Peristaltic transport of a power-law fluid: Application to the ductus efferentes of the reproductive tract

The problem of peristaltic transport of a non-Newtonian (power-law) fluid in uniform and non-uniform two-dimensional channels has been investigated under zero Reynolds number with long wavelength approximation. A comparison of the results with those for a Newtonian fluid model shows that the magnitude of pressure rise, under a given set of conditions, is smaller in the case of the non-Newtonian fluid (power-law indexn < 1) at zero flow rate. Further, the pressure rise is smaller asn decreases from 1 at zero flow rate, is independent ofn at a certain value of flow rate and becomes greater if flow rate increases further. Also, at a given flow rate, an increase in wavelength leads to a decrease in pressure rise and increase in the influence of non-Newtonian behaviour. Pressure rise in the case of non-uniform geometry, is found to be much smaller than the corresponding value in the case of uniform geometry. Finally, the analysis is applied and compared with observed flow rates in the ductus efferentes of the male reproductive tract.     

Peristaltic motion of a third-order fluid in a planar channel

In order to determine the characteristics of the peristaltic transport of shear thinning non-Newtonian materials, the motion of a third-order fluid in a planar channel having walls that are transversely displaced by an infinite, harmonic traveling wave of large wavelength and negligibly small Reynolds number was analyzed using a perturbation expansion in terms of a variant of the Deborah number. Within the range of validity of this analysis, we found the pumping rate of a shear-thinning fluid is less than that for a Newtonian fluid having a shear viscosity the same as the lower-limiting viscosity of the nonNewtonian material. Also, the space of variables for which trapping of a bolus of fluid occurs is reduced for the shear-thinning fluid investigated here.     

Orientation and rheology of rodlike particles with weak Brownian diffusion in a second-order fluid under simple shear flow

An analysis of particle orientation in a dilute suspension of rodlike particles in a second-order fluid was performed to examine the effects of the elasticity of the fluid and of weak Brownian diffusion of the particle on its orientation. Distributions of particle orientation under a simple shear flow with rate of shearg have been obtained as a function of a single nondimensional parameter, ^* =/r _e ² (D/g), which combines the effects of the particle aspect ratior _e , the weak fluid elasticity, and the weak Brownian rotation diffusion coefficientD of the particle. In the limit of larger _e , when the fluid elasticity is strong enough to overcome the rotational diffusion effect on the particle motion, most of the particles will orient close to the vorticity axis. A new shear-thinning mechanism of the shear viscosity of such systems is predicted by the theory.     

Pressure driven flow of dilute polymer solutions in a channel: Analytic results for very small and for very large channels

Kinetic theory of dilute macromolecular solutions is applied to pressure driven flow in a small channel where wall- (and interfacial) layers have to be reckoned with. The complete rheology is studied. It turns out that for very small channels both the shear stress and the normal stress are an order of magnitude larger than corresponding quantities in simple shear. On the other hand, when the channel is so wide that the wall layers are very thin in comparison, agreement with results appropriate for simple shear is found. The volume flow rate-pressure difference relation is derived and compared to the prediction which utilizes the slip velocity concept. For very small channels, this concept is five orders of magnitude off, but reproduces asymptotically correct results for very large channels.     

Taylor-Couette stability analysis for a Doi-Edwards fluid

The stability of Taylor-Couette flow of entangled polymeric solutions to small axisymmetric stationary disturbances is analyzed using the Doi-Edwards constitutive equation in the small gap limit. A previous analysis of Karlsson, Sokolov, and Tanner for the general K-BKZ equation, of which the Doi-Edwards equation is a special case, reduces the problem to one of numerically evaluating seven viscoelastic functions of the shear rate in the gap. Of these seven, only three — two of which are related to the second normal stress difference, and one of them to shear thinning — significantly affect the flow stability. The negative second normal stress difference of the Doi-Edwards fluid stabilizes the flow at low values of the Weissenberg number ₁ , while shear thinning produces strong destabilization at moderate Weissenberg number. Here ₁ is the longest relaxation time. Non-monotonic effects of viscoelasticity on Taylor-Couette stability analogous to those predicted here have been observed in experiments of Giesekus. The extreme shear thinning of the Doi-Edwards fluid is also predicted to produce a large growth in the height of the Taylor cells, a phenomenon that has been seen experimentally by Beavers and Joseph.     

