherland, K.R.; Dabiri, J.O.; Koehl, M.A.R. Simultaneous field measurements of ostracod swimming behavior and background flow. Limnol. Oceanogr.
**2011**, 1, 135–146. [Google Scholar] [CrossRef] - Jones, S.K.; Yun, Y.J.J.; Hedrick, T.L.; Griffith, B.E.; Miller, L.A. Bristles reduce the force required to ‘fling’ wings apart in the smallest insects. J. Exp. Biol.
**2016**, 219, 3759–3772. [Google Scholar] [CrossRef] [PubMed] - Koch, E.W. Hydrodynamics, diffusion-boundary layers and photosynthesis of the seagrasses, thalassia testudinum and cymodocea nodosa. Mar. Biol.
**1994**, 118, 767–776. [Google Scholar] [CrossRef] - Hurd, C.L. Water motion, marine macroalgal physiology, and production. J. Phycol.
**2000**, 36, 453–472. [Google Scholar] [CrossRef] - Ludeman, D.; Farrar, N.; Riesgo, A.; Paps, J.; Leys, S. Evolutionary origins of sensation in metazoans: Functional evidence for a new sensory organ in sponges. BMC Evol. Biol.
**2014**, 14, 3. [Google Scholar] [CrossRef] [PubMed] - Babu, D.; Roy, S. Left-right asymmetry: Cilia stir up new surprises in the node. Open Biol.
**2013**, 3, 130052. [Google Scholar] [CrossRef] [PubMed] - Lighthill, J. Acoustic streaming in the ear itself. J. Fluid Mech.
**1992**, 239, 551–606. [Google Scholar] [CrossRef] - Bornschlogl, T. How filopodia pull: What we know about the mechanics and dynamics of filopodia. Cryoskeleon
**2013**, 70, 590–603. [Google Scholar] [CrossRef] [PubMed] - Jiang, H.; Osborn, T. Hydrodynamics of copepods: A review. Surv. Geophys.
**2004**, 25, 339–370. [Google Scholar] [CrossRef] - Geierman, C.; Emlet, R. Feeding behavior, cirral fan anatomy, reynolds numbers, and leakiness of balanus glandula, from post-metamophic juvenile to the adult. J. Exp. Mar. Biol. Ecol.
**2009**, 379, 68–76. [Google Scholar] [CrossRef] - Alexander, D. The biomechanics of solids and fluids: the physics of life. Eur. J. Phys.
**2001**, 37, 053001. [Google Scholar] [CrossRef] - Feitl, K.; Millett, A.; Colin, S.; Dabiri, J.; Costello, J. Functional morphology and fluid interactions during early development of the scyphomedusa aurelia aurita. Biol. Bull.
**2009**, 217, 283–291. [Google Scholar] [CrossRef] [PubMed] - Wilson, M.; Peng, J.; Dabiri, J.; Eldredge, J. Lagrangian coherent structures in low reynolds number swimming. J. Phys. Condens. Matter
**2009**, 21, 204105. [Google Scholar] [CrossRef] [PubMed] - Miller, L.A.; Peskin, C.S. A computational fluid dynamics of clap and fling in the smallest insects. J. Exp. Biol.
**2009**, 208, 3076–3090. [Google Scholar] [CrossRef] [PubMed] - Summarell, C.G.; Ingole, S.; Fish, F.; Marshall, C. Comparative analysis of the flexural stiffness of pinniped vibrissae. PLoS ONE
**2015**. [Google Scholar] [CrossRef] - Vogel, S. Life in Moving Fluids: The Physical Biology of Flow, 2nd ed.; Princeton University Press: Princeton, NJ, USA, 1994. [Google Scholar]
- Schreuder, M.D.J.; Brewer, C.A.; Heine, C. Modelled influences of non-exchanging trichomes on leaf boundary layers and gas exchange. J. Theor. Biol.
**2001**, 210, 23–32. [Google Scholar] [CrossRef] [PubMed] - Huwaldt, J.A. Plot Digitizer. Available online: http://plotdigitizer.sourceforge.net/ (accessed on 23 August 2017).
- Hedrick, T.L. Software techniques for two- and three-dimensional kinematic measurements of biological and biomimetic systems. Bioinspir. Biomimetics
