Drag reduction in the turbulent pipe flow of polymers

The paper concerns an experimental study of the fully developed turbulent pipe flow of several different aqueous polymer solutions: 0.25%, 0.3% and 0.4% carboxymethylcellulose (CMC), 0.2% xanthan gum (XG), a 0.09%/0.09% CMC/XG blend, 0.125% and 0.2% polyacrylamide (PAA). The flow data include friction factor vs. Reynolds number, mean velocity and near-wall shear rate distributions, and axial velocity fluctuation intensity u′ at a fixed radial location as a laminar/turbulent transition indicator. For each fluid we also include measurements of shear viscosity, first normal-stress difference and extensional viscosity. At high shear rates we find that the degree of viscoelasticity increases with concentration (0.3% CMC is an exception) for a given polymer, and in the sequence XG, CMC/XG, CMC, PAA, whilst at low shear rates the ranking changes to CMC, CMC/XG, XG, PAA. The extensional viscosity ranking is XG/CMC, XG, CMC, PAA at high strain rates and the same as that for the viscoelasticity at low shear rates. We find that the observed drag-reduction behaviour is consistent for most part with the viscoelastic and extensional-viscosity behaviour at the low shear and strain rates typical of those occurring in the outer zone of the buffer region.Although laminar/turbulent transition is practically indiscernible from the friction factor vs. Reynolds number plots, particularly for PAA and XG, the u′ level provides a very clear indicator and it is found that the transition delay follows much the same trend with elasticity/extensional viscosity as the drag reduction.     

The effect of dissolved polymers on the rheological properties of coating colours

Coating colours used for the coating of paper and board consist mainly of a mineral pigment, which is very often clay, a synthetic binder such as a styrenebutadiene latex, dispersion agents and water retention aids. The latter are often water soluble polymers. These polymers have a very strong influence on the rheological properties of the coating colours, both on the strain rate dependence of the apparent viscosity and on the viscoelasticity. The effects of two different grades of carboxymethylcellulose (CMC) and one grade of hydroxyethylcellulose (HEC), on the rheological properties at room temperature of a clay-based coating colour at pH 8, have been investigated. It is concluded that the high values of the dynamic modulus of the colours are due to interactions between the cellulose derivatives and the solid particles, i.e. mainly the clay particles. For HEC this interaction is associated with adsorption of the polymeric molecules on the clay particles. In the case of CMC, the adsorption is strongly retarded by the presence of the dispersant (a polyacrylate salt). It is suggested that the marked elasticity of the CMC-containing colour in addition to a possible polymer adsorption may be due to charge interactions and/or depletion flocculation. The effect of CMC and HEC on the water-retention properties of the colour is also discussed.     

The minimum in-line coalescence height of bubbles in non-Newtonian fluid

The minimum in-line coalescence height of bubbles generated from a submerged nozzle was investigated experimentally in shear thinning non-Newtonian fluid at lower Reynolds number (2∼60). Carboxymethyl cellulose sodium (CMC) aqueous solution and carbon dioxide were used as the liquid phase and the gas phase, respectively. The process of the formation, movement and in-line coalescence of bubbles was visualized and recorded by a high-speed digital camera. The influences of bubble size, bubble generation frequency and liquid property on the minimum in-line coalescence height of bubbles were investigated by changing nozzle diameter, gas flow rate and the mass concentration of CMC aqueous solutions. For a given liquid, the generating frequency and size of bubbles increased but the minimum coalescence height of in-line bubbles decreased when the nozzle diameter and gas flow rate were increased. When the nozzle diameter and gas flow rate were fixed, the shear-thinning effect of CMC aqueous solution became stronger