Response of an elastic Bingham fluid to oscillatory shear

The response of an elastic Bingham fluid to oscillatory strain has been modeled and compared with experiments on an oil-in-water emulsion. The newly developed model includes elastic solid deformation below the yield stress (or strain), and Newtonian flow above the yield stress. In sinusoidal oscillatory deformations at low strain amplitudes the stress response is sinusoidal and in phase with the strain. At large strain amplitudes, above the yield stress, the stress response is non-linear and is out of phase with strain because of the storage and release of elastic recoverable strain. In oscillatory deformation between parallel disks the non-uniform strain in the radial direction causes the location of the yield surface to move in-and-out during each oscillation. The radial location of the yield surface is calculated and the resulting torque on the stationary disk is determined. Torque waveforms are calculated for various strains and frequencies and compared to experiments on a model oil-in-water emulsion. Model parameters are evaluated independently: the elastic modulus of the emulsion is determined from data at low strains, the yield strain is determined from the phase shift between torque and strain, and the Bingham viscosity is determined from the frequency dependence of the torque at high strains. Using these parameters the torque waveforms are predicted quantitatively for all strains and frequencies. In accord with the model predictions the phase shift is found to depend on strain but to be independent of frequency.Notation A plate strain amplitude (parallel plates) - A _R plate strain amplitude at disk edge (parallel disks) - G elastic modulus - m torque (parallel disks) - M normalized torque (parallel disks) = 2m/R ³₀ - N ratio of viscous to elastic stresses (parallel plates) =µ A/ ₀ ratio of viscous to elastic stresses (parallel disks) =µ A _R/₀ - r normalized radial position (parallel disks) =r/R - r radial position (parallel disks) - R disk radius (parallel disks) - t normalized time = t — /2 - t time - _E elastic strain - _P plate strain (displacement of top plate or disk divided by distance between plates or disks) - _PR plate strain at disk edge (parallel disks) - ₀ yield strain - _E normalized elastic strain = _E/₀ - _P normalized plate strain = _P/₀ - _PR normalized plate strain at disk edge (parallel disks) = _PR/₀ - ₀ normalized plate strain amplitude (parallel plates) =A/ ₀ — normalized plate strain amplitude at disk edge (parallel disks) =A _R/₀ - phase shift between _P andT (parallel plates) — phase shift between _PR andM (parallel disks) - µ Bingham viscosity - stress - ₀ yield stress - T normalized stress =/ ₀ - frequency     

Drilling fluid shear stress overshoot behavior

Shear stress overshoot behavior was studied in four drilling fluid systems and ten bentonite dispersions. These overshoot properties, also described by the American Petroleum Institute as gel strengths, were measured after gelation times of 10 s to 24 h at temperatures of 20–80 °C. Two different rheometers were used to measure overshoot behavior. Gel strength development with time followed a first-order model. Gel development rates at 20 °C varied from 0.005 to 0.01 min^–1 for drilling fluid systems and from 0.0004 to 0.02 min^–1 for bentonite dispersions. Increasing the gelation temperature for each drilling fluid system caused an increase in the gelling rate constant. Comparison of gel strengths in bentonite dispersions were made using a Fann 35 A viscometer and a Weissenberg Rheogoniometer. Higher gel strength values observed using the Rheogoniometer were believed to be due to differences in instrument spring stiffness and fixture inertia.     

Peristaltic motion of a power-law fluid in an asymmetric channel

The flow of a power-law fluid is investigated in an asymmetric channel caused by the movement of peristaltic waves with the same speed but with different amplitudes and phases on the flexible walls of the channel. The differential equation governing the flow is non-linear and can admit non-unique solutions. There exist two different physically meaningful solutions one satisfying the boundary conditions at the upper wall and the other at the lower wall. The effects of the power-law nature of the fluid on the pumping characteristics and axial velocity are studied in detail.     

A study of swelling and collapse in polyelectrolyte gels

Swelling and collapse of a polyelectrolyte are shown to be the result of a delicate balance of three thermodynamic forces, viz. osmotic pressure, polymer elasticity, and van der Waals attraction. The behaviour of the polyelectrolyte gel is studied under isotropic pressure and under uniaxial and biaxial loading.     