**2008**, 3, 034001. [Google Scholar] [CrossRef] [PubMed] - Weinbaum, S.; Zhang, X.; Han, Y.; Vink, H.; Cowin, S. Mechanotransduction and flow across the endothelial glycocalyx. Proc. Natl. Acad. Sci. USA
**2003**, 100, 7988–7995. [Google Scholar] [CrossRef] [PubMed] - Guo, J.; Zhang, J. Velocity distributions in laminar and turbulent vegetated flows. J. Hydraul. Res.
**2016**, 54, 117–130. [Google Scholar] [CrossRef] - Waldrop, L.D. Ontogenetic scaling of the olfactory antennae and flicking behavior of the shore crab Hemigrapsus oregonensis. Chem. Sens.
**2013**, 38, 541–550. [Google Scholar] [CrossRef] [PubMed] - Waldrop, L.D.; Nguyen, Q.; Bantay, R. Scaling of olfactory antennae and kinematics of antennule flicking of the terrestrial hermit crabs coenobita rugosus and coenobita perlatus during ontogeny. PeerJ
**2014**, 2, e535. [Google Scholar] [CrossRef] [PubMed][Green Version] - Szymanski, D.B.; Marks, M.D.; Wicks, S.M. Organzied f-actin is essential for normal tricome morphogenesis in arabidopsis. Plant Cell
**1999**, 11, 2331–2347. [Google Scholar] [CrossRef] [PubMed] - Koehl, M. Small-scale fluid dynamics of olfactory antennae. Mar. Freshw. Behav. Physiol.
**1996**, 27, 127–141. [Google Scholar] [CrossRef] - Koehl, M. Biomechanics of microscopic appendages: Functional shifts caused by changes in speed. J. Biomech.
**2004**, 37, 789–795. [Google Scholar] [CrossRef] [PubMed] - Koehl, M. Transitions in function at low reynolds number: Hair-bearing animal appendages. Math. Methods Appl. Sci.
**2001**, 24, 1523–1532. [Google Scholar] [CrossRef] - Loudon, C.; Koehl, M. Sniffing by a silkworm moth: Wing fanning enhances air penetration through and pheromone interception by antennae. J. Exp. Biol.
**2000**, 203, 2977–2990. [Google Scholar] [PubMed] - Blough, T.; Colin, S.; Costello, J.; Marques, A. Ontogenetic changes in the bell morphology and kinematics and swimming behavior of rowing medusae: The special case of the limnomedusa liriope tetraphylla. Biol. Bull.
**2011**, 220, 6–14. [Google Scholar] [CrossRef] [PubMed] - Nepf, H.M. Flow and transport in regions with aquatic vegetation. Annu. Rev. Fluid Mech.
**2013**, 44, 123–142. [Google Scholar] [CrossRef] - Finnigan, J. Turbulence in plant canopies. Annu. Rev. Fluid Mech.
**2000**, 32, 519–571. [Google Scholar] [CrossRef] - Brinkman, H.C. A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res.
**1949**, 1, 27–34. [Google Scholar] [CrossRef] - Damiano, E.R.; Long, D.S.; Smith, M.L. Estimation of viscosity profiles using velocimetry data from parallel flows of linearly viscous: Application to microvascular hemodynamics. J. Fluid Mech.
**2004**, 512, 1–19. [Google Scholar] [CrossRef] - Smith, M.L.; Long, D.S.; Damiano, E.R.; Ley, K. Near-wall micro-piv reveals a hydrodynamically relevant endothelial surface layer in venules in vivo. Biophys. J.
**2003**, 85, 637–645. [Google Scholar] [CrossRef] - Vincent, P.E.; Sherwin, S.J.; Weinberg, P.D. Viscous flow over outflow slits covered by an anisotropic brinkman medium: A model of flow above interendothelial cell clefts. Phys. Fluids
**2008**, 20, 063106. [Google Scholar] [CrossRef] - Ferko, M.C.; Bhatnagar, A.; Garcia, M.B.; Butler, P.J. Finite-element stress analysis of a multicomponent model of sheared and focally-adhered endothelial cells. Ann. Biomed. Eng.
**2007**, 35, 208–223. [Google Scholar] [CrossRef] [PubMed] - Leiderman, K.M.; Miller, L.A.; Fogelson, A.L. The effects of spatial inhomogeneities on flow through the endothelial surface layer. J. Theor. Biol.
**2008**, 252, 313–325. [Google Scholar] [CrossRef] [PubMed] - Darcy, H. Les Fontaines Publiques de la Ville de Dijon; Dalmont: Paris, France, 1856. [Google Scholar]
- Bejan, A. Convection Heat Transfer; John Wiley & Sons: Hoboken, NJ, USA, 1984. [Google Scholar]
- Shavit, U.; Bar-Yosef, G.; Rosenzweig, R.; Assouline, S. Modified brinkman equation for a free flow problem at the interface of porous surfaces: The cantor-taylor brush configuration case. Water Resour. J.
