Undulating Seal Whiskers Evolved Optimal Wavelength-to-Diameter Ratio for Efficient Reduction in Vortex-Induced Vibrations

Seals are well-known for their remarkable hydrodynamic trail-following capabilities made possible by undulating flow-sensing whiskers that enable the seals to detect fish swimming as far as 180 m away. In this work, the form-function relationship in the undulating whiskers of two different phocid seal species, viz. harbor and gray seals, is studied. The geometry and material properties of excised harbor and grey seal whiskers are systematically characterized using blue light 3D scanning, optical and scanning electron microscopy, and nanoindentation. The effect of the undulating geometry on the whiskers' vibration in uniform water flow is studied using both experimental (piezoelectric MEMS and 3D-printed piezoresistive sensors developed in-house) and numerical (finite element method) techniques. The results indicate that the dimensionless ratio of undulation wavelength to mean whisker diameter (λ/Dm ) in phocid seals may have evolved to be in the optimal range of 4.4-4.6, enabling an order-of-magnitude reduction in vortex-induced vibrations (compared to a similarly-shaped circular cylinder) and, consequently, an enhanced flow sensing capability with minimal self-induced noise. The results highlight the importance of the dimensionless λ/Dm ratio in the biomimetic design of seal whisker-inspired vibration-resistant structures, such as marine risers and wake detection sensors for submarines.

Two-fluid dynamics and micron-thin boundary layers shape cytoplasmic flows in early Drosophila embryos

Cytoplasmic flows are widely emerging as key functional players in development. In early Drosophila embryos, flows drive the spreading of nuclei across the embryo. Here, we combine hydrodynamic modeling with quantitative imaging to develop a two-fluid model that features an active actomyosin gel and a passive viscous cytosol. Gel contractility is controlled by the cell cycle oscillator, the two fluids being coupled by friction. In addition to recapitulating experimental flow patterns, our model explains observations that remained elusive and makes a series of predictions. First, the model captures the vorticity of cytosolic flows, which highlights deviations from Stokes' flow that were observed experimentally but remained unexplained. Second, the model reveals strong differences in the gel and cytosol motion. In particular, a micron-sized boundary layer is predicted close to the cortex, where the gel slides tangentially while the cytosolic flow cannot slip. Third, the model unveils a mechanism that stabilizes the spreading of nuclei with respect to perturbations of their initial positions. This self-correcting mechanism is argued to be functionally important for proper nuclear spreading. Fourth, we use our model to analyze the effects of flows on the transport of the morphogen Bicoid and the establishment of its gradients. Finally, the model predicts that the flow strength should be reduced if the shape of the domain is more round, which is experimentally confirmed in Drosophila mutants. Thus, our two-fluid model explains flows and nuclear positioning in early Drosophila, while making predictions that suggest novel future experiments.

A one-dimensional model for the pulsating flow of cerebrospinal fluid in the spinal canal

The monitoring of intracranial pressure (ICP) fluctuations, which is needed in the context of a number of neurological diseases, requires the insertion of pressure sensors, an invasive procedure with considerable risk factors. Intracranial pressure fluctuations drive the wave-like pulsatile motion of cerebrospinal fluid (CSF) along the compliant spinal canal. Systematically derived simplified models relating the ICP fluctuations with the resulting CSF flow rate can be useful in enabling indirect evaluations of the former from non-invasive magnetic resonance imaging (MRI) measurements of the latter. As a preliminary step in enabling these predictive efforts, a model is developed here for the pulsating viscous motion of CSF in the spinal canal, assumed to be a linearly elastic compliant tube of slowly varying section, with a Darcy pressure-loss term included to model the fluid resistance introduced by the trabeculae, which are thin collagen-reinforced columns that form a web-like structure stretching across the spinal canal. Use of Fourier-series expansions enables predictions of CSF flow rate for realistic anharmonic ICP fluctuations. The flow rate predicted using a representative ICP waveform together with a realistic canal anatomy is seen to compare favourably with in vivo phase-contrast MRI measurements at multiple sections along the spinal canal. The results indicate that the proposed model, involving a limited number of parameters, can serve as a basis for future quantitative analyses targeting predictions of ICP temporal fluctuations based on MRI measurements of spinal-canal anatomy and CSF flow rate.

