A computational fluid dynamics model of viscous coupling of hairs

Arrays of arthropod filiform hairs form highly sensitive mechanoreceptor systems capable of detecting minute air disturbances, and it is unclear to what extent individual hairs interact with one another within sensor arrays. We present a computational fluid dynamics model for one or more hairs, coupled to a rigid-body dynamics model, for simulating both biological (e.g., a cricket cercal hair) and artificial MEMS-based systems. The model is used to investigate hair–hair interaction between pairs of hairs and quantify the extent of so-called viscous coupling. The results show that the extent to which hairs are coupled depends on the mounting properties of the hairs and the frequency at which they are driven. In particular, it is shown that for equal length hairs, viscous coupling is suppressed when they are driven near the natural frequency of the undamped system and the damping coefficient at the base is small. Further, for certain configurations, the motion of a hair can be enhanced by the presence of nearby hairs. The usefulness of the model in designing artificial systems is discussed.     

Interaction between arthropod filiform hairs in a fluid environment

Many arthropods use filiform hairs as mechanoreceptors to detect air motion. In common house crickets (Acheta domestica) the hairs cover two antenna-like appendages called cerci at the rear of the abdomen. The biomechanical stimulus-response properties of individual filiform hairs have been investigated and modeled extensively in several earlier studies. However, only a few previous studies have considered viscosity-mediated coupling between pairs of hairs, and only in particular configurations. Here, we present a model capable of calculating hair-to-hair coupling in arbitrary configurations. We simulate the coupled motion of a small group of mechanosensory hairs on a cylindrical section of cercus. We have found that the coupling effects are non-negligible, and likely constrain the operational characteristics of the cercal sensory array.     

Modeling arthropod filiform hair motion using the penalty immersed boundary method

Crickets are able to sense their surrounding environment through about 2000 filiform hairs located on a pair of abdominal cerci. The mechanism by which the cricket is able to sense a wide range of input signals using these filiform hairs of different length and orientation is of great interest. Most of the previous filiform hair models have focused on a single, rigid hair in an idealized air field. Here, we present a model of the cercus and filiform hairs that are mechanically coupled to the surrounding air, and the model equations are based on the penalty immersed boundary method. The key difference between the penalty immersed boundary method and the traditional immersed boundary method is the addition of forces to account for density differences between the immersed solid (the filiform hairs) and the surrounding fluid (air). The model is validated by comparing the model predictions to experimental results, and then the model is used to examine the interactions between multiple hairs. With multiple hairs, there is little interaction when the hairs are separated by more than 1mm, and, as they move closer, they interact through viscous coupling, which reduces the deflection of the hairs due to the air movement. We also examine the computational scalability of the algorithm and show that the computational costs grow linearly with the number of hairs being modeled.     

Structural scaling and functional design of the cercal wind-receptor hairs of cricket

We have estimated the intrinsic mechanical parameters of cricket cercal wind-receptor hairs. The hairs were modeled as an inverted pendulum, and mechanical parameters of the equation of motion were determined from data given by a systematic measurement of mobility by the least-square error method. The theoretical torque which turns the hair shaft is given by the drag force due to the moving air. The drag force is given by the method of Stokes' mechanical impedance of an oscillating cylinder in viscous fluid. The effect of the boundary layer in which air is stagnating on the substrate surface is also taken into account. The moment of inertia of a hair shaft shows a clear length dependency to the power of 4.32 of the hair length. The torsional resistance within the hair base and the stiffness of hair-supporting spring also show clear length dependencies to the power of 2.77 and 1.67, respectively. The torsional resistance within the hair base is so large that the hair is a strongly damped non-oscillatory second-order system. The large resistance within the hair base represents an efficient energy absorption by the sensory cell. The resistance seems to match with the source impedance, i.e., the frictional resistance at the site of air-hair contact. The impedance matching provides the condition of maximum power transmission from the moving air to the sensory cell. Structural scaling is discussed in relation to the functional scaling of the frequency-range fractionation of the mechanical filter array with a common biological design. Accepted: 24 February 1998     

Ventricular motion during the ejection phase: a computational analysis.

In the present paper, the study of the ventricular motion during systole was addressed by means of a computational model of ventricular ejection. In particular, the implications of ventricular motion on blood acceleration and velocity measurements at the valvular plane (VP) were evaluated. An algorithm was developed to assess the force exchange between the ventricle and the surrounding tissue, i.e., the inflow and outflow vessels of the heart. The algorithm, based on the momentum equation for a transitory flowing system, was used in a fluid-structure model of the ventricle that includes the contractile behavior of the fibers and the viscous and inertial forces of the intraventricular fluid. The model calculates the ventricular center of mass motion, the VP motion, and intraventricular pressure gradients. Results indicate that the motion of the ventricle affects the noninvasive estimation of the transvalvular pressure gradient using Doppler ultrasound. The VP motion can lead to an underestimation equal to 12.4 +/- 6.6%.     