**2002**, 38, 1320–1334. [Google Scholar] [CrossRef] - Shavit, U.; Rosenzweig, R.; Assouline, S. Free flow at the interface of porous surfaces: Generalization of the taylor brush configuration. Transp. Porous Media
**2004**, 54, 345–360. [Google Scholar] [CrossRef] - Grunbaum, D.; Strathmann, R.R. Form, performance and trade-offs in swimming and stability of armed larvae. J. Mar. Res.
**2003**, 61, 659–691. [Google Scholar] [CrossRef] - Reidenbach, M.A.; Koseff, J.R.; Koehl, M.A.R. Hydrodynamic forces on larvae affect their settlement on coral reefs in turbulent, wave-driven flow. Limnol. Oceanogr.
**2009**, 54, 318–330. [Google Scholar] [CrossRef] - Koehl, M.A.R.; Hadfield, M. Hydrodynamics of larval settlement from a larva’s point of view. Integr. Comp. Biol.
**2010**, 50, 539–551. [Google Scholar] [CrossRef] [PubMed] - Cheer, A.; Cheung, S.; Hung, T.; Piedrahita, R.H.; Sanderson, S.L. Computational fluid dynamics of fish gill rakers during crossflow filtration. Bull. Math. Biol.
**2012**, 74, 981–1000. [Google Scholar] [CrossRef] [PubMed] - Waldrop, L.D.; Miller, L.A.; Khatri, S. A tale of two antennules: The performance of crab odour-capture organs in air and water. R. Soc. Interface
**2016**. [Google Scholar] [CrossRef] [PubMed] - Winkler, R. Low reynolds number hydrodynamics and mesoscale simulations. Eur. Phys. J. Spec. Top.
**2016**, 225, 2079–2097. [Google Scholar] [CrossRef] - Atzberger, P.J.; Kramer, P.R.; Peskin, C.S. A stochastic immersed boundary method for fluid–structure dynamics at microscopic length scales. J. Comp. Phys.
**2007**, 224, 1255–1292. [Google Scholar] [CrossRef] - Strychalski, W.; Guy, R.D. A computational model of bleb formation. Math. Med. Biol.
**2013**, 30, 115–130. [Google Scholar] [CrossRef] [PubMed] - Peskin, C.S.; McQueen, D.M. Fluid Dynamics of the Heart and Its Valves. In Case Studies in Mathematical Modeling: Ecology, Physiology, and Cell Biology; Adler, F.R., Lewis, M.A., Dalton, J.C., Eds.; Prentice-Hall: Upper Saddle River, NJ, USA, 1996; pp. 309–338. [Google Scholar]
- Griffith, B.E. Immersed boundary model of aortic heart valve dynamics with physiological driving and loading conditions. Int. J. Numer. Meth. Biomed. Eng.
**2012**, 28, 317–345. [Google Scholar] [CrossRef] - Fauci, L.; Peskin, C. A computational model of aquatic animal locomotion. J. Comput. Phys.
**1988**, 77, 85–108. [Google Scholar] [CrossRef] - Hoover, A.P.; Miller, L.A. A numerical study of the benefits of driving jellyfish bells at their natural frequency. J. Theor. Biol.
**2015**, 374, 13–25. [Google Scholar] [CrossRef] [PubMed] - Jones, S.K.; Laurenza, R.; Hedrick, T.L.; Griffith, B.E.; Miller, L.A. Lift- vs. drag-based for vertical force production in the smallest flying insects. J. Theor. Biol.
**2015**, 384, 105–120. [Google Scholar] [CrossRef] [PubMed] - Battista, N.A.; Baird, A.J.; Miller, L.A. A mathematical model and matlab code for muscle-fluid–structure simulations. Integr. Comp. Biol.
**2015**, 55, 901–911. [Google Scholar] [CrossRef] [PubMed] - Hamlet, C.; Fauci, L.J.; Tytell, E.D. The effect of intrinsic muscular nonlinearities on the energetics of locomotion in a computational model of an anguilliform swimmer. J. Theor. Biol.
**2015**, 385, 119–129. [Google Scholar] [CrossRef] [PubMed] - Zhu, L.; He, G.; Wang, S.; Miller, L.A.; Zhang, X.; You, Q.; Fang, S. An immersed boundary method by the lattice boltzmann approach in three dimensions. Comput. Math. Appl.
**2011**, 61, 3506–3518. [Google Scholar] [CrossRef] - Miller, L.A.; Santhanakrishnan, A.; Jones, S.K.; Hamlet, C.; Mertens, K.; Zhu, L. Reconfiguration and the reduction of vortex-induced vibrations in broad leaves. J. Exp. Biol.
**2012**, 215, 2716–2727. [Google Scholar] [CrossRef] [PubMed] - Kim, Y.; Peskin, C.S. 2d parachute simulation by the immersed boundary method. SIAM J. Sci. Comput.
**2006**, 28, 2294–2312. [Google Scholar] [CrossRef] - Peskin, C. The immersed boundary method. Acta Numer.
**2002**, 11, 479–517. [Google Scholar] [CrossRef] - Kempe, T.; Frohlich, J. An improved immersed boundary method with direct forcing for the simulation of particle laden flows. J. Comput. Phys.