Flow-signal correlation in seal whisker array sensing

Phocid seals detect and track artificial or biogenic hydrodynamic trails based on mechanical signals of their whisker arrays. In this paper, we investigated the correlations between flow structures and whisker array signals using a simplified numerical model of fluid-structure interaction (FSI). Three-dimensional (3D) wakes of moving paddles in three different shapes (rectangular plate, undulated plate, and circular cylinder) were simulated using an in-house immersed-boundary-method-based computational fluid dynamics solver. One-way FSI was then simulated to obtain the dynamic behavior and root signal of each whisker in the two whisker arrays on a seal head in each wake. The position, geometry, and material properties of each whisker were modeled based on the measurements reported in literatures. The correlations between the wake structures and whisker array signals were analyzed. It was found that the patterns of the signals on the whisker arrays can reflect the strength, timing, and moving trajectories of the jets induced by the vortices in the wakes. Specifically, the rectangular plate generates the strongest starting vortex ring as well as the strongest jets, while the undulated plate generates the weakest ones. These flow features are fully reflected by the largest whisker signal magnitude in the rectangular plate sensing and the smallest one in the undulated plate sensing. Moreover, the timing of the signal initiation and the maximum signal agree well with the timing of the jet reaching the arrays and the maximum flow speed, respectively. The correlation coefficient between the moving trajectories of the jet and the movement of the high signal level region in the array was found to be higher than 0.9 in the rectangular plate case. The results provide a physical insight into the mechanisms of seal whisker flow sensing.

The art of coarse Stokes: Richardson extrapolation improves the accuracy and efficiency of the method of regularized stokeslets

The method of regularized stokeslets is widely used in microscale biological fluid dynamics due to its ease of implementation, natural treatment of complex moving geometries, and removal of singular functions to integrate. The standard implementation of the method is subject to high computational cost due to the coupling of the linear system size to the numerical resolution required to resolve the rapidly varying regularized stokeslet kernel. Here, we show how Richardson extrapolation with coarse values of the regularization parameter is ideally suited to reduce the quadrature error, hence dramatically reducing the storage and solution costs without loss of accuracy. Numerical experiments on the resistance and mobility problems in Stokes flow support the analysis, confirming several orders of magnitude improvement in accuracy and/or efficiency.

Collective dynamics of sperm cells

While only a single sperm may fertilize the egg, getting to the egg can be facilitated, and possibly enhanced, by sperm group dynamics. Examples range from the trains formed by wood mouse sperm to the bundles exhibited by echidna sperm. In addition, observations of wave-like patterns exhibited by ram semen are used to score prospective sample fertility for artificial insemination in agriculture. In this review, we discuss these experimental observations of collective dynamics, as well as describe recent mechanistic models that link the motion of individual sperm cells and their flagella to observed collective dynamics. Establishing this link in models involves negotiating the disparate time- and length scales involved, typically separated by a factor of 1000, to capture the dynamics at the greatest length scales affected by mechanisms at the shortest time scales. Finally, we provide some outlook on the subject, in particular, the open questions regarding how collective dynamics impacts fertility. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

Three-dimensional scaling laws of cetacean propulsion characterize the hydrodynamic interplay of flukes' shape and kinematics

Cetaceans convert dorsoventral body oscillations into forward velocity with a complex interplay between their morphological and kinematic features and the fluid environment. However, it is unknown to what extent morpho-kinematic features of cetaceans are intertwined to maximize their efficiency. By interchanging the shape and kinematic variables of five cetacean species, the interplay of their flukes morpho-kinematic features is examined by characterizing their thrust, power and propulsive efficiency. It is determined that the shape and kinematics of the flukes have considerable influence on force production and power consumption. Three-dimensional heaving and pitching scaling laws are developed by considering both added mass and circulatory-based forces, which are shown to closely model the numerical data. Using the scaling relations as a guide, it is determined that the added mass forces are important in predicting the trend between the efficiency and aspect ratio, however, the thrust and power are driven predominately by the circulatory forces. The scaling laws also reveal that there is an optimal dimensionless heave-to-pitch ratio h* that maximizes the efficiency. Moreover, the optimal h* varies with the aspect ratio, the amplitude-to-chord ratio and the Lighthill number. This indicates that the shape and kinematics of propulsors are intertwined, that is, there are specific kinematics that are tailored to the shape of a propulsor in order to maximize its propulsive efficiency.