Viscosity-mediated motion coupling between pairs of trichobothria on the leg of the spider <Emphasis Type="Italic">Cupiennius salei</Emphasis>

Arachnids and insects use long, thin hairs as motion sensors to detect signals contained in the movement of the surrounding air. These hairs often form groups with a small spacing of tens to hundreds of micrometers between them. For air oscillation frequencies of biological interest, the potential exists for viscosity-mediated coupling among hairs in a group affecting their response characteristics. Even a small diameter hair can, in principle, affect the flow field around it and the dynamics of the hairs in its neighborhood. The viscosity-mediated coupling between a pair of hairs is investigated here both experimentally and theoretically. The conditions for the existence of the coupling effect, and its magnitude as a function of relevant parameters, are determined. In the range of biologically relevant frequencies (30–300 Hz), viscous coupling between pairs of hairs is only very small in the case of the spider Cupiennius salei. Theoretical analysis points to the relatively large spacing between hairs (20 to 50 hair diameters) and the tuning of the hairs to the above-mentioned frequencies to explain the practical absence of coupling.     

Mobilities of the cercal wind-receptor hairs of the cricket, Gryllus bimaculatus

The deflection sensitivities of cercal filiform hairs of the cricket, Gryllus bimaculatus, were determined by direct measurement. The tangential velocity of deflecting hair shafts in response to stimulus air motion was measured in situ by a laser-Doppler velocimeter with surface scattering of the shaft. The velocity of the stimulus air motion in a small wind tunnel was calibrated by the same velocimeter with smoke from a joss-stick. The mobility of the hair was obtained from former measurements with reference to the latter calibration of the single apparatus. A Gaussian white noise signal was employed as a stimulus waveform, and the stimulus-response transfer function was calculated through a cross-correlation method, which provides greater precision and wider frequency for a longer period of measurement. The mobility of hair was expressed in deflection amplitudes and phase shifts in reference to the velocity sinusoid of a stimulus at various frequencies. The measurements established the following conclusions. The wind receptor hairs comprise an array of mechanical band-pass filters whose best frequencies are inversely proportional to the length. The motion dynamics of the wind-receptor hairs have strong damping. Accepted: 24 February 1998     

Hydrodynamics of bolus flow—An analytical approach to blood flow in capillaries

A model which was used by Prothero and Burton to simulate a particular configuration in capillary blood vessels is investigated from a hydrodynamic point of view. In this model, the erythrocytes are approximated by rigid pistons, and plasma is assumed to be an incompressible Newtonian fluid. An order of magnitude analysis using the physiologically realistic values for various parameters reduces the exact equations of motion to an equation describing the creeping motion of the fluid. An analytical approach to the solution of the equation is proposed and some results are reported here. The solution of the flow field is given in terms of a stream function which is represented by two infinite series composed of known functions. Two coupled infinite systems of algebraic equations determining the coefficients of the two series have been derived. This method of solution is proposed as an alternative to the entirely numerical procedure of solving the similar problem proposed by Bugliarelloet al. A limiting case of large aspect ratio (the ratio of the axial spacing of the two successive erythrocytes to the capillary diameter) is studied and the solution, valid away from the erythrocyte surface, has been obtained in simple form. It resembles the classical Poisenille flow, but the pressure gradient is related to the erythrocyte speed.     

Relative contributions of organ shape and receptor arrangement to the design of cricket’s cercal system

Understanding the relative contributions of the shape of a sensory organ and the arrangement of receptors to the overall performance of the organ has long been a challenge for sensory biologists. We tackled this issue using the wind-sensing system of crickets, the cerci, two conical abdominal appendages covered with arrays of filiform hairs. Scanning electron microscopy coupled with 3D reconstruction methods were used for mapping of all cercal filiform hairs. The hairs are arranged according to their diameter in a way that avoids collisions with neighbours during hair deflection: long hairs are regularly spaced, whereas short hairs are both randomly and densely distributed. Particle image velocimetry showed that the variation in diameter of the cercus along its length modifies the pattern of fluid velocities. Hairs are subject to higher air flow amplitudes at the base than at the apex of the cercus. The relative importance of interactions between receptors and the air flow around the organ may explain the performance of the cricket's cercal system: it is characterised by a high density of statistically non-interacting short hairs located at the base of the cercus where sensitivity to air currents is the highest.     

Elastic expulsion from a long tube

The paper presents a theoretical analysis of elastic expulsion from a long pressurized tube following instantaneous severance. The governing equations of motion and continuity are used, together with an equation representing radial equilibrium of the tube wall, to construct a differential equation for the inner radius of the tube as a function of axial position and time. Similarity solutions for this equation are presented for a Newtonian fluid. The effect of the relaxed/fully-extended radius ratio on wall dilatation and expulsion rate is discussed.     

