Mathematica numerical simulation of peristaltic biophysical transport of a fractional viscoelastic fluid through an inclined cylindrical tube

This paper studies the peristaltic transport of a viscoelastic fluid (with the fractional second-grade model) through an inclined cylindrical tube. The wall of the tube is modelled as a sinusoidal wave. The flow analysis is presented under the assumptions of long wave length and low Reynolds number. Caputo's definition of fractional derivative is used to formulate the fractional differentiation. Analytical solutions are developed for the normalized momentum equations. Expressions are also derived for the pressure, frictional force, and the relationship between the flow rate and pressure gradient. Mathematica numerical computations are then performed. The results are plotted and analysed for different values of fractional parameter, material constant, inclination angle, Reynolds number, Froude number and peristaltic wave amplitude. It is found that fractional parameter and Froude number resist the flow pattern while material constant, Reynolds number, inclination of angle and amplitude aid the peristaltic flow. Furthermore, frictional force and pressure demonstrate the opposite behaviour under the influence of the relevant parameters emerging in the equations of motion. The study has applications in uretral biophysics, and also potential use in peristaltic pumping of petroleum viscoelastic bio-surfactants in chemical engineering and astronautical applications involving conveyance of non-Newtonian fluids (e.g. lubricants) against gravity and in conduits with deformable walls.     

Slip Effects on Mixed Convective Peristaltic Transport of Copper-Water Nanofluid in an Inclined Channel

Peristaltic transport of copper-water nanofluid in an inclined channel is reported in the presence of mixed convection. Both velocity and thermal slip conditions are considered. Mathematical modelling has been carried out using the long wavelength and low Reynolds number approximations. Resulting coupled system of equations is solved numerically. Quantities of interest are analyzed through graphs. Numerical values of heat transfer rate at the wall for different parameters are obtained and examined. Results showed that addition of copper nanoparticles reduces the pressure gradient, axial velocity at the center of channel, trapping and temperature. Velocity slip parameter has a decreasing effect on the velocity near the center of channel. Temperature of nanofluid increases with increase in the Grashoff number and channel inclination angle. It is further concluded that the heat transfer rate at the wall increases considerably in the presence of copper nanoparticles.     

A mathematical model for the peristaltic flow of chyme movement in small intestine

A mathematical model based on viscoelastic fluid (fractional Oldroyd-B model) flow is considered for the peristaltic flow of chyme in small intestine, which is assumed to be in the form of an inclined cylindrical tube. The peristaltic flow of chyme is modeled more realistically by assuming that the peristaltic rush wave is a sinusoidal wave, which propagates along the tube. The governing equations are simplified by making the assumptions of long wavelength and low Reynolds number. Analytical approximate solutions of problem are obtained by using homotopy analysis method and convergence of the obtained series solution is properly checked. For the realistic values of the emerging parameters such as fractional parameters, relaxation time, retardation time, Reynolds number, Froude number and inclination of tube, the numerical results for the pressure difference and the frictional force across one wavelength are computed and discussed the roles played by these parameters during the peristaltic flow. On the basis of this study, it is found that the first fractional parameter, relaxation time and Froude number resist the movement of chyme, while, the second fractional parameter, retardation time, Reynolds number and inclination of tube favour the movement of chyme through the small intestine during pumping. It is further revealed that size of trapped bolus reduces with increasing the amplitude ratio whereas it is unaltered with other parameters.     