A design principle of root length distribution of plants

Shaping a plant root into an ideal structure for water capture is increasingly important for sustainable agriculture in the era of global climate change. Although the current genetic engineering of crops favours deep-reaching roots, here we show that nature has apparently adopted a different strategy of shaping roots. We construct a mathematical model for optimal root length distribution by considering that plants seek maximal water uptake at the metabolic expenses of root growth. Our theory finds a logarithmic decrease of root length density with depth to be most beneficial for efficient water uptake, which is supported by biological data as well as our experiments using root-mimicking network systems. Our study provides a tool to gauge the relative performance of root networks in transgenic plants engineered to endure a water deficit. Moreover, we lay a fundamental framework for mechanical understanding and design of water-absorptive growing networks, such as medical and industrial fluid transport systems and soft robots, which grow in porous media including soils and biotissues.

Experimental Study of Body-Fin Interaction and Vortex Dynamics Generated by a Two Degree-Of-Freedom Fish Model

Oscillatory modes of swimming are used by a majority of aquatic swimmers to generate thrust. This work seeks to understand the phenomenological relationship between the body and caudal fin for fast and efficient thunniform swimming. Phase-averaged velocity data was collected and analyzed in order to understand the effects of body-fin kinematics on the wake behind a two degree-of-freedom fish model. The model is based on the yellowfin tuna (Thunnus albacares) which is known to be both fast and efficient. Velocity data was obtained along the side of the tail and caudal fin region as well as in the wake downstream of the caudal fin. Body-generated vortices were found to be small and have an insignificant effect on the caudal fin wake. The evolution of leading edge vortices formed on the caudal fin varied depending on the body-fin kinematics. The circulation produced at the trailing edge during each half-cycle was found to be relatively insensitive to the freestream velocity, but also varied with body-fin kinematics. Overall, the generation of vorticity in the wake was found to dependent on the trailing edge motion profile and velocity. Even relatively minor deviations from the commonly used model of sinusoidal motion is shown to change the strength and organization of coherent structures in the wake, which have been shown in the literature to be related to performance metrics such as thrust and efficiency.

Hotspots of boundary accumulation: dynamics and statistics of micro-swimmers in flowing films

Biological flows over surfaces and interfaces can result in accumulation hotspots or depleted voids of microorganisms in natural environments. Apprehending the mechanisms that lead to such distributions is essential for understanding biofilm initiation. Using a systematic framework, we resolve the dynamics and statistics of swimming microbes within flowing films, considering the impact of confinement through steric and hydrodynamic interactions, flow and motility, along with Brownian and run-tumble fluctuations. Micro-swimmers can be peeled off the solid wall above a critical flow strength. However, the interplay of flow and fluctuations causes organisms to migrate back towards the wall above a secondary critical value. Hence, faster flows may not always be the most efficacious strategy to discourage biofilm initiation. Moreover, we find run-tumble dynamics commonly used by flagellated microbes to be an intrinsically more successful strategy to escape from boundaries than equivalent levels of enhanced Brownian noise in ciliated organisms.

Squirmers with swirl: a model for Volvox swimming

Colonies of the green alga Volvox are spheres that swim through the beating of pairs of flagella on their surface somatic cells. The somatic cells themselves are mounted rigidly in a polymeric extracellular matrix, fixing the orientation of the flagella so that they beat approximately in a meridional plane, with axis of symmetry in the swimming direction, but with a roughly [Formula: see text] azimuthal offset which results in the eponymous rotation of the colonies about a body-fixed axis. Experiments on colonies of Volvox carteri held stationary on a micropipette show that the beating pattern takes the form of a symplectic metachronal wave (Brumley et al. Phys. Rev. Lett., vol. 109, 2012, 268102). Here we extend the Lighthill/Blake axisymmetric, Stokes-flow model of a free-swimming spherical squirmer (Lighthill Commun. Pure Appl. Maths, vol. 5, 1952, pp. 109-118; Blake J. Fluid Mech., vol. 46, 1971b, pp. 199-208) to include azimuthal swirl. The measured kinematics of the metachronal wave for 60 different colonies are used to calculate the coefficients in the eigenfunction expansions and hence predict the mean swimming speeds and rotation rates, proportional to the square of the beating amplitude, as functions of colony radius. As a test of the squirmer model, the results are compared with measurements (Drescher et al. Phys. Rev. Lett., vol. 102, 2009, 168101) of the mean swimming speeds and angular velocities of a different set of 220 colonies, also given as functions of colony radius. The predicted variation with radius is qualitatively correct, but the model underestimates both the mean swimming speed and the mean angular velocity unless the amplitude of the flagellar beat is taken to be larger than previously thought. The reasons for this discrepancy are discussed.

