Metachronal Wave Research Articles

Free surface waves induced by vortex shedding in cylinder arrays

ABSTRACTThe paper presents and discusses the results of an accurate and extensive experimental investigation on the waves that originate when an array of cylinders is placed in an otherwise uniform open channel flow. The study was originally aimed at providing new and more accurate data to calibrate and validate a theoretical model recently proposed in the literature. However, two interesting characteristics of the phenomenon at hand, never reported in the literature, also emerged from the experimental study, i.e. (i) the wave produced by the cylinders is actually a metachronal wave rather than a pure transverse seiche; (ii) two different resonant conditions exist for each single wave mode, i.e. two typical Strouhal numbers characterize the resonance frequency of vortex shedding from the cylinders in an array. Evidence of the metachronal wave, as well as some preliminary data on the longitudinal component of this wave, are given in the paper. Also, the double resonance condition is experimentally demonstrated and discussed.

Study of heat transfer on physiological driven movement with CNT nanofluids and variable viscosity

Background and objectivesWith ongoing interest in CNT nanofluids and materials in biotechnology, energy and environment, microelectronics, composite materials etc., the current investigation is carried out to analyze the effects of variable viscosity and thermal conductivity of CNT nanofluids flow driven by cilia induced movement through a circular cylindrical tube. Metachronal wave is generated by the beating of cilia and mathematically modeled as elliptical wave propagation by Blake (1971). Methods, results and conclusionsThe problem is formulated in the form of nonlinear partial differential equations, which are simplified by using the dimensional analysis to avoid the complicacy of dimensional homogeneity. Lubrication theory is employed to linearize the governing equations and it is also physically appropriate for cilia movement. Analytical solutions for velocity, temperature and pressure gradient and stream function are obtained. The analytical results are numerically simulated by using the Mathematica Software and plotted the graphs for velocity profile, temperature profile, pressure gradient and stream lines for better discussion and visualization. This model is applicable in physiological transport phenomena to explore the nanotechnology in engineering the artificial cilia and ciliated tube/pipe.

MHD dissipative flow and heat transfer of Casson fluids due to metachronal wave propulsion of beating cilia with thermal and velocity slip effects under an oblique magnetic field

A theoretical investigation of magnetohydrodynamic (MHD) flow and heat transfer of electrically-conducting viscoplastic fluids through a channel is conducted. The robust Casson model is implemented to simulate viscoplastic behavior of fluids. The external magnetic field is oblique to the fluid flow direction. Viscous dissipation effects are included. The flow is controlled by the metachronal wave propagation generated by cilia beating on the inner walls of the channel. The mathematical formulation is based on deformation in longitudinal and transverse velocity components induced by the ciliary beating phenomenon with cilia assumed to follow elliptic trajectories. The model also features velocity and thermal slip boundary conditions. Closed-form solutions to the non-dimensional boundary value problem are obtained under physiological limitations of low Reynolds number and large wavelength. The influence of key hydrodynamic and thermo-physical parameters i.e. Hartmann (magnetic) number, Casson (viscoplastic) fluid parameter, thermal slip parameter and velocity slip parameter on flow characteristics are investigated. A comparative study is also made with Newtonian fluids (corresponding to massive values of plastic viscosity). Stream lines are plotted to visualize trapping phenomenon. The computations reveal that velocity increases with increasing the magnitude of Hartmann number near the channel walls whereas in the core flow region (center of the channel) significant deceleration is observed. Temperature is elevated with greater Casson parameter, Hartmann number, velocity slip, eccentricity parameter, thermal slip and also Brinkmann (dissipation) number. Furthermore greater Casson parameter is found to elevate the quantity and size of the trapped bolus. In the pumping region, the pressure rise is reduced with greater Hartmann number, velocity slip, and wave number whereas it is enhanced with greater cilia length.

Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion

In this article, the influence of magnetohydrodynamics (MHD) on cilia motion of particle–fluid suspension through a porous planar channel has been investigated. The governing equations of Casson fluid model for fluid phase and particulate phase are solved by taking the assumption of long wavelength and neglecting the inertial forces due to laminar flow. The solutions for the resulting differential equations have been obtained analytically and a close form of solutions is presented. The expression for pressure rise along the whole length of the channel is evaluated numerically. The influences of all the physical parameters are demonstrated graphically. Trapping mechanism has also been discussed with the help of streamlines. It is observed that due to the influence of magnetohydrodynamics and particle volume fraction, velocity of the fluid decreases. It is also found that pressure rise shows similar behaviour for particle volume fraction and Casson fluid parameter.

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 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.

Metachronal wave of artificial cilia array actuated by applied magnetic field

In this paper, a biomimetic microstructure related to cilia, which are effective fluidic and conveying systems in nature, is described. Authors have already reported that a magnetic elastomer pillar actuated by a rotating magnetic field can work like a natural cilium. In the present work, we show examples of a cilia array with a metachronal wave as the next step. A metachronal wave is a sequential action of a number of cilia. It is theoretically known that a metachronal wave gives a higher fluidic efficiency; however, there has been no report on a metachronal wave by artificial cilia. We prepared magnetic elastomer pillars that contain chainlike clusters of magnetic particles. The orientation of chains was set to be different in each pillar so that each pillar will deform with a different phase.

Coordinated beating of algal flagella is mediated by basal coupling

Cilia and flagella often exhibit synchronized behavior; this includes phase locking, as seen in Chlamydomonas, and metachronal wave formation in the respiratory cilia of higher organisms. Since the observations by Gray and Rothschild of phase synchrony of nearby swimming spermatozoa, it has been a working hypothesis that synchrony arises from hydrodynamic interactions between beating filaments. Recent work on the dynamics of physically separated pairs of flagella isolated from the multicellular alga Volvox has shown that hydrodynamic coupling alone is sufficient to produce synchrony. However, the situation is more complex in unicellular organisms bearing few flagella. We show that flagella of Chlamydomonas mutants deficient in filamentary connections between basal bodies display markedly different synchronization from the wild type. We perform micromanipulation on configurations of flagella and conclude that a mechanism, internal to the cell, must provide an additional flagellar coupling. In naturally occurring species with 4, 8, or even 16 flagella, we find diverse symmetries of basal body positioning and of the flagellar apparatus that are coincident with specific gaits of flagellar actuation, suggesting that it is a competition between intracellular coupling and hydrodynamic interactions that ultimately determines the precise form of flagellar coordination in unicellular algae.

Influences of slip and Cu-blood nanofluid in a physiological study of cilia

ObjectiveThe objective of this paper is to study the influences of slip and Cu-blood nanofluid in a physiological study of cilia. Design/methodology/approachThe right wall and the left wall possess metachronal wave that is traveling along the outer boundary of the channel. The features of ciliary structures are determined by the dominance of viscous effects over inertial effects using the long wavelength approximation. The flow possessions for the viscous nano fluid are solved as a function of the cilia and metachronal wave velocity. Mathematica numerical simulation is used to calculate pressure rise. Exact solutions are calculated for the temperature and for the velocity profile. FindingsNumerical integration has been implemented to get the expression of pressure rise. Graphical results have been offered for pressure rise, temperature and stream function for various physical parameters of interest. Symmetry of the curved channel is recovered for larger values of the curvature parameter. It is found from the graphs of the pressure rise that reflux case occurs for increasing values of curvature and velocity slip parameter. It is also found that temperature decreases for increasing values of nanoparticle volume fraction. Basically, higher thermal conductivity of the nanoparticles plays a key role for quick heat dissipation, and this justifies the use of the copper nanoparticles in different situations as a coolant. ConclusionsVelocity graph near the right wall of the channel decreases when we add nanoparticles into our base fluid, whereas an opposite behavior is depicted near the left wall due to ciliated tips. Pressure gradient increases for increasing values of Gr (Grashof number), L (velocity slip parameter) ϕ (nanoparticle volume fraction) and k (curvature parameter). Pressure rise shows increasing behavior for the increasing values of ε, Gr and ϕ throughout the region. Shear stress graphs show the increasing behavior for increasing values of curvature parameter and volume fraction of the nanoparticle. Temperature profile decreases when we add nanoparticles into our base fluid. Temperature profile also increases with an increase in thermal slip parameter γ and heat absorption parameter. It is observed that the number of trapped bolus increases when we increase the value of the cilia length parameter.