Evaluation of Adult White Sturgeon Swimming Capabilities and Applications to Fishway Design

Concern over passage of sturgeon barriers, has focused attention on fishway design that accommodates its swimming performance. In order to evaluate swimming performance, regarding fish ladder type partial barriers, wild adult sturgeons, Acipenser transmontanus; 121–76m fork length, were captured in the San Francisco Bay Estuary and Yolo Bypass toe drain. Hydrodynamic forces and kinematic parameters for swimming performance data were collected in a laboratory flume under three flow conditions through barriers and ramp. The experiments were conducted in a 24.4 m long, 2.1 m wide, and 1.62 m deep aluminum channel. Two geometric configurations of the laboratory model were designed based on channel characteristics that have been identified in natural river systems. At a given swimming speed and fish size, the highest guidance efficiencies of successful white sturgeon passage as a function of flow depth, flow velocity, turbulence intensity, Reynolds number, Froude number and shear velocity observed in the steady flow condition, tested with the horizontal ramp structure, occurred at an approach velocity of 0.33 ms^-1. The guidance efficiency of successful sturgeon passage increased both with increasing flow velocity and Froude number, and decreased both with the flow depth and the turbulence intensity. This study also provides evidence that tail beat frequency increases significantly with swimming speed, but tail beat frequency decreases with fish total length. Stride length increases both with swimming speed and fish total length. The importance of unsteady forces is expressed by the reduced frequency both with swimming speed and fish total length. Regression analysis indicates that swimming kinematic variables are explained by the swimming speed, the reduced frequency and the fish total length. The results emphasize the importance of fish ladder type patchiness when a fishway is designed for the passage of sturgeon.     

A long wavelength solution for a microorganism swimming in a channel

The hydrodynamics of a microorganism swimming in a channel is investigated. The microorganism is modeled as a two-dimensional sheet swimming at low Reynolds numbers between two rigid walls. The wavelengths of the propulsive waves passing down the sheet are assummed to be very large compared to the channel spacing, but the amplitude of the propulsive waves is arbitrary. Explicit analytical solutions for the propulsive velocity and the rate of energy dissipated in terms of the wave amplitude, channel spacing, wave number, and wave speeds are given.     

Human cochlear hydrodynamics: A high-resolution μCT-based finite element study

Measurements of perilymph hydrodynamics in the human cochlea are scarce, being mostly limited to the fluid pressure at the basal or apical turn of the scalae vestibuli and tympani. Indeed, measurements of fluid pressure or volumetric flow rate have only been reported in animal models. In this study we imaged the human ear at 6.7 and 3-µm resolution using µCT scanning to produce highly accurate 3D models of the entire ear and particularly the cochlea scalae. We used a contrast agent to better distinguish soft from hard tissues, including the auditory canal, tympanic membrane, malleus, incus, stapes, ligaments, oval and round window, scalae vestibule and tympani. Using a Computational Fluid Dynamics (CFD) approach and this anatomically correct 3D model of the human cochlea, we examined the pressure and perilymph flow velocity as a function of location, time and frequency within the auditory range. Perimeter, surface, hydraulic diameter, Womersley and Reynolds numbers were computed every 45° of rotation around the central axis of the cochlear spiral. CFD results showed both spatial and temporal pressure gradients along the cochlea. Small Reynolds number and large Womersley values indicate that the perilymph fluid flow at auditory frequencies is laminar and its velocity profile is plug-like. The pressure was found 102–106° out of phase with the fluid flow velocity at the scalae vestibule and tympani, respectively. The average flow velocity was found in the sub-µm/s to nm/s range at 20–100 Hz, and below the nm/s range at 1–20 kHz.     

Hydraulic microhabitats and the distribution of macroinvertebrate assemblages in riffles

1. The influence of hydraulic conditions on the spatial distribution of macroinvertebrate assemblages was investigated in three riffles in a perennial Australian river. 2. Velocity, depth and variability of substrate roughness were measured at each of 56 macroinvertebrate sampling locations. Complex hydraulic variables (roughness Reynolds number, shear velocity, Froude number) were calculated from combinations of two or three of the directly measured variables. The biological significance of directly measured and complex hydraulic variables was determined by a combination of univariate and multivariate statistical procedures. 3. Macroinvertebrate abundance, number of taxa and community composition were significantly different between the identified roughness Reynolds number, Froude number, velocity and shear velocity microhabitats throughout the studied riffles. 4. Regression analysis showed macroinvertebrate abundance and number of taxa were negatively related to roughness Reynolds number, shear velocity, velocity and Froude number. Depth was negatively related to abundance. In general, the majority of the macroinvertebrate community preferred the areas of riffles with the lowest near‐bed turbulence. 5. Roughness Reynolds number explained more of the spatial variation in invertebrate abundance, number of taxa and community composition than the other hydraulic variables, either directly measured or calculated. Of the directly measured variables, velocity had the greatest explanatory power, which was marginally less than roughness Reynolds number and shear velocity. 6. This study demonstrated that small‐scale differences in hydraulic conditions created by combinations of velocity, depth and substrate roughness have an important role in the spatial distribution of macroinvertebrate assemblages in riffle habitats.     