Flow disturbances generated by feeding and swimming zooplankton

Interactions between planktonic organisms, such as detection of prey, predators, and mates, are often mediated by fluid signals. Consequently, many plankton predators perceive their prey from the fluid disturbances that it generates when it feeds and swims. Zooplankton should therefore seek to minimize the fluid disturbance that they produce. By means of particle image velocimetry, we describe the fluid disturbances produced by feeding and swimming in zooplankton with diverse propulsion mechanisms and ranging from 10-µm flagellates to greater than millimeter-sized copepods. We show that zooplankton, in which feeding and swimming are separate processes, produce flow disturbances during swimming with a much faster spatial attenuation (velocity u varies with distance r as u ∝ r(-3) to r(-4)) than that produced by zooplankton for which feeding and propulsion are the same process (u ∝ r(-1) to r(-2)). As a result, the spatial extension of the fluid disturbance produced by swimmers is an order of magnitude smaller than that produced by feeders at similar Reynolds numbers. The "quiet" propulsion of swimmers is achieved either through swimming erratically by short-lasting power strokes, generating viscous vortex rings, or by "breast-stroke swimming." Both produce rapidly attenuating flows. The more "noisy" swimming of those that are constrained by a need to simultaneously feed is due to constantly beating flagella or appendages that are positioned either anteriorly or posteriorly on the (cell) body. These patterns transcend differences in size and taxonomy and have thus evolved multiple times, suggesting a strong selective pressure to minimize predation risk.

A physical explanation of the temperature dependence of physiological processes mediated by cilia and flagella

The majority of biological rates are known to exhibit temperature dependence. Here I reveal a direct link between temperature and ecologically relevant rates such as swimming speeds in Archaea, Bacteria, and Eukaryotes as well as fluid-pumping and filtration rates in many metazoans, and show that this relationship is driven by movement rates of cilia and flagella. I develop models of the temperature dependence of cilial and flagellar movement rates and evaluate these with an extensive compilation of data from the literature. The model captures the temperature dependence of viscosity and provides a mechanistic and biologically interpretable explanation for the temperature dependence of a range of ecologically relevant processes; it also reveals a clear dependence on both reaction rate-like processes and the physics of the environment. The incorporation of viscosity allows further insight into the effects of environmental temperature variation and of processes, such as disease, that affect the viscosity of blood or other body fluids.

Blood flow in small tubes: quantifying the transition to the non-continuum regime

In small vessels blood is usually treated as a Newtonian fluid down to diameters of ~200 μm. We investigate the flow of red blood cell (RBC) suspensions driven through small tubes (diameters 10-150 μm) in the range marking the transition from arterioles and venules to the largest capillary vessels. The results of the simulations combined with previous simulations of uniform shear flow and experimental data show that for diameters less than ~100 μm the suspension's stress cannot be described as a continuum, even a heterogeneous one. We employ the dissipative particle dynamics (DPD) model, which has been successfully used to predict human blood bulk viscosity in homogeneous shear flow. In tube flow the cross-stream stress gradient induces an inhomogeneous distribution of RBCs featuring a centreline cell density peak, and a cell-free layer (CFL) next to the wall. For a neutrally buoyant suspension the imposed linear shear-stress distribution together with the differentiable velocity distribution allow the calculation of the local viscosity across the tube section. The viscosity across the section as a function of the strain rate is found to be essentially independent of tube size for the larger diameters and is determined by the local haematocrit (H) and shear rate. Other RBC properties such as asphericity, deformation, and cell-flow orientation exhibit similar dependence for the larger tube diameters. As the tube size decreases below ~100 μm in diameter, the viscosity in the central region departs from the large-tube similarity function of the shear rate, since H increases significantly towards the centreline. The dependence of shear stress on tube size, in addition to the expected local shear rate and local haematocrit, implies that blood flow in small tubes cannot be described as a heterogeneous continuum. Based on the analysis of the DPD simulations and on available experimental results, we propose a simple velocity-slip model that can be used in conjunction with continuum-based simulations.

Modelling single- and tandem-bubble dynamics between two parallel plates for biomedical applications

Carefully timed tandem microbubbles have been shown to produce directional and targeted membrane poration of individual cells in microfluidic systems, which could be of use in ultrasound-mediated drug and gene delivery. This study aims at contributing to the understanding of the mechanisms at play in such an interaction. The dynamics of single and tandem microbubbles between two parallel plates is studied numerically and analytically. Comparisons are then made between the numerical results and the available experimental results. Numerically, assuming a potential flow, a three-dimensional boundary element method (BEM) is used to describe complex bubble deformations, jet formation, and bubble splitting. Analytically, compressibility and viscous boundary layer effects along the channel walls, neglected in the BEM model, are considered while shape of the bubble is not considered. Comparisons show that energy losses modify the bubble dynamics when the two approaches use identical initial conditions. The initial conditions in the boundary element method can be adjusted to recover the bubble period and maximum bubble volume when in an infinite medium. Using the same conditions enables the method to recover the full dynamics of single and tandem bubbles, including large deformations and fast re-entering jet formation. This method can be used as a design tool for future tandem-bubble sonoporation experiments.

