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.     

The influence of flow-induced non-Newtonian fluid properties on turbulent drag reduction

Friction factors and velocity profiles in turbulent drag reduction can be compared to Newtonian fluid turbulence when the shear viscosity at the wall shear rate is used for the Reynolds number and the local shear viscosity is used for the non-dimensional wall distance. On this basis, an apparent maximum drag reduction asymptote is found which is independent of Reynolds number and type of drag reducing additive. However, no shear viscosity is able to account for the difference between the measured Reynolds stress and the Reynolds stress calculated from the mean velocity profile (the Reynolds stress deficit). If the appropriate local viscosity to use with the velocity fluctuation correlations includes an elongational component, the problem can be resolved. Taking the maximum drag reduction asymptote as a non-Newtonian flow, with this effective viscosity, leads to agreement with the concept of an asymptote only when the solvent viscosity is used in the non-dimensional wall distance.     

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.     

Newtonian stratified flow through an abrupt expansion

The present paper is devoted to the numerical simulations of stratified generalized Newtonian flow. The main feature of the algorithm is to include the coordinates of the interface as primitive unknowns of the flow problem together with the nodal velocity components and pressures calculated on a deformable finite element mesh. Newton-Raphson's iterative method is used for solving the non-linear problem. Special attention is given to the location of the intersection between the interface and a solid boundary, when it is unknown at the outset. The method is applied to the creeping stratified flow of two Newtonian fluids through a one-to-four abrupt expansion; the results remain valid for the contraction problem. It is found that, under appropriate conditions, the small Newtonian corner vortex can increase by a large factor, both in size and intensity; the degrees of freedom of the simulation are the ratios of flow-rate and of viscosity of the fluids.     

Zur Strömung viskoser Flüssigkeiten durch gewellte Rohre

Zusammenfassung Die Strömung zäher nicht-newtonscher Flüssigkeiten durch gewellte kreisförmige Rohre unter Wirkung eines axialen Druckgefälles wurde theoretisch und experimentell untersucht. Das besondere Interesse galt dabei der durch die Wellung veränderten Druck-Durchsatz-Charakteristik. Die theoretischen Betrachtungen beziehen sich auf den Grenzfall hinreichend langer Wellen, wo sich der Querschnitt des Flüssigkeitsfadens in Strömungsrichtung nur langsam ändert und sowohl die Trägheit als auch die Elastizität der Flüssigkeit eine untergeordnete Rolle spielen. In dieser Näherung konnte der Durchsatz in Abhängigkeit vom Druckgefälle und den nichtlinearen Fließeigenschaften der Flüssigkeit für beliebige Wellenamplituden analytisch bestimmt werden. Die theoretischen Ergebnisse sind in Einklang mit experimentellen Befunden.

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The flow of a non-Newtonian fluid driven by an axial pressure gradient through a corrugated pipe of circular cross-section is analysed theoretically and realized experimentally. Particular attention is directed toward the pressure-discharge characteristics of the pipe which change due to the corrugation. A theory is developed for the limiting case of corrugations with long wave-length, where the cross-section of the pipe varies gradually in the flow direction and the effects of inertia and elasticity of the fluid are of minor importance. Within such a long-wave approximation the flow rate, in dependence on the applied pressure drop and on the non-linear viscous properties of the fluid, is determined analytically for arbitrary wave amplitudes. The theoretical results are in agreement with experimental findings.

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On the orientation of ellipsoidal particles in a turbulent shear flow

Direct numerical simulation (DNS) of small prolate ellipsoidal particles suspended in a turbulent channel flow is reported. The coupling between the fluid and the particles is one-way. The particles are subjected to the hydrodynamic drag force and torque valid for creeping flow conditions. Six different particle cases with varying particle aspect ratios and equivalent response times are investigated. Results show that, in the near-wall region, ellipsoidal particles tend to align with the mean flow direction, and the alignment increases with increasing particle aspect ratio. When the particle inertia increases, the particles are less oriented in the spanwise direction and more oriented in the wall-normal direction. In the core region of the channel, the orientation becomes isotropic.     

