A note on the translation of a thin rod inside a cylinder

The problem of a thin rod moving longitudinally along the axis of symmetry of a cylindrical vessel is examined for Newtonian and non-Newtonian liquids. For non-Newtonian fluids, the inelastic power-law type solution predicts the experimental results particularly well. On account of wall effects, the induced pressure gradients are much greater for a Newtonian fluid than for a viscoelastic fluid. In fact, in the latter case, they may be considered negligible when the radius of the inner cylinder is small compared to the one of the outer cylinder.     

The stratified flow of gas and non-newtonian liquid in horizontal pipes

Predictions of pressure drop and holdup are presented for the stratified flow of gas and non-Newtonian liquid obeying the Ostwald-de Waele power law model. The model of Taitel & Dukler (1976) for gas/Newtonian liquid flow is extended to liquids possessing either shear-thinning or shear-thickening laminar flow behaviour and computed results are given for flow behaviour indices in the range $\text{0.1 ≤ n ≤ 2}$. In particular, conditions are defined for drag reduction of the liquid flow by the presence of the gas. It is concluded that drag reduction occurs over the largest ranges of liquid and gas flow rates at the lowest $\text{n}$ values, provided that liquid flow remains laminar, but that maximum drag reduction may be expected for shear-thickening liquids with $\text{n}$ values of 2 or greater. Ratios of the liquid flow rate in the presence of gas to that for liquid flow alone under a constant pressure gradient are also presented. These ratios frequently exceed unity and are greatest for highly shear-thinning liquids.Although the Taitel & Dukler approach is consistent with experiments on gas/Newtonian liquid flow, and, in addition, appears to be valid for immiscible Newtonian liquid-liquid systems, provided that the viscosity ratio of the two phases is at least five, experiments are required to confirm its applicability for gas/non-Newtonian systems.     

Dynamic scaling of unsteady shear-thinning non-Newtonian fluid flows in a large-scale model of a distal anastomosis

Dimensional analysis has been applied to an unsteady pulsatile flow of a shear-thinning power-law non-Newtonian liquid. An experiment was then designed in which both Newtonian and non-Newtonian liquids were used to model blood flow through a large-scale (38.5 mm dia.), simplified, rigid arterial junction (a distal anastomosis of a femorodistal bypass). The flow field within the junction was obtained by Particle Imaging Velocimetry and near-wall velocities were used to calculate the wall shear stresses. Dimensionless wall shear stresses were obtained at different points in the cardiac cycle for two different but dynamically similar non-Newtonian fluids; the good agreement between the measured dimensionless wall shear stresses confirm the validity of the dimensional analysis. However, blood exhibits a constant viscosity at high-shear rates and to obtain complete dynamic similarity between large-scale experiments and life-scale flows, the high-shear viscosity also needs to be included in the analysis. How this might be done is discussed in the paper.     

Turbulent flow of Newtonian and shear-thinning liquids through a sudden axisymmetric expansion

 Measurements are reported for the turbulent flow through a sudden expansion of a moderately elastic shear-thinning liquid and also for two Newtonian liquids. The differences in the mean velocity fields for the two fluid types are relatively small, including the length of the recirculation region which is essentially unaffected by the fluid rheology. Although turbulent kinetic energy levels for the non-Newtonian fluids are always lower than for the Newtonian fluids, no significant difference is found in the relative contributions to the turbulent kinetic energy of the axial, radial and tangential normal stresses. Since the vorticity thicknesses are much the same for all flows, viscoelasticity appears to be responsible for the reduced levels of turbulent kinetic energy for the non-Newtonian fluids. Received: 6 November 1998/Accepted: 27 January 1999     

Experimental and theoretical aspects of the two-roll mill problem

Experiments are described in which two cylinders of the same radii-rotate with identical speeds in a bath of Newtonian or non-Newtonian liquid. The torque on one of the cylinders is measured as a function of rotational speed for various values of the cylinder separation and the flow patterns are observed by a dye-injection technique.The observed experimental results for a Newtonian liquid correlate well with the theoretical predictions but a similar correlation in the case of elastic liquids is made difficult by the strongly three-dimensional nature of the flow in this case and the difficulty in estimating the amount of liquid passing through the rollers. The possibility of flow reversal effects due to the high Trouton ratios in the case of the elastic liquids is investigated both experimentally and theoretically.     