Instabilities on Prey Dynamics in Jellyfish Feeding

We study the dynamics of plankton in the wake of a jellyfish. Using an analytical approach, we derive a reduced-order equation that governs the prey motion which is modeled as neutrally-buoyant inertial particle. This modified equation takes into account both the effects of prey inertia and self-propulsion and enables us to calculate both the attracting and repelling Lagrangian coherent structures for the prey motion. For the case of zero self-propulsion, it is simplified to the equation of motion for infinitesimal fluid particles. Additionally, we determine the critical size of prey over which instabilities on its motion occur resulting in different dynamics from those predicted by the reduced-order equation even for the case of zero self-propulsion. We illustrate our theoretical findings through an experimentally measured velocity field of a jellyfish. Using the inertial equation, we calculate the Lagrangian coherent structures that characterize prey motion as well as the instability regions over which larger prey will have different dynamics even for the case of zero self-propulsion.     

Effects of velocity jumps on blood flow in large arteries

When a human being experiences a sudden velocity change, the blood flow is disturbed. A theoretical analysis to predict the effects of sudden velocity changes on blood flow in large arteries is presented. The situations is modelled as a one-dimensional flow problem in a viscoelastic tube where the fluid viscosity convective term in the equation of motion and nonlinearity in the elastic modulus of the tube wall are neglected. The governing equations of the model are solved by Laplace transformation. The computed results show that relatively high blood pressures, capable of harming circulation, are produced even by relatively moderate velocity jumps.     

Two-Fluid Model of Biofilm Disinfection

We consider a dynamic model of biofilm disinfection in two dimensions. The biofilm is treated as a viscous fluid immersed in a fluid of less viscosity. The bulk fluid moves due to an imposed external parabolic flow. The motion of the fluid is coupled to the biofilm inducing motion of the biofilm. Both the biofilm and the bulk fluid are dominated by viscous forces, hence the Reynolds number is negligible and the appropriate equations are Stokes equations. The governing partial differential equations are recast as boundary integral equations using a version of the Lorenz reciprocal relationship. This allows for robust treatment of the simplified fluid/biofilm motion. The transport of nutrients and antimicrobials, which depends directly on the velocities of the fluid and biofilm, is also included. Disinfection of the bacteria is considered under the assumption that the biofilm growth is overwhelmed by disinfection. Supported by NSF award DMS-0612467.     

Fluid Forces Enhance the Performance of an Aspirant Leader in Self-Organized Living Groups

In this paper, the performance of an individual aiming at guiding a self-organized group is numerically investigated. A collective behavioural model is adopted, accounting for the mutual repulsion, attraction and orientation experienced by the individuals. Moreover, these represent a set of solid particles which are supposed to be immersed in a fictitious viscous fluid. In particular, the lattice Boltzmann and Immersed boundary methods are used to predict the fluid dynamics, whereas the effect of the hydrodynamic forces on particles is accounted for by solving the equation of the solid motion through the time discontinuous Galerkin scheme. Numerical simulations are carried out by involving the individuals in a dichotomous process. On the one hand, an aspirant leader (AL) additional individual is added to the system. AL is forced to move along a prescribed direction which intersects the group. On the other hand, these tend to depart from an obstacle represented by a rotating lamina which is placed in the fluid domain. A numerical campaign is carried out by varying the fluid viscosity and, as a consequence, the hydrodynamic field. Moreover, scenarios characterized by different values of the size of the group are investigated. In order to estimate the AL''s performance, a proper parameter is introduced, depending on the number of individuals following AL. Present findings show that the sole collective behavioural equations are insufficient to predict the AL''s performance, since the motion is drastically affected by the presence of the surrounding fluid. With respect to the existing literature, the proposed numerical model is enriched by accounting for the presence of the encompassing fluid, thus computing the hydrodynamic forces arising when the individuals move.     

Blood flow: Theory,effective viscosity and effects of particle distribution

A study is made of blood flow by assuming that the blood constitutes a suspension of cells in plasma instead of a simple homogeneous fluid. A macroscopic theory governing the motion of plasma in a plasma-cell system is derived from the local volume averaging method for a system without mass transfer between the phases, and its characteristic length is much larger than the size of the cells. The equations governing the motion of the local averaged fluid quantities include one additional term in the equation of motion and two additional terms in the energy equation. These terms represent, respectively, the force exerted upon the fluid by the particles, and the rate of heat transfer and work kone upon the fluid by the particles. The theory is applied to obtain the effective viscosity as the explicit function of the volume concentration of the cells by assuming that the cells behave like rigid spherical particles with slip-collision, and the plasma is an compressible Newtonian fluid. Comparison with existing experimental results shows a good agreement. The theory is also used to obtain the effects of cell distribution upon the overall effective viscosity in a circular tube. The quantitative result shows that there is a decrease in overall effective viscosity as the concentration of cells increases toward the center of the tube, and the overall effective viscosity is smaller than the flow with evenly distributed cells.     