**2012**, 231, 3663. [Google Scholar] [CrossRef] - Pinelli, A.; Naqavi, I.Z.; Piomelli, U.; Favier, J. Immersed boundary method for generalised finite volume and finite difference navier-stokes solvers. J. Comput. Phys.
**2010**, 229, 9073–9091. [Google Scholar] [CrossRef][Green Version] - Battista, N.A.; Strickland, W.C.; Miller, L.A. Ib2d: A python and matlab implementation of the immersed boundary method. Bioinspir. Biomim.
**2017**, 12, 036003. [Google Scholar] [CrossRef] [PubMed] - Battista, N.A.; Strickland, W.C.; Barrett, A.; Miller, L.A. Ib2d reloaded: A more powerful python and matlab implementation of the immersed boundary method. arXiv
**2017**, arXiv:1707.06928. [Google Scholar] - Griffith, B.E. An Adaptive and Distributed-Memory Parallel Implementation of the Immersed Boundary (ib) Method. Available online: https://github.com/IBAMR/IBAMR (accessed on 21 October 2014).
- Berger, M.J.; Oliger, J. Adaptive mesh refinement for hyperbolic partial-differential equations. J. Comput. Phys.
**1984**, 53, 484–512. [Google Scholar] [CrossRef] - Berger, M.J.; Colella, P. Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys.
**1989**, 82, 64–84. [Google Scholar] [CrossRef] - Cheer, A.Y.L.; Koehl, M.A.R. Paddles and rakes: Fluid flow through bristled appendages of small organisms. J. Theor. Biol.
**1987**, 129, 17–39. [Google Scholar] [CrossRef] - Lai, M.-C.; Peskin, C.S. An immersed boundary method with formal second-order accuracy and reduced numerical viscosity. J. Comp. Phys.
**2000**, 160, 705–719. [Google Scholar] [CrossRef] - Griffith, B.E.; Luo, X. Hybrid finite differencec/finite element immersed boundary method. Int. J. Numer. Methods Biomed. Eng.
**2017**. [Google Scholar] [CrossRef] - Adrian, R.J. Particle–imaging techniques for experimental fluid mechanics. Ann. Rev. Fluid Mech.
**1991**, 23, 261–304. [Google Scholar] [CrossRef] - Willert, C.E.; Gharib, M. Digital particle image velocimetry. Exp. Fluids
**1991**, 10, 181–193. [Google Scholar] [CrossRef] - Secomb, T.; Hsu, R.; Pries, A. A model for red blood cell motion in glycocalyx-lined capillaries. Am. J. Physiol. Heart Circ. Physiol.
**1998**, 274, H1016–H1022. [Google Scholar] - Secomb, T.; Hsu, R.; Pries, A. Effect of the endothelial surface layer on transmission of fluid shear stress to endothelial cells. J. Biorheol.
**2001**, 38, 143–150. [Google Scholar] - Damiano, E. The effect of the endothelial-cell glycocalyx on the motion of red blood cells through capillaries. Microvasc. Res.
**1998**, 55, 77–91. [Google Scholar] [CrossRef] [PubMed] - Feng, J.; Weinbaum, S. Lubrication theory in highly compressible porous media: The mechanics of skiing, from red cells to humans. J. Fluid Mech.
**2000**, 422, 281–317. [Google Scholar] [CrossRef] - Kim, S.J. 3d Numerical Simulation of Turbulent Open-Channel Flow through Vegetation. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA, USA, 2011. [Google Scholar]
- Bazilevs, Y.; Korobenko, A.; Deng, X.; Yan, J.; Kinzel, M.; Dabiri, J.O. Fsi modeling of vertical-axis wind turbines. J. App. Mech.
**2014**, 81, 081006. [Google Scholar] [CrossRef] - Kinzel, M.; Araya, D.B.; Dabiri, J.O. Turbulence in vertical axis wind turbine canopies. Phys. Fluids
**2015**, 27, 115102. [Google Scholar] [CrossRef] - Chen, N.; Gunzburger, M.; Wang, X. Asymptotic analysis of the differences between the stokes-darcy system with different interface conditions and the stokes-brinkman system. J. Math. Anal. Appl.
**2010**, 368, 658–676. [Google Scholar] [CrossRef] - Whitaker, S. Flow in porous media i: A theoretical derivation of darcy’s law. Transp. Porous Med.
**1986**, 1, 3–25. [Google Scholar] [CrossRef] - Whitaker, S. The forchheimer equation: A theoretical development. Transp. Porous Med.
**1996**, 25, 27–61. [Google Scholar] [CrossRef]