Simulation and modelling of slip flow over surfaces grafted with polymer brushes and glycocalyx fibres

Fabrication of functionalized surfaces using polymer brushes is a relatively simple process and parallels the presence of glycocalyx filaments coating the luminal surface of our vasculature. In this paper, we perform atomistic-like simulations based on dissipative particle dynamics (DPD) to study both polymer brushes and glycocalyx filaments subject to shear flow, and we apply mean-field theory to extract useful scaling arguments on their response. For polymer brushes, a weak shear flow has no effect on the brush density profile or its height, while the slip length is independent of the shear rate and is of the order of the brush mesh size as a result of screening by hydrodynamic interactions. However, for strong shear flow, the polymer brush is penetrated deeper and is deformed, with a corresponding decrease of the brush height and an increase of the slip length. The transition from the weak to the strong shear regime can be described by a simple 'blob' argument, leading to the scaling γ̇₀ ∝ σ^3/2, where γ̇₀ is the critical transition shear rate and σ is the grafting density. Furthermore, in the strong shear regime, we observe a cyclic dynamic motion of individual polymers, causing a reversal in the direction of surface flow. To study the glycocalyx layer, we first assume a homogeneous flow that ignores the discrete effects of blood cells, and we simulate microchannel flows at different flow rates. Surprisingly, we find that, at low Reynolds number, the slip length decreases with the mean flow velocity, unlike the behaviour of polymer brushes, for which the slip length remains constant under similar conditions. (The slip length and brush height are measured with respect to polymer mesh size and polymer contour length, respectively.) We also performed additional DPD simulations of blood flow in a tube with walls having a glycocalyx layer and with the deformable red blood cells modelled accurately at the spectrin level. In this case, a plasma cell-free layer is formed, with thickness more than three times the glycocalyx layer. We then find our scaling arguments based on the homogeneous flow assumption to be valid for this physiologically correct case as well. Taken together, our findings point to the opposing roles of conformational entropy and bending rigidity - dominant effects for the brush and glycocalyx, respectively - which, in turn, lead to different flow characteristics, despite the apparent similarity of the two systems.

Modeling the hydrodynamics of Phloem sieve plates

Sieve plates have an enormous impact on the efficiency of the phloem vascular system of plants, responsible for the distribution of photosynthetic products. These thin plates, which separate neighboring phloem cells, are perforated by a large number of tiny sieve pores and are believed to play a crucial role in protecting the phloem sap from intruding animals by blocking flow when the phloem cell is damaged. The resistance to the flow of viscous sap in the phloem vascular system is strongly affected by the presence of the sieve plates, but the hydrodynamics of the flow through them remains poorly understood. We propose a theoretical model for quantifying the effect of sieve plates on the phloem in the plant, thus unifying and improving previous work in the field. Numerical simulations of the flow in real and idealized phloem channels verify our model, and anatomical data from 19 plant species are investigated. We find that the sieve plate resistance is correlated to the cell lumen resistance, and that the sieve plate and the lumen contribute almost equally to the total hydraulic resistance of the phloem translocation pathway.

Filter feeders and plankton increase particle encounter rates through flow regime control

Collisions between particles or between particles and other objects are fundamental to many processes that we take for granted. They drive the functioning of aquatic ecosystems, the onset of rain and snow precipitation, and the manufacture of pharmaceuticals, powders and crystals. Here, I show that the traditional assumption that viscosity dominates these situations leads to consistent and large-scale underestimation of encounter rates between particles and of deposition rates on surfaces. Numerical simulations reveal that the encounter rate is Reynolds number dependent and that encounter efficiencies are consistent with the sparse experimental data. This extension of aerosol theory has great implications for understanding of selection pressure on the physiology and ecology of organisms, for example filter feeders able to gather food at rates up to 5 times higher than expected. I provide evidence that filter feeders have been strongly selected to take advantage of this flow regime and show that both the predicted peak concentration and the steady-state concentrations of plankton during blooms are approximately 33% of that predicted by the current models of particle encounter. Many ecological and industrial processes may be operating at substantially greater rates than currently assumed.