Beating of Gill Lateral Cell Cilia of Crassostrea virginica Involves Neuronal Innervation and the Presence of Gap Junctions

Suspension feeding bivalves use a ciliary pump driven by tracts of lateral cilia located on gill filaments to provide water currents for feeding, respiration and waste elimination. These cells in Crassostrea virginica are controlled by serotonergic‐dopaminergic innervation via the branchial nerve. Serotonin is the neurotransmitter that increases beating rates, while dopamine decreases it. Lateral cells of many other bivalves are similarly innervated. The cellular mechanism regulating lateral cell cilia beating is not fully explained. Second messenger systems are involved and gill filaments receive branches from the branchial nerve, but it is not known if each lateral cell is individually innervated. Lateral cells have gap junctions and questions arise if gap junctions are involved in controlling the rate or coordination of beating. We hypothesize gap junctions are involved in the cellular mechanism regulating lateral cell cilia beating rate and/or coordination to maintain the metachronal wave pattern. To study this we made immunohistological preparations using antibodies to visualize the gap junctions, and used the Sihler Whole Mount method to visualize nerves in gill filaments. We also set up preparations in which gill filaments were divided into proximal and distal sections by a petroleum jelly barrier, which prevented diffusion of drugs from one side to the other. We stimulated the branchial nerve entering the proximal end of the filaments with suction electrodes (5 hz, 2 msec, 10 – 20 volts), which causes a release of serotonin and increases beating rates. Stroboscopic microscopy was used to measure cilia beating rates. Drugs were applied to distal ends of preparations and beating rates of proximal and distal ends were compared. Applying procaine, a local anesthetic, to distal ends prevented electrical stimulation from increasing beating rates, while beating rate at the proximal ends increased, showing nerve impulse propagation is necessary for ciliary response. Using the same procedure, we tested 3 gap junction blockers, each with different mechanisms of action: lindane, diphenylborinic anhydride and mefloquine. Mefloquine had no effect on beating, but lindane and diphenylborinic anhydride both produced statistically significant reductions in beating rates at the distal ends. Lindane was stronger, causing a dose‐dependent (10−7 – 10−4M) blockage of beating, while diphenylborinic anhydride only blocked at a high dose of 10−4M. Results show neuronal innervation to filaments in conjunction with functioning gap junctions in the cells are necessary to increase cilia beating rates. Disrupting either one interfered with the normal response. Lindane is reported to reduce gap junction permeability and increase intracellular Ca2+ levels; either one or both of these actions may be responsible for the results we obtained.Support or Funding InformationThis work was supported in part by 2R25GM06003 of the Bridge Program of NIGMS and 0516041071 of NYSDOE.

Hydrodynamic interactions of cilia on a spherical body.

Microorganisms develop coordinated beating patterns on surfaces lined with cilia known as metachronal waves. For a chain of cilia attached to a flat ciliate, it has been shown that hydrodynamic interactions alone can lead the system to synchronize. However, several microorganisms possess a curve-shaped ciliate body and so to understand the effect of this geometry on the formation of metachronal waves, we evaluate the hydrodynamic interactions of cilia near a large spherical body. Using a minimal model, we show that for a chain of cilia around the sphere, the natural periodicity in the geometry leads the system to synchronize. We also report an emergent wavelike behavior when an asymmetry is introduced to the system.

Realizing the Physics of Motile Cilia Synchronization with Driven Colloids

Cilia and flagella in biological systems often show large scale cooperative behaviors such as the synchronization of their beats in “metachronal waves.” These are beautiful examples of emergent dynamics in biology, and are essential for life, allowing diverse processes from the motility of eukaryotic microorganisms, to nutrient transport and clearance of pathogens from mammalian airways. How these collective states arise is not fully understood, but it is clear that individual cilia interact mechanically, and that a strong and long-ranged component of the coupling is mediated by the viscous fluid. We review here the work by ourselves and others aimed at understanding the behavior of hydrodynamically coupled systems, and particularly a set of results that have been obtained both experimentally and theoretically by studying actively driven colloidal systems. In these controlled scenarios, it is possible to selectively test aspects of living motile cilia, such as the geometrical arrangement, the effects of the driving profile and the distance to no-slip boundaries. We outline and give examples of how it is possible to link model systems to observations on living systems, which can be made on microorganisms, on cell cultures or on tissue sections. This area of research has clear clinical application in the long term, as severe pathologies are associated with compromised cilia function in humans.