Effects of fish size, time‐to‐fatigue and turbulence on swimming performance: a case study of Galaxias maculatus

A simple relationship for the inanga Galaxias maculatus swimming velocity is suggested and tested in low and high turbulence channels. The relationship connects the swimming velocity with fish Reynolds and Froude numbers and can be used in both ecological analysis ( e.g . habitat requirements) and management strategies ( e.g . fishways design). Contrary to some previous studies and intuition, effects of turbulence on swimming performance appeared to be negligible. The most likely explanation for this result relates to mechanics of fish–turbulence interactions that may be dependent on both the turbulence scales and energy. The data suggest that future studies of turbulence effects on fish behaviour should involve, in addition to turbulence energetics, consideration of fish dimensions in relation to the spectrum of turbulence scales.     

Hydrodynamic modeling of cerebrospinal fluid motion within the spinal cavity

The fluid that resides within cranial and spinal cavities, cerebrospinal fluid (CSF), moves in a pulsatile fashion to and from the cranial cavity. This motion can be measured hy magnetic resonance imaging (MRI) and may he of clinical importance in the diagnosis of several brain and spinal cord disorders such as hydrocephalus, Chiari malformation, and syringomyelia. In the present work, a geometric and hydrodynamic characterization of an anatomically relevant spinal canal model is presented. We found that inertial effects dominate the flow field under normal physiological flow rates. Along the length of the spinal canal, hydraulic diameter was found to vary significantly from 5 to 15 mm. The instantaneous Reynolds number at peak flow rate ranged from 150 to 450, and the Womersle number ranged from 5 to 17. Pulsatile flow calculations are presented for an idealized geometric representation of the spinal cavity. A linearized Navier-Stokes model of the pulsatile CSF flow was constructed based on MRI flow rate measurements taken on a healthy volunteer. The numerical model was employed to investigate effects of cross-sectional geometry and spinal cord motion on unsteady velocity, shear stress, and pressure gradientfields. The velocity field was shown to be blunt, due to the inertial character of the flow, with velocity peaks located near the boundaries of the spinal canal rather than at the midpoint between boundaries. The pressure gradient waveform was found to be almost exclusively dependent on the flow waveform and cross-sectional area. Characterization of the CSF dynamics in normal and diseased states may be important in understanding the pathophysiology of CSF related disorders. Flow models coupled with MRI flow measurements mnay become a noninvasive tool to explain the abnormal dynamics of CSF in related brain disorders as well as to determine concentration and local distribution of drugs delivered into the CSF space.     

Hydrodynamic detection by cupulae in a lateral line canal: functional relations between physics and physiology