Numerical simulation of the non-Newtonian blood flow through a mechanical aortic valve

This work focuses on the comparison between Newtonian and non-Newtonian blood flows through a bileaflet mechanical heart valve in the aortic root. The blood, in fact, is a concentrated suspension of cells, mainly red blood cells, in a Newtonian matrix, the plasma, and consequently its overall behavior is that of a non-Newtonian fluid owing to the action of the cells’ membrane on the fluid part. The common practice, however, assumes the blood in large vessels as a Newtonian fluid since the shear rate is generally high and the effective viscosity becomes independent of the former. In this paper, we show that this is not always the case even in the aorta, the largest artery of the systemic circulation, owing to the pulsatile and transitional nature of the flow. Unexpectedly, for most of the pulsating cycle and in a large part of the fluid volume, the shear rate is smaller than the threshold level for the blood to display a constant effective viscosity and its shear thinning character might affect the system dynamics. A direct inspection of the various flow features has shown that the valve dynamics, the transvalvular pressure drop and the large-scale features of the flow are very similar for the Newtonian and non-Newtonian fluid models. On the other hand, the mechanical damage of the red blood cells (hemolysis), induced by the altered stress values in the flow, is larger for the non-Newtonian fluid model than for the Newtonian one.     

Phase split of nitrogen/non-Newtonian fluid two-phase flow at a micro-T-junction

An experimental investigation has been undertaken to understand the phase split of nitrogen gas/non-Newtonian liquid two-phase flow passing through a 0.5 mm T-junction that oriented horizontally. Four different liquids, including water and aqueous solutions of carboxymethyl cellulose (CMC) with different mass concentrations of 0.1, 0.2 and 0.3 wt%, were employed. Rheology experiments showed that different from water, CMC solutions in this study are pseudoplastic non-Newtonian fluid whose viscosity decreases with increasing the shear rate. The inlet flow patterns were observed to be slug flow, slug–annular flow and annular flow. The fraction of liquid taken off at the side arm for nitrogen gas/non-Newtonian liquid systems is found to be higher than that for nitrogen gas/Newtonian liquid systems in all inlet flow patterns. In addition, with increasing the pseudoplasticity of the liquid phase, the side arm liquid taken off increases, but the increasing degree varies with each flow pattern. For annular flow, the increasing degree is much greater than those for slug and slug–annular flows.     

New model for single spherical particle settling velocity in power law (visco-inelastic) fluids

Particle settling in a non-Newtonian power law fluid is of interest to many industrial applications, including chemical, food, pharmaceutical, and petroleum industry. Conventionally, the Newtonian model for the drag coefficient prediction is extended to non-Newtonian fluids. The approach of merely replacing a viscosity term in Newtonian correlation with a power law apparent viscosity is reported to be inadequate.     

Displacement of a Newtonian fluid by a non-Newtonian fluid in a porous medium

This paper presents an analytical Buckley-Leverett-type solution for one-dimensibnal immiscible displacement of a Newtonian fluid by a non-Newtonian fluid in porous media. The non-Newtonian fluid viscosity is assumed to be a function of the flow potential gradient and the non-Newtonian phase saturation. To apply this method to field problems a practical procedure has been developed which is based on the analytical solution and is similar to the graphic technique of Welge. Our solution can be regarded as an extension of the Buckley-Leverett method to Non-Newtonian fluids. The analytical result reveals how the saturation profile and the displacement efficiency are controlled not only by the relative permeabilities, as in the Buckley-Leverett solution, but also by the inherent complexities of the non-Newtonian fluid. Two examples of the application of the solution are given. One application is the verification of a numerical model, which has been developed for simulation of flow of immiscible non-Newtonian and Newtonian fluids in porous media. Excellent agreement between the numerical and analytical results has been obtained using a power-law non-Newtonian fluid. Another application is to examine the effects of non-Newtonian behavior on immiscible displacement of a Newtonian fluid by a power-law non-Newtonian fluid.     

A boundary integral method for two-dimensional (non)-Newtonian drops in slow viscous flow

A boundary integral method for the simulation of the time-dependent deformation of Newtonian or non-Newtonian drops suspended in a Newtonian fluid is developed. The boundary integral formulation for Stokes flow is used and the non-Newtonian stress is treated as a source term which yields an extra integral over the domain of the drop. The implementation of the boundary conditions is facilitated by rewriting the domain integral by means of the Gauss divergence theorem. To apply the divergence theorem smoothness assumptions are made concerning the non-Newtonian stress tensor. The correctness of these assumptions in actual simulations is checked with a numerical validation procedure. The method appears mathematically correct and the numerical algorithm is second order accurate. Besides this validation we present simulation results for a Newtonian drop and a drop consisting of an Oldroyd-B fluid. The results for Newtonian and non-Newtonian drops in two dimensions indicate that the steady state deformation is quite independent of the drop-fluid. The deformation process, however, appears to be strongly dependent on the drop-fluid. For the non-Newtonian drop a mechanical model is developed to describe the time-dependent deformation of the cylinder for small capillary numbers.     