**Figure 2.**Depicting the physical setup for the Brinkman equations. There are two regions of interest, namely where ${\alpha}^{2}=0$ and ${\alpha}^{2}>0$.

**Figure 3.**Schematic of the experimental setup showing the side view (

**a**) and the top view (

**b**) of the flow chamber with the physical model inserted. Flow direction in the tank is from right to left.

**Figure 4.**Some example velocity vector fields obtained from particle image velocimetry (PIV) measurements of the flow around the physical models from the side view. The lengths of the vectors are proportional to the magnitude of the velocity. Flow is from right to left. The background of the images are snapshots of the flow. The regions where the model pins are located are enclosed by yellow boxes. In (

**a**–

**c**) the length of the layer was kept constant, while the height was varied, while in (

**d**) the length was decreased and the height was equal to that of (

**c**).

**Figure 5.**Experimentally and theoretically determined velocity profiles between two model layers. The experimental data is labeled with open circles, and the theoretical predictions are denoted with a solid line. The values of $\alpha $ have units of cm${}^{-1}$. In the top row, the spacing between pins and layer length is fixed at 5 mm and 43 mm, respectively. The height is varied from 8 mm (

**a**) to 22 mm (

**b**). The middle rows shows the velocity profiles are shown between two layers with a constant height of 28 mm and spacing of 7.5 mm. The length of the layer was varied from 1 mm (

**c**) to 52 mm (

**d**). The bottom row shows profiles for a constant layer height of 22 mm and lengths of 46.5 and 34 mm. The spacing was varied from 2.5 mm (

**e**) to 10 mm (

**f**).

**Figure 6.**Velocity vectors along planes taken parallel to the direction of flow and through the center of the cylinder (

**a,c**) and between periodic cylinders (

**b,d**) at $Re\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1,10$ ($R{e}_{d}=0.03125,0.3125$). Reynolds numbers were changed by varying the dynamic viscosity. The height of the cylinder is set to 0.156, such that $H/D$ = 5 and $G/D$ = 4.

**Figure 7.**Velocity vectors taken within planes parallel to the direction of flow at $Re=1$ ($R{e}_{d}=0.03125$). The planes are taken through the center of the cylinder (

**a**), one diameter from the center of a cylinder (

**b**), and halfway between rows of cylinders (

**c**). The height of the cylinders were set to 0.156, such that $H/D$ = 5 and $G/D$ = 4.

**Figure 8.**Magnitude of flow velocity averaged across planes perpendicular to the z-axis for four different cylinder heights. The top of each cylinder is plotted as a vertical dashed line. A dotted line has been used in the y and z plots where the average fluid velocity in the direction of the flow (x) falls below $1\times {10}^{-5}$.