In the present review, signal-processing capabilities of the canal lateral line organ imposed by its peripheral architecture are quantified in terms of a limited set of measurable physical parameters. It is demonstrated that cupulae in the lateral line canal organ can only partly be described as canal fluid velocity detectors. Deviation from velocity detection may result from resonance, and can be characterized by the extent to which a single dimensionless resonance number, N _r , exceeds 1. This number depends on four physical parameters: it is proportional to cupular size, cupular sliding stiffness and canal fluid density, and inversely proportional to the square of fluid viscosity. Situated in a canal, a cupula may benefit from its resonance by compensating for the limited frequency range of water motion that is efficiently transferred into the lateral line canal. The peripheral transfer of hydrodynamic signals, via canal and cupula, leads to a nearly constant sensitivity to outside water acceleration in a bandwidth that ranges from d.c. to a cut-off frequency of up to several hundreds of Hertz, significantly exceeding the cut-off frequency of the lateral line canal. Threshold values of hydrodynamic detection by the canal lateral line organ are derived in terms of water displacement, water velocity, water acceleration and water pressure gradients and are shown to be close to the detection limits imposed by hair cell mechano-transduction in combination with the physical constraints of peripheral lateral line signal transfer. The notion that the combination of canal- and cupular hydrodynamics effectively provides the lateral line canal organ with a constant sensitivity to water acceleration at low frequencies so that it consequently functions as a low-pass detector of pressure gradients, supports the appropriateness of describing it as a sense organ that “feels at a distance” (Dijkgraaf in Biol Rev 38:51–105, 1963)     

Thermoregulation and the determinants of heat transfer in Colias butterflies

Summary As a means of exploring behavioral and morphological adaptations for thermoregulation in Colias butterflies, convective heat transfer coefficients of real and model butterflies were measured in a wind tunnel as a function of wind speed and body orientation (yaw angle). Results are reported in terms of a dimensionless heat transfer coefficient (Nusselt number, Nu) and a dimensionless wind speed (Reynolds number, Re), for a wind speed range typical of that experienced by basking Colias in the field. The resultant Nusselt-Reynolds (Nu-Re) plots thus indicate the rates of heat transfer by forced convection as a function of wind speed for particular model geometries.For Reynolds numbers throughout the measured range, Nusselt numbers for C. eurytheme butterflies are consistently lower than those for long cylinders, and are independent of yaw angle. There is significant variation among individual butterflies in heat transfer coefficients throughout the Re range. Model butterflies without artificial fur have Nu-Re relations similar to those for cylinders. Heat transfer in these models depends upon yaw angle, with higher heat transfer at intermediate yaw angles (30–60°); these yaw effects increase with increasing Reynolds number. Models with artificial fur, like real Colias, have Nusselt numbers which are consistently lower than those for models without fur at given Reynolds numbers throughout the Re range. Unlike real Colias, however, the models with fur do show yaw angle effects similar to those for models without fur.The independence of heat loss from yaw angle for real Colias is consistent with field observations indicating no behavioral orientation to wind direction. The presence of fur on the models reduces heat loss but does not affect yaw dependence. The large individual variation in heat transfer coefficients among butterflies is probably due to differences in fur characteristics rather than to differences in wing morphology.Finally, a physical model of a butterfly was constructed which accurately simulates the body temperatures of basking Colias in the field for a variety of radiation and wind velocity conditions. The success of the butterfly simulator in mimicking Colias thermal characteristics confirms our preliminary understanding of the physical bases for and heat transfer mechanisms underlying thermoregulatory adaptations in these butterflies.     

Radiation Effects on the Flow of Powell-Eyring Fluid Past an Unsteady Inclined Stretching Sheet with Non-Uniform Heat Source/Sink

This study investigates the unsteady flow of Powell-Eyring fluid past an inclined stretching sheet. Unsteadiness in the flow is due to the time-dependence of the stretching velocity and wall temperature. Mathematical analysis is performed in the presence of thermal radiation and non-uniform heat source/sink. The relevant boundary layer equations are reduced into self-similar forms by suitable transformations. The analytic solutions are constructed in a series form by homotopy analysis method (HAM). The convergence interval of the auxiliary parameter is obtained. Graphical results displaying the influence of interesting parameters are given. Numerical values of skin friction coefficient and local Nusselt number are computed and analyzed.     