Classical wave equation in fluid filament stretching

Extension of thin non-uniform filaments of Newtonian and power-law fluids was fond to be governed by the classical wave equation, when inertia, surface tension, air drag and gravitational pull are negligible and the filaments are isothermal. The extension can either be one-shot transient stretching or transients in continuous drawing. Several simple transients in melt spinning were discussed as solutions of the classical wave equation with the phenomenon of wave reflection found to occur at the spinneret. The sustained oscillation of draw resonance encountered in unstable melt spinning was found to be mathematically equivalent to a repetitive whip lashing movement of a string under tension.     

Different behaviors of concentrated polysaccharide systems in large-amplitude oscillating shear fields

Two series of oscillatory flow tests were carried out on two concentrated polysaccharide systems (hydroxyethyl guar gum and scleroglucan) in order to evaluate the possibilities offered by large-deformation techniques for the discrimination between different classes of macromolecular systems (i.e., entanglement networks and weak gels). Frequently and strain sweeps, as well as combined steady and oscillatory shear tests were performed to analyze the influence of strain amplitude and of the superposed parallel shear rate on the dynamic properties of these materials.     

Steady and modulated flow of an Ostwald fluid around a rotating disk

The solutions of the continuity equation and the equations of motion of the flow in the vicinity of a rotating disk have been established for an Ostwald fluid under steady-state conditions and in modulated flow around a mean value. Under steady-state conditions, the kinematics of the flow is scarcely dependent on the rheological parameters close to the disk, however, for n < 1 long-range effects have been put forward. For modulated flow, in the high-frequency range, a behaviour very different from that observed for a Newtonian fluid was found. In the low-frequency range an asymptotic solution has been proposed which is of special interest in mass transfer problems.Presented at the Second Conference of European Rheologists, Prague, June 17–20, 1986     

Stress optical measurement of the third normal stress difference in polymer melts under oscillatory shear

Results are reported for the dynamic moduli,G andG, measured mechanically, and the dynamic third normal stress difference, measured optically, of a series bidisperse linear polymer melts under oscillatory shear. Nearly monodisperse hydrogenated polyisoprenes of molecular weights 53000 and 370000 were used to prepare blends with a volume fraction of long polymer, _L, of 0.10, 0.20, 0.30, 0.50, and 0.75. The results demonstrate the applicability of birefringence measurements to solve the longstanding problem of measuring the third normal stress difference in oscillatory flow. The relationship between the third normal stress difference and the shear stress observed for these entangled polymer melts is in agreement with a widely predicted constitutive relationship: the relationship between the first normal stress difference and the shear stress is that of a simple fluid, and the second normal stress difference is proportional to the first. These results demonstrate the potential use of 1,3-birefringence to measure the third normal stress difference in oscillatory flow. Further, the general constitutive equation supported by the present results may be used to determine the dynamic moduli from the measured third normal stress difference in small amplitude oscillatory shear. Directions for future research, including the use of birefringence measurements to determineN ₂/N ₁ in oscillatory shear, are described.     

A boundary element simulation of the unsteady motion of a sphere in a cylindrical tube containing a viscoelastic fluid

A boundary element method is used to simulate the unsteady motion of a sphere falling under gravity along the centreline of a cylindrical tube containing a viscoelastic fluid. The fluid is modelled by the upper-convected Maxwell constitutive equation. Results show that the viscoelasticity of the liquid leads to a damped oscillation in sphere velocity about its terminal value. The maximum sphere velocity, which occurs in the first overshoot, is approximately proportional to the square root of the Weissenberg number when the ratio of the sphere radius to the tube radius is sufficiently small. Particular attention is also paid to the wall effects. It is shown that a closer wall reduces the oscillatory amplitude of the sphere velocity but increases its frequency. The results suggest that the falling-ball technique, which is now widely used for viscosity measurement, might also be used for the determination of a relaxation time for a viscoelastic fluid.     

A numerical comparison of two decoupled methods for the simulation of viscoelastic fluid flows

Two decoupled methods for the finite element solution of the planar stick-slip transition problem with Oldroyd-B fluids, namely the method of characteristics and the Lesaint-Raviart technique, are presented and compared. These procedures are used for the local treatment of the stress transport equation imbedded in the constitutive law and allow the approximation of stresses with discontinuous shape functions. Computations are carried out up to a Deborah number of 4 and the methods are shown to yield fairly similar results.     

On the derivation of the constitutive equation of the simple fluid from that of a simple material

It is shown, by the use of the symmetry group of a fluid, that the stress in an elastic fluid depends on the current value of the deformation gradient through the absolute value of the determinant of the deformation gradient, whereas in a simple fluid it depends on this value of the determinant and the history of the relative deformation gradient and nothing else. This derivation is then used to show some of the errors in the criticisms made about the work of Noll. Additionally, a lacuna in Noll's derivation is removed to overcome the only valid criticism made about this theory of simple fluids.     