**Figure 9.**Magnitude of flow velocity averaged across planes perpendicular to the z-axis for different cylinder spacing. The top of the cylinders is plotted as a vertical dashed line.

**Figure 10.**Average velocity in the direction of the flow for various cylinder heights, compared with the best fitting Brinkman model. The top of each cylinder is marked with a black X. The best choice of porosity is given for each cylinder height in the figure legend.

**Table 1.**Measured morphological parameters of various cylinder-like structures in biology. The diameter based Reynolds Number, $R{e}_{d}$, was computed using the length scale as the diameter of the cylinders, kinematic viscosity of air or water. The characteristic velocity was chosen as the free stream velocity (approximately 1–2 m/s for trichomes, the wind speed on an average day). Trichome measurements were taken from images using [21,22].

Structure | Diameter | Height | Gap | $\mathit{G}/\mathit{D}$ | H/D | ${\mathit{Re}}_{\mathit{d}}$ | References |
---|---|---|---|---|---|---|---|

Glycocalyx | 10–12 nm | 150–400 nm | 20 nm | 2 | 12–40 | $\mathcal{O}(-3)$ | [23] |

Microvilli | 90 nm | 2.5 $\mathsf{\mu}$m | 165 nm | 1.83 | 28 | $\mathcal{O}(-3)$ | [24] |

Aesthetascs | 5.69–8.1 $\mathsf{\mu}$m | 347–648 $\mathsf{\mu}$m | - | 2–30 | 61–80 | $\mathcal{O}(-2)-\mathcal{O}\left(1\right)$ | [25,26] |

Bristled wings | 0.3–2.5 $\mathsf{\mu}$m | 25–200 $\mathsf{\mu}$m | 2–16 $\mathsf{\mu}$m | 5–10 | 10–150 | $\mathcal{O}(-2)$ | [5] |

Trichomes | 28.1 $\mathsf{\mu}$m | 96.5 $\mathsf{\mu}$m | 65.6 $\mathsf{\mu}$m | 2.33 | 3.4 | $\mathcal{O}\left(1\right)$ | [27] |

Parameter | Value |
---|---|

L | 1.0 m |

$dx$ | $L/512$ m |

$ds$ | $L/1024$ m |

dt | ${10}^{-4}$ s |

$\mathsf{\rho}$ | 1000 kg/m${}^{3}$ |

$\mu $ | varied |

V | 0.1 m/s |

k${}_{targ}$ | 3.186 × 10${}^{2}$ kg/s${}^{2}$ |

tower spacing | $L/8-L$ |

end time | 10–200 s |

$Re$ | 0.1–10 |

$R{e}_{d}$ | $3.124\times {10}^{-3}-{10}^{-1}$ |

$G/D$ | 4–32 |

$H/D$ | 5–20 |

Parameter | Value |
---|---|

H | 0.05 m |

d${}_{pin}$ | 0.001 m |

V${}_{exp}$ | 0.002 m/s |

${\mathsf{\mu}}_{exp}$ | 1.229 kg/(ms) |

${\mathsf{\rho}}_{exp}$ | 1340 kg/m${}^{3}$ |

layer length | {0.01, 0.043} m |

layer height | {0.008, 0.022, 0.028} m |

pin spacing | {0.0025, 0.005, 0.01} m |

$R{e}_{d}$ | ∼0.001 |

$Re$ | ∼0.01 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Strickland, C.; Miller, L.; Santhanakrishnan, A.; Hamlet, C.; Battista, N.A.; Pasour, V. Three-Dimensional Low Reynolds Number Flows near Biological Filtering and Protective Layers. *Fluids* **2017**, *2*, 62.
https://doi.org/10.3390/fluids2040062

**AMA Style**

Strickland C, Miller L, Santhanakrishnan A, Hamlet C, Battista NA, Pasour V. Three-Dimensional Low Reynolds Number Flows near Biological Filtering and Protective Layers. *Fluids*. 2017; 2(4):62.
https://doi.org/10.3390/fluids2040062

**Chicago/Turabian Style**

Strickland, Christopher, Laura Miller, Arvind Santhanakrishnan, Christina Hamlet, Nicholas A. Battista, and Virginia Pasour. 2017. "Three-Dimensional Low Reynolds Number Flows near Biological Filtering and Protective Layers" *Fluids* 2, no. 4: 62.
https://doi.org/10.3390/fluids2040062