Numerical simulations of the pulsating flow of cerebrospinal fluid flow in the cervical spinal canal of a Chiari patient

The flow of cerebrospinal fluid (CSF) in a patient-specific model of the subarachnoid space in a Chiari I patient was investigated using numerical simulations. The pulsating CSF flow was modeled using a time-varying velocity pulse based on peak velocity measurements (diastole and systole) derived from a selection of patients with Chiari I malformation. The present study introduces the general definition of the Reynolds number to provide a measure of CSF flow instability to give an estimate of the possibility of turbulence occurring in CSF flow. This was motivated by the fact that the combination of pulsating flow and the geometric complexity of the spinal canal may result in local Reynolds numbers that are significantly higher than the commonly used global measure such that flow instabilities may develop into turbulent flow in these regions. The local Reynolds number was used in combination with derived statistics to characterize the flow. The results revealed the existence of both local unstable regions and local regions with velocity fluctuations similar in magnitude to what is observed in fully turbulent flows. The results also indicated that the fluctuations were not self-sustained turbulence, but rather flow instabilities that may develop into turbulence. The case considered was therefore believed to represent a CSF flow close to transition.     

Modeling pulsatile flow in aortic aneurysms: effect of non-Newtonian properties of blood

Pulsatile flow in an axisymmetric rigid-walled model of an abdominal aorta aneurysm was analyzed numerically for various aneurysm dilations using physiologically realistic resting waveform at time-averaged Reynolds number of 300 and peak Reynolds number of 1607. Discretization of the governing equations was achieved using a finite element scheme based on the Galerkin method of weighted residuals. Comparisons with previously published work on the basis of special cases were performed and found to be in excellent agreement. Our findings indicate that the velocity fields are significantly affected by non-Newtonian properties in pathologically altered configurations. Non-Newtonian fluid shear stress is found to be greater than Newtonian fluid shear stress during peak systole. Further, the maximum shear stress is found to occur near the distal end of AAA during peak systole. The impact of non-Newtonian blood flow characteristics on pressure compared to Newtonian model is found insignificant under resting conditions. Viscous and inertial forces associated with blood flow are responsible for the changes in the wall that result in thrombus deposition and dilation while rupture of AAA is more likely determined by much larger mechanical stresses imposed by pulsatile pressure on the wall of AAA.     

Hydraulic parameters and benthic invertebrate distributions in two gravel-bed New Zealand rivers

1. Benthic invertebrates were sampled over a matrix of about eighty combinations of mean velocity (10–150cms^?1) and depth (10–150cm) in two rivers that differed in substrate size variability. Vertical velocity profiles were measured at each sample site and substratum roughness was measured and estimated from percentage cover by stone size classes. The influence of depth on periphyton biomass was also measured. 2. The hydraulic and substrate data were used to investigate the correlations between conventional (mean velocity, depth, substrate size) and complex hydraulic variables (Froude number, shear velocity, and water column and boundary Reynolds number) that were either calculated from direct measurements or inferred from mean velocity, depth, kinematic viscosity and substrate roughness. The ecological relevance of these hydraulic variables was investigated by comparing their degree of correlation with invertebrate densities and community metrics. 3. The invertebrate variables had similar correlations with mean velocity and the complex near-bed hydraulic variables in the river with uniform cobble substrates. In the river with diverse substrates, however, average correlations with Froude number, and inferred shear velocity and boundary Reynolds number were 25–45% higher than with velocity. Of all the individual hydraulic parameters, the boundary Reynolds number, calculated from simple measures, was most strongly correlated with benthic invertebrate distributions and taxa richness. However, invertebrate distributions were more strongly correlated with predictions of multiple regression models, incorporating substrate size, depth and mean velocity, than with any single hydraulic variable. 4. Hydraulic influences on food availability and oxygen concentration in the benthos are likely mechanisms affecting the hydraulic preferences of several taxa. Lower periphyton biomass with depth, partly attributable to light attenuation, appeared to have a non-hydraulic influence on a collector-browser species.     