Influence of Magnetic Field and Slip on Jeffrey Fluid in a Ciliated Symmetric Channel with Metachronal Wave Pattern

Influence of Magnetic Field and Slip on Jeffrey Fluid in a Ciliated Symmetric Channel with Metachronal Wave Pattern

Entropy generation analysis for metachronal beating of ciliated Cu-water nanofluid with magnetic field

In this paper, we have discussed the entropy generation on the ciliary motion of the peristaltic transport of fluids in human body. On the basis of the mathematical model of the copper nanofluids with pure water as the base fluid, the complete problem is analysed under the presence of heat transfer. It is considered that a group of cilia operate together and produce metachronal waves to transport the fluid. The governing flow equations are non-linear partial differential equations, which are reduced to ordinary differential equation form using the dimensionless variables. The investigation is carried out under the assumptions of low Reynold's number and long wavelength. Exact solutions of the fluid velocity, pressure gradient and temperature profile have been obtained. The obtained expressions are then described through graphs for various relevant parameters. Trapping phenomenon for nanoparticle volume fraction is presented at the end of the paper.

Ciliary metachronal wave propagation on the compliant surface of Paramecium cells

Ciliary movements in protozoa exhibit metachronal wave-like coordination, in which a constant phase difference is maintained between adjacent cilia. It is at present generally thought that metachronal waves require hydrodynamic coupling between adjacent cilia and the extracellular fluid. To test this hypothesis, we aspirated a Paramecium cell using a micropipette which completely sealed the surface of the cell such that no fluid could pass through the micropipette. Thus, the anterior and the posterior regions of the cell were hydrodynamically decoupled. Nevertheless, we still observed that metachronal waves continued to propagate from the anterior to the posterior ends of the cell, suggesting that in addition to hydrodynamic coupling, there are other mechanisms that can also transmit the metachronal waves. Such transmission was also observed in computational modeling where the fluid was fully decoupled between two partitions of a beating ciliary array. We also imposed cyclic stretching on the surface of live Paramecium cells and found that metachronal waves persisted in the presence of cyclic stretching. This demonstrated that, in addition to hydrodynamic coupling, a compliant substrate can also play a critical role in mediating the propagation of metachronal waves.

Trapping study of nanofluids in an annulus with cilia

The main purpose of this article is to present a mathematical model of ciliary motion in an annulus. In this analysis, two dimensional flow of a viscous nanofluid is observed in an annulus with ciliated tips. The current theoretical model may be supposed as mathematical illustration to the movement of ciliary motion in the presence of an endoscopic tube (or catheter tube). The inner tube is rigid, while the outer tube takes a metachronal wave. The features of ciliary structures are determined by the dominance of viscous effects over inertial effects using the long-wavelength approximation. Exact solutions have been established for both velocity and temperature profiles, which include nanoparticle effects. The features of the ciliary motion are analyzed by plotting graphs and discussed in detail.

Swimming with stiff legs at low Reynolds number.

Locomotion at low Reynolds number is not possible with cycles of reciprocal motion, an example being the oscillation of a single pair of rigid paddles or legs. Here, I demonstrate the possibility of swimming with two or more pairs of legs. They are assumed to oscillate collectively in a metachronal wave pattern in a minimal model based on slender-body theory for Stokes flow. The model predicts locomotion in the direction of the traveling wave, as commonly observed along the body of free-swimming crustaceans. The displacement of the body and the swimming efficiency depend on the number of legs, the amplitude, and the phase of oscillations. This study shows that paddling legs with distinct orientations and phases offers a simple mechanism for driving flow.

Metachronal waves in the flagellar beating of Volvox and their hydrodynamic origin.