Evaluating Herschel-Bulkley fluids with the back extrusion (annular pumping) technique

A mathematical model was developed to describe the behavior of Herschel-Bulkley fluids in a back extrusion (annular pumping) device. A technique was also developed to determine the rheological properties (yield stress, flow behavior index, and consistency coefficient) of these fluids. Mathematical terms were expressed in four dimensionless terms, and graphical aids and tables were prepared to facilitate the handling of the expressions.Nomenclature a radius of the plunger, m - dv/dr shear rate, s^–1 - F force applied to the plunger, N - F _b buoyancy force, N - F _cb force corrected for buoyancy, N - F _T recorded force just before the plunger is stopped, N - F _Te recorded force after the plunger is stopped, N - g acceleration due to gravity, m/s² - H(t) momentary height between plunger and container bottom, m - K a/R, dimensionless - L length of annular region, m - L(t) depth of plunger penetration, m - n flow behavior index, dimensionless - p static pressure, Pa - P _L pressure in excess of hydrostatic pressure at the plunger base, Pa - p ₀ pressure at entrance to annulus, Pa - P pressure drop per unit of length, Pa/m - Q total volumetric flow rate through the annulus, m³/s - r radial coordinate, measured from common axis of cylinder forming annulus, m - R radius of outer cylinder of annulus, m - s reciprocal of n, dimensionless - t time, s - T dimensionless shear stress, defined in Eq. (3) - T ₀ dimensionless yield stress, defined in Eq. (4) - T _w dimensionless shear stress at the plunger wall - _p velocity of plunger, m/s - velocity, m/s - mass density of fluid, kg/m³ - Newtonian viscosity, Pa s - P p ₀ –p _L , Pa - consistency coefficient, Pa sⁿ - value of where shear stress is zero - _–, ₊ limits of the plug flow region (Fig. 1) - r/R - shear stress, Pa - _y yield stress, Pa - _w shear stress at the plunger wall, Pa - dimensionless flow rate defined in Eq. (24) - dimensionless velocity defined by Eq. (5) - _–, ₊ dimensionless velocity outside the plug flow region - _max dimensionless maximum velocity in the plug flow region - _p dimensionless velocity at the plunger wall     

Peristaltic motion of third grade fluid in curved channel

＂ Analysis is performed to study the slip effects on the peristaltic flow of non-Newtonian fluid in a curved channel with wall properties. The resulting nonlinear partial differential equations are transformed to a single ordinary differential equation in a stream function by using the assumptions of long wavelength and low Reynolds number. This differential equation is solved numerically by employing the built-in routine for solving nonlinear boundary value problems （BVPs） through the software Mathematica. In addition, the analytic solutions for small Deborah number are computed with a regular perturbation technique. It is noticed that the symmetry of bolus is destroyed in a curved channel. An intensification in the slip effect results in a larger magnitude of axial velocity. Further, the size and circulation of the trapped boluses increase with an increase in the slip parameter. Different from the case of planar channel, the axial velocity profiles are tilted towards the lower part of the channel. A comparative study between analytic and numerical solutions shows excellent agreement.     

Prediction of stress in a linear viscoelastic solid strained while cooling

A simplethermoelastic method is proposed, and justified, for predicting stresses arising during cooling of a linearviscoelastic solid. It is equivalent to representing the material by an array of spring-switch thermoelastic elements. The final stress resulting from an increment of strain is calculated using isochronal modulus data applying to the temperature at which the strain was applied, modified to accommodate temperature dependence of the limiting moduli. The method is exact for a material whose relaxation times and limiting moduli scale uniformly with change in temperature, with time-temperature shift factora _T obeying the Arrhenius equation, cooled such that reciprocal absolute temperature is linear in time. For other cooling sequences it is useful as an approximation. In particular, it assists the computational prediction of stresses arising during cooling in polymer processes.     

The rheology of a smectic main-chain polymer

The rheology of a smectic main-chain polymer shows that the activation energy is high (275 kJ mol^–1) and that the viscosity is more than two orders of magnitude higher than in the isotropic phase. The recoverable strain in the smectic phase is largely due to the reformation of the smectic layers after cessation of shear.