Lubrication of the human ankle joint in walking with the synovial fluid filtrated by the cartilage with the surface zone worn out: steady pure sliding motion.

A mixture model of synovial fluid filtration by cartilage in the human ankle joint during walking is presented for steady sliding motion of the articular surfaces. In the paper the cartilage surface zone is assumed worn out. The same model has been recently applied to the squeeze-film problem for the human hip joint loaded by the body weight during standing (Hlavácek, Journal of Biomechanics 26, 1145-1150, 1151-1160, 1993; Hlavácek and Novák, Journal of Biomechanics 28, 1193-1198, 1199-1205, 1995). The linear biphasic model for cartilage (elastic porous matrix + ideal fluid) due to Prof. V. C. Mow and his co-workers and the biphasic model for synovial fluid (viscous fluid + ideal fluid), as used in the above-mentioned squeeze-film problem, are applied. For the physiologic parameters of the ankle joint during walking, a continuous synovial fluid film about 1 microm thick is maintained under steady entraining motion according to the classical model without the fluid transport across the articular surface. This is not the case in the filtration model with the cartilage surface zones worn out. On the contrary, this filtration model indicates that synovial fluid is intensively filtrated by such cartilage, so that no continuous fluid film is maintained and a synovial gel layer, about 10(-8) m thick, develops over the majority of the contact. Thus, if the cartilage surface zones are worn out, boundary lubrication should prevail in the ankle joint under steady sliding motion for the mean values of loading and the sliding velocity encountered in walking cycle.     

Peristaltic Transport of Carreau-Yasuda Fluid in a Curved Channel with Slip Effects

The wide occurrence of peristaltic pumping should not be surprising at all since it results physiologically from neuro-muscular properties of any tubular smooth muscle. Of special concern here is to predict the rheological effects on the peristaltic motion in a curved channel. Attention is focused to develop and simulate a nonlinear mathematical model for Carreau-Yasuda fluid. The progressive wave front of peristaltic flow is taken sinusoidal (expansion/contraction type). The governing problem is challenge since it has nonlinear differential equation and nonlinear boundary conditions even in the long wavelength and low Reynolds number regime. Numerical solutions for various flow quantities of interest are presented. Comparison for different flow situations is also made. Results of physical quantities are interpreted with particular emphasis to rheological characteristics.     

Wave Propagation through a Newtonian Fluid Contained within a Thick-Walled,Viscoelastic Tube

The propagation of harmonic pressure waves through a Newtonian fluid contained within a thick-walled, viscoelastic tube is considered as a model of arterial blood flow. The fluid is assumed to be homogeneous and Newtonian, and its motion to be laminar and axisymmetric. The wall is assumed to be isotropic, incompressible, linear, and viscoelastic. It is also assumed that the motion is such that the convective acceleration is negligible. The motion of the fluid is described by the linearized form of the Navier-Stokes equations and the motion of the wall by classical elasticity theory. The frequency dependence of the wall mechanical properties are represented by a three parameter, relaxation-type model. Using boundary conditions describing the continuity of stress and velocity components in the fluid and the wall, explicit solutions for the system of equations of the model have been obtained. The longitudinal fluid impedance has been expressed in terms of frequency and the system parameters. The frequency equation has been solved and the propagation constant also expressed in terms of frequency and system parameters. The results indicate that the fluid impedance is smaller than predicted by the rigid tube model or by Womersley''s constrained elastic tube model. Also, the velocity of propagation is generally slower and the transmission per wavelength less than predicted by Womersley''s elastic tube model. The propagation constant is very sensitive to changes in the degree of wall viscoelasticity.     

The morphology of the wet, non-papillate (WN) stigma form in the tribe Caesalpinieae (Gaesalpinioideae: Leguminosae)

Stigmas in 24 species of 11 genera, classified in three groups, Caesalpinia, Peltophorum and Dimorphandra, of the tribe Caesalpinieae have been examined by light and scanning electron microscopy. All are of the WN (wet, non-papillate) form. The receptive stigmatic surface comprises a crater, generally at the apex of the style, which in fresh flowers is full of clear fluid. The crater rim may be fringed with non-receptive hairs, and its shape and depth are variable between species. Extreme forms are found in Caesalpinia sappan where crater hairs are short and the crater funnel-shaped and very deep, and C. vesicaria and Delonix regia where crater hairs are long and the crater appears very shallow.