Groups of eukaryotic cilia and flagella are capable of coordinating their beating over large scales, routinely exhibiting collective dynamics in the form of metachronal waves. The origin of this behaviour—possibly influenced by both mechanical interactions and direct biological regulation—is poorly understood, in large part due to a lack of quantitative experimental studies. Here we characterize in detail flagellar coordination on the surface of the multicellular alga Volvox carteri, an emerging model organism for flagellar dynamics. Our studies reveal for the first time that the average metachronal coordination observed is punctuated by periodic phase defects during which synchrony is partial and limited to specific groups of cells. A minimal model of hydrodynamically coupled oscillators can reproduce semi-quantitatively the characteristics of the average metachronal dynamics, and the emergence of defects. We systematically study the model's behaviour by assessing the effect of changing intrinsic rotor characteristics, including oscillator stiffness and the nature of their internal driving force, as well as their geometric properties and spatial arrangement. Our results suggest that metachronal coordination follows from deformations in the oscillators' limit cycles induced by hydrodynamic stresses, and that defects result from sufficiently steep local biases in the oscillators' intrinsic frequencies. Additionally, we find that random variations in the intrinsic rotor frequencies increase the robustness of the average properties of the emergent metachronal waves.

Cilium height difference between strokes is more effective in driving fluid transport in mucociliary clearance: A numerical study.

Mucociliary clearance is the first line of defense in our airway. The purpose of this study is to identify and study key factors in the cilia motion that influence the transport ability of the mucociliary system. Using a rod-propel-fluid model, we examine the effects of cilia density, beating frequency, metachronal wavelength, and the extending height of the beating cilia. We first verify that asymmetry in the cilia motion is key to developing transport in the mucus flow. Next, two types of asymmetries between the effective and recovery strokes of the cilia motion are considered, the cilium beating velocity difference and the cilium height difference. We show that the cilium height difference is more efficient in driving the transport, and the more bend the cilium during the recovery stroke is, the more effective the transport would be. It is found that the transport capacity of the mucociliary system increases with cilia density and cilia beating frequency, but saturates above by a threshold value in both density and frequency. The metachronal wave that results from the phase lag among cilia does not contribute much to the mucus transport, which is consistent with the experimental observation of Sleigh (1989). We also test the effect of mucus viscosity, whose value is found to be inversely proportional to the transport ability. While multiple parts have to interplay and coordinate to allow for most effective mucociliary clearance, our findings from a simple model move us closer to understanding the effects of the cilia motion on the efficiency of this clearance system.

Development of actuation system for artificial cilia with magnetic elastomer

In this paper, we describe the development of magnetically actuated artificial cilia. Natural cilia are a highly efficient device that produces flow under a small-Reynolds-number state. There are two important characteristics of natural cilia; one is asymmetric movement, which is composed of effective and recovery strokes, and the other is the phase difference of a stroke in each cilium in an array that will produce a metachronal wave. In this paper, we propose an actuation system for artificial cilia composed of a silicone elastomer and multiparticle chains of a magnetic material. The applied magnetic field is controlled by rotation of a permanent magnet. This rotating magnetic field induced an asymmetric movement similar to that of a natural cilium. We also changed the orientation angle of multiparticle chains to control the phase difference of a stroke in each artificial cilium. This technique would help to realize metachronal waves of artificial cilia.

An investigation of non-Newtonian fluid flow due to metachronal beating of cilia in a tube

The aim of this study is to explain theoretically the role of ciliary motion on the transport of epididymal fluid through the ductus efferentes of the male reproductive track. For this purpose, a mathematical model has been developed for the flow of a non-Newtonian fluid in an axisymmetric tube due to metachronal wave of cilia motion for the more realistic consequences. Carreau viscous fluid model is considered to see the rheological effects on the pumping characteristics of the flow. Regular perturbation method has been employed to obtain the analytical expressions for the stream function, the velocity field and a relation between the pressure difference and the volume flow rate. It is found that the volume flow rate is influenced significantly by Weissenberg number We and the cilia length parameter ε. The computational results are presented graphically to see the effects of various physical parameters. Finally, the analysis is applied and compared with the observed value of the flow rate of spermatic fluid in the ductus efferentes of the male reproductive track. The volume flow rate is reported closed to the estimated value 6 × 10-3 ml/h in the human ductus efferentes when We = 0.5 and ε is near by 0.25.