Viscosity measurements with a magnetoviscometer in the zero-shear and transition region of polypropylenes

Streaming of a non-Newtonian fluid around a sphere is of special importance not only for measuring viscosities with falling spheres, but also for many problems connected with polymer processing. Using the mentioned measuring principle, attention has to be paid to the following points: The sphere is moving in a fluid (melt) of finite extension which requires the application of wall and perhaps end corrections. These are possibly not the same for Newtonian and non-Newtonian fluids. To calculate the viscosity with the help of Stokes law the steady-state velocity is necessary, and it is essential, how long it takes the sphere to reach it. To compare our results with other data, an average shear rate has to be calculated, since there is no uniform shear rate around the sphere. Velocities being very low in our experiments result in very small Reynolds numbers (Re < 10^–3), which allows the application of Stokes law practically without corrections.The experiments were performed at zero shear and in the transition region above. It turned out, that it is usually not possible to extrapolate from shear-dependent viscosity data to zero-shear viscosity.Dedicated to Prof. A. Neckel on the occasion of his 60th birthday     

The flow past a sphere in a cylindrical tube: effects of intertia,shear-thinning and elasticity

Numerical solutions are presented for the flow past a sphere placed at the centreline of a cylindrical tube for Reynolds numbers ranging from 0 to 150, using a boundary element method. Fluids are modelled by a variety of constitutive equations including the Newtonian, the Carreau and the Phan-Thien-Tanner models. The influence of inertia, shear-thinning and fluid elasticity on the flow field, drag and the pressure drop force-drag ratio is examined. Some results are compared with available experimental data.     

A note on the periodic motion of a visco-elastic fluid in a radially non-symmetric constricted tube

An approximate solution for the problem of fluid flow through a rigid tube with a mild constriction is given. It is assumed that the fluid is visco-elastic (Maxwell fluid) and the constriction is non-symmetric with respect to the radial distance. A theoretical result is given for the wall shear stress and numerical solutions are shown graphically for different values of the relaxation time and the shape parameter of the constriction profile.     

The squeezing flow of a model suspension fluid

The paper is concerned with the squeezing flow of a model suspension fluid. The numerical solution obtained by a time-dependent Boundary Element Method is compared to an asymptotic solution at large radius. It is found that the kinematics are Newtonian in character, and the fibres quickly align themselves radially. Consequently, the squeezing force is only weakly dependent on the initial orientations of the fibres and the device can be used for measuring the effective viscosity of the suspension. The effective viscosity found from the squeezing flow agrees surprisingly well with experimental data and numerical data derived from the falling sphere geometry at low volume fractions ( < 0.1).     

Hydrodynamisch induzierte Eigenbewegung von idealisierten Teilchen in newtonschen Suspensionsmedien unter Berücksichtigung begrenzender Wände.

Zusammenfassung Die hydrodynamisch induzierte Bewegung starrer und elastischer Teilchen, die in einem newtonschen Medium suspendiert sind, wird unter Berücksichtigung gekrümmter Begrenzungen untersucht. Die Geometrie eines hyperbolischen Zylinders erlaubt die Simulation einer Strömung durch poröse Medien und in den Grenzfällen durch ein Rohr und ein kreisförmiges Loch. Der Einfluß des inhomogenen Strömungsfeldes verursacht eine Migrationsbewegung der suspendierten Teilchen quer zu den Stromlinien. Es wird gezeigt, daß eine inhomogene Suspension elastischer Teilchen nach Durchlauf eines konvergenten und eines divergenten Bereiches eine Entmischung erfährt. Die Konzentration der Teilchen nimmt sowohl im Kernbereich der Strömung als auch aufgrund des abstoßenden Wandeffektes in unmittelbarer Wandnähe ab.

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The flow-induced migration of rigid and elastic particles suspended in a Newtonian medium is studied. The flow situation investigated consists of pressuredriven flow through a hyperbolic cylinder. This permits the simulation of flow through a porous medium and in the limiting cases through a tube and through a circular hole. The influence of the nonhomogeneous flow field effects a migration of the suspended particles across the streamlines. It is deduced that nonhomogeneous suspensions of elastic particles receive a separation according to size after passing a converging and diverging section. The results imply that the particle concentration decreases in the centre of the flow as well as in the immediate neighbourhood of the walls.

Zweiter Teil einer vom Fachbereich Chemietechnik der Universität Dortmund genehmigten Dissertation     

A photographic study of shapes of bubbles and coalescence in non-Newtonian polymer solutions

This communication reports (photographically) on the shapes of bubbles of different gases in several Newtonian and polymer solutions, encompassing a wide range of rheological behavior. Effects of a surface active agent are mentioned and bubble coalescence in viscoelastic solutions is visualized.     