Oxygen and CO2 transfer of a polypropylene dimpled membrane lung with variable secondary flows

Gas transfer performance is presented for one form of the Oxford membrane lung in which vortex mixing is induced in blood flow across a dimpled polypropylene membrane. Dimensional analysis has been used to define the parameters characterizing mass transfer in the device, and of three fluid mechanical parameters: Reynolds number based on peak pulsation velocity, Strouhal number, and ratio of mean to oscillatory flows, only the first has been found to affect mass transfer significantly in the ranges studied. For oxygenation, rated flows in excess of 5 l min⁻¹ m⁻² are measured. A new definition is presented for a rated flow for extracorporeal CO₂ removal and values in excess of 1.2 l min⁻¹ m⁻² are obtained.     

Influence of Hall Current and Viscous Dissipation on Pressure Driven Flow of Pseudoplastic Fluid with Heat Generation: A Mathematical Study

In this paper, we study the influence of heat sink (or source) on the peristaltic motion of pseudoplastic fluid in the presence of Hall current, where channel walls are non-conducting in nature. Flow analysis has been carried out under the approximations of a low Reynolds number and long wavelength. Coupled equations are solved using shooting method for numerical solution for the axial velocity function, temperature and pressure gradient distributions. We analyze the influence of various interesting parameters on flow quantities. The present study can be considered as a mathematical presentation of the dynamics of physiological organs with stones.     

Analysis of oscillatory flow disturbances and thermal characteristics inside fluidic cells due to fluid leakage and wall slip conditions

The effects of both fluid leakage and wall slip conditions are studied analytically and numerically on the fluctuation rate in the flow inside non-isothermal disturbed thin films supported by soft seals within a fluidic cell. Flow disturbances due to internal pressure pulsations and external squeezing are considered in this work. The main controlling parameters are found to be the dimensionless leakage parameter, softness of the seal, squeezing number, dimensionless slip parameter, the thermal squeezing parameter and the power law index. Accordingly, their influences on the fluctuation rate and heat transfer characteristics inside disturbed thin films are determined and discussed. It is found that an increase in the dimensionless leakage parameter, softness of the seal-upper plate assembly and the wall slip parameter result in more cooling and an increase in the fluctuation level in the flow. However, an increase in the squeezing number and the fluid power index decrease flow fluctuations. Finally, a suggested design to alleviate a number of problems in fluidic cells is presented.     

Effect of Joule Heating and Thermal Radiation in Flow of Third Grade Fluid over Radiative Surface

This article addresses the boundary layer flow and heat transfer in third grade fluid over an unsteady permeable stretching sheet. The transverse magnetic and electric fields in the momentum equations are considered. Thermal boundary layer equation includes both viscous and Ohmic dissipations. The related nonlinear partial differential system is reduced first into ordinary differential system and then solved for the series solutions. The dependence of velocity and temperature profiles on the various parameters are shown and discussed by sketching graphs. Expressions of skin friction coefficient and local Nusselt number are calculated and analyzed. Numerical values of skin friction coefficient and Nusselt number are tabulated and examined. It is observed that both velocity and temperature increases in presence of electric field. Further the temperature is increased due to the radiation parameter. Thermal boundary layer thickness increases by increasing Eckert number.     

The Effects of Thermal Radiation on an Unsteady MHD Axisymmetric Stagnation-Point Flow over a Shrinking Sheet in Presence of Temperature Dependent Thermal Conductivity with Navier Slip

In this paper, the magnetohydrodynamic (MHD) axisymmetric stagnation-point flow of an unsteady and electrically conducting incompressible viscous fluid in with temperature dependent thermal conductivity, thermal radiation and Navier slip is investigated. The flow is due to a shrinking surface that is shrunk axisymmetrically in its own plane with a linear velocity. The magnetic field is imposed normally to the sheet. The model equations that describe this fluid flow are solved by using the spectral relaxation method. Here, heat transfer processes are discussed for two different types of wall heating; (a) a prescribed surface temperature and (b) a prescribed surface heat flux. We discuss and evaluate how the various parameters affect the fluid flow, heat transfer and the temperature field with the aid of different graphical presentations and tabulated results.