Dynamics of formation and dripping of drops of deformation-rate-thinning and -thickening liquids from capillary tubes

Dynamics of formation of drops of non-Newtonian liquids from capillary tubes is studied computationally. The rheology of the drop liquids is described by a constitutive relation that accounts for both deformation-rate-thinning and -thickening. The analysis is expedited by reducing the original system of three-dimensional but axisymmetric equations to a system of one-dimensional slender-jet equations. The slender-jet equations are solved by a method of lines using a finite element method for spatial discretization and an adaptive finite difference method for time integration. The simulations follow the formation in time of thousands of drops in sequence, including any satellites that may be produced upon the breakup of a thin thread connecting an about-to-form primary drop to the rest of the liquid attached to the tube. Rate-thickening is shown to produce bead-on-string patterns, which are typically attributed to viscoelastic effects, along the thin threads as they near pinch-off. Rate-thinning, on the other hand, is demonstrated to reduce the length of such thin threads. Simulations are used to identify conditions that may lead to minimization and/or elimination of unwanted satellites. Analysis of dripping or leaky faucets of non-Newtonian liquids reveals rich nonlinear dynamical behavior. As with Newtonian liquids, simple periodic or P-1, where P stands for period, dripping at low flow rates gives way to more complex responses as flow rate is increased. In addition to P-1, P-2, and P-4 responses seen in recent computational analyses of dripping faucets of Newtonian liquids, the new non-Newtonian simulations have also uncovered difficult-to-find P-3 responses as well as chaotic states. Rate-thinning and low viscosities are shown to enhance the complexity of observed responses. Rate-thickening, on the other hand, lowers the critical value of the flow rate for the onset of complexity but narrows the range of flow rates over which the dynamics is complex. The possibility of hysteresis is demonstrated and the effect of fluid rheology on the value of the flow rate for transition from dripping to jetting is determined.     

Models for the deformation of a single ellipsoidal drop: a review

Dilute polymer blends and immiscible liquid emulsions are characterized by a globular morphology. The dynamics of a single drop subjected to an imposed flow field has been considered to be a valuable model system to get information on dilute blends. This problem has been studied either theoretically by developing exact theories for small drop deformations or by developing simplified models often based on phenomenological assumptions. In this paper, a critical overview of the available models for the dynamics of a single drop is presented, discussing four different systems, namely the Newtonian system, where a single Newtonian drop is immersed in an infinite Newtonian matrix; the non-Newtonian system, where at least one of the components, the drop fluid or the matrix one, is non-Newtonian; the confined Newtonian system, where the matrix is confined and wall effects alter the drop dynamics; and the confined non-Newtonian system.     

An experimental study of drop migration in shear flow between concentric cylinders

The phenomenon of migration of liquid drops in Couette flow between concentric cylinders due to non-Newtonian fluid properties and shape deformation has been studied experimentally. The results agree very well with the theory of Chan and Leal, which included the effect of hydrodynamic interaction with the bounding walls, and that of velocity profile curvature in a Couette device. Significant observations that were not reported in previous studies include the migration of a deformable Newtonian drop to an equilibrium position between the centerline and the inner rotor, and the competition between normal stresses and shape deformation effects for the case of a Newtonian drop in a non-Newtonian fluid.     

Technique for extensional rheology characterization of highly reactive viscoelastic liquids

This paper reports the elaboration and validation of a new technique for extensional rheological characterization of highly reactive viscous liquids contained in a transparent reaction bulb where considered liquids have been prepared. For measurements, a rod was pulled down to contact with a surface of the liquid and then instantly raised up resulting in formation of a liquid thread. During the test, the studied fluid is in an equilibrium state with an inherent vapor. The elasto-capillary self-thinning and ultimate breakup of threads was registered by a high-speed camera. From the fluid dynamics point of view, the main distinguishing feature of the proposed method from the standard capillary breakup extensional rheometer (CaBER) method is the presence of unlimited amount of a test liquid under the endplate of a moving rod. In the present work, the presumable effect of this distinguishing factor on the final test data was estimated. In the frame of this research, the simulations of Newtonian and non-Newtonian fluids were conducted for the tests performed with the newly elaborated and CaBER-like configurations of devices. The test and computed data show some difference in the results obtained only at low values of Hencky strain (at the initial period of thread evolution). Increase in Hencky strain leads to a good agreement between results obtained with the help of CaBER-like devices as well as the proposed one.     