Transient rheology in a new type of couette apparatus. Application to blood

By using our new air-bearing viscometer different types of transient flow can be studied; in the present work this viscometer was specifically applied to non-Newtonian blood suspensions. To observe the influence of both the aggregation and the deformation of red blood cells (RBCs), different concentrations of fibrinogen and dextran were used: the suspended RBCs were simply washed or rigidified with diamide. From the data three rheological parameters were determined, two of which are relative to the behavior at low and at high shear gradients, respectively. Their values were related to the theory of Taylor, and the internal viscosity of RBCs was estimated to be around 3 cp.     

Response of blood flow through an artery under stenotic conditions

This paper presents an analytical study on the behavoiur of blood flow in an artery having a stenosis. This is basically formulated through the use of a suitable mathematical model. The arterial segment under consideration is simulated by an anisotropically elastic cylindrical tube filled with a viscous incompressible fluid representing blood. The analysis is carried out for an artery with mild local narrowing in its lumen forming a stenosis. Particular emphasis has been paid to the effect of the surrounding connective tissues on the motion of the arterial wall. Blood is treated as a Newtonian fluid. The analysis is restricted to propagation of small amplitude harmonic waves, generated due to the flow of blood whose wave length is large compared to the radius of the arterial segment. The effect of the shape of stenosis on the resistance to blood flow has been well illustrated quantitatively through numerical computations of the resulting expressions. A quantitative analysis is also made for the variation of the phase velocity, as well as the velocity of wave propagation and the flow rate, in order to illustrate the applicability of the model.     

Direct numerical simulation of the turbulent energy balance and the shear stresses in power-law fluid flows in pipes

The results of direct numerical simulation of turbulent flows of non-Newtonian pseudoplastic fluids in a straight pipe are presented. The data on the distributions of the turbulent stress tensor components and the shear stress and turbulent kinetic energy balances are obtained for steady turbulent flows at the Reynolds numbers of 10⁴ and 2×10⁴. As distinct from Newtonian fluid flows, the viscous shear stresses turn out to be significant even far from the wall. In power-law fluid flows the mechanism of the energy transport from axial to transverse component fluctuations is suppressed. It is shown that with decrease in the fluid index the turbulent transfer of the momentum and the velocity fluctuations between the wall layer and the flow core reduces, while the turbulent energy flux toward the wall increases. The earlier-proposed models for the average viscosity and the non-Newtonian one-point correlations are in good agreement with the data of direct numerical simulation.     

Estimates of apparent elongational viscosity using the fibre spinning and “pure” methods

A comparison is made between elongational properties predicted by the FENE-P dumbbell rheological equation of state for fibre spinning and true values obtained from pure elongation. True values are found by using a constant elongation rate in the model and calculating the steady state values of the stresses. Two new averaging procedures are suggested for extracting elongational properties from fibre spinning experiments. It is suggested that a variety of flow conditions be used and care be taken when determining these properties. The rheological model is found to predict adequately shear and elongational flow properties of the polyacrylamide-type of Boger or ideal elastic fluid.     

Computational Modelling of Non-Newtonian Effects on Flow in Channels with Moving Wall Indentations

Non-Newtonian effects in a channel with moving wall indentations are assessed numerically by a finite volume method for solving the unsteady incompressible Navier-Stokes equations and using a power-law model exhibiting shear thinning viscosity and Casson's model as the constitutive equations for the non-Newtonian fluid. The computations show that for a non-Newtonian fluid, there are differences in the velocity profiles and in the structure and size of the reversed flow regions as compared with the corresponding Newtonian fluid. The comparison of non-Newtonian and Newtonian wall shear stress reveals a slight decrease in the magnitude on the average for the non-Newtonian case, eventually resulting in the strength of the “wave train” being slightly weaker than those corresponding to a Newtonian fluid.     

A procedure for calculation of the Wagner model velocity profile in the Poiseuille flow

The presented procedure enables calculation of a velocity profile for the Wagner fluid in the Poiseuille flow. The velocity profile can be approximated with a prescribed accuracy, thus enabling boundary conditions for the Wagner model, that are required for numerical simulations to be defined. Convergence analysis for the procedure indicates the existence of a critical Deborah number, which limits the validity of the approximation.     

On the flow resistance of viscoelastic fluids through packed beds

Analytical solutions are obtained for the free surface cell model of packed beds using a third order fluid. Second order perturbed results indicate a substantial increase in resistance to the flow of a viscoelastic fluid through a packed bed. This predicted increase is in good agreement with experimental findings.     

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.