Dispersion of linear gravity waves on a viscoelastic fluid in an horizontal canal

A characteristic equation is derived that describes the spatial decay of linear surface gravity waves on Maxwell fluids. Except at small frequencies, the derived dispersion relation is different from the temporal decay dispersion relation which is normally studied within fluid mechanics. The implications for waves on viscous Newtonian fluids using the two different dispersion relations is briefly discussed. The wave number is measured experimentally as function of the frequency in a horizontal canal. Seven Newtonian fluids and four viscoelastic liquids with constant viscosity have been used in the experiments. The spatial decay theory for Newtonian fluids fits well to the experimental data. The model and experiments are used to determine limits for the Maxwell fluid time numbers for the four viscoelastic liquids. As a result of low viscosity it was not possible within this study to obtain these time numbers from oscillatory experiments. Therefore, a comparison of surface gravity wave experiments with theory is applicable as a method to evaluate memory times of low viscosity viscoelastic fluids.     

A modified SPH method to model the coalescence of colliding non-Newtonian liquid droplets

This paper develops a modified smoothed particle hydrodynamics (SPH) method to model the coalescence of colliding non-Newtonian liquid droplets. In the present SPH, a van der Waals (vdW) equation of state is particularly used to represent the gas-to-liquid phase transition similar to that of a real fluid. To remove the unphysical behavior of the particle clustering, also known as tensile instability, an optimized particle shifting technique is implemented in the simulations. To validate the numerical method, the formation of a Newtonian vdW droplet is first tested, and it clearly demonstrates that the tensile instability can be effectively removed. The method is then extended to simulate the head-on binary collision of vdW liquid droplets. Both Newtonian and non-Newtonian fluid flows are considered. The effect of Reynolds number on the coalescence process of droplets is analyzed. It is observed that the time up to the completion of the first oscillation period does not always increase as the Reynolds number increases. Results for the off-center binary collision of non-Newtonian vdW liquid droplets are lastly presented. All the results enrich the simulations of the droplet dynamics and deepen understandings of flow physics. Also, the present SPH is able to model the coalescence of colliding non-Newtonian liquid droplets without tensile instability.     

Lubricated and nonlubricated squeezing flow of a double layered array of two power law liquids

Force-time relationships of a double-layered array of two power law liquids in squeezing flow at a constant displacement rate were generated with a computer. As in the case of a single layer, lubrication, or lack of it, has the strongest influence on the magnitude of the forces and the flow pattern. Transient flow regimes that were prominent in the behavior of Newtonian liquid arrays in lubricated squeezing flow were also found in the behavior of the power law liquids. Their prominence was influenced by the liquid's flow index and it was drastically magnified as the differences in the liquid's consistency increased.     

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.     

Oscillations of a gas bubble in a non-Newtonian liquid under the action of an acoustic field

The evolution of the radius of a spherical cavitation bubble in an incompressible non-Newtonian liquid under the action of an external acoustic field is investigated. Non-Newtonian liquids having relaxation properties and also pseudoplastic and dilatant liquids with powerlaw equation of state are studied. The equations for the oscillation of the gas bubble are derived, the stability of its radial oscillation and its spherical form are investigated, and formulas are given for the characteristic frequency of oscillations of the cavitation hollow in a relaxing liquid. The equations are integrated numerically. It is shown that in a relaxing non-Newtonian liquid the viscosity may lead to the instability of the radial oscillations and the spherical form of the bubble. The results obtained here are compared with the behavior of a gas bubble in a Newtonian liquid.     

A physically based correlation for the effects of power law rheology and inclination on slug bubble rise velocity

A study has been made of the motion of long bubbles in inclined pipes containing viscous Newtonian and non-Newtonian liquids. A semi-theoretical expression for the rise velocity of air bubbles in water is derived on the hypothesis that the dominant factor is the momentum exchange of the bubble underflow, i.e. the bubble nose shape. The correlation calls on empirical inputs from established literature on bubble rise speeds at high Reynolds number. The effects of increasing Newtonian viscosity are analysed with reference to the momentum exchange and it is shown how viscosity reduces the inclination dependence of the bubble Froude number. Results from an experimental survey in seven different non-Newtonian liquids in three different diameter pipes are presented. These data are correlated so as to decouple the effects of surface tension and viscosity. An empirical relation is proposed for the effective shear rate in the fluid travelling around the bubble nose. Our correlation is compared to literature data from a broad range of Reynolds numbers with excellent agreement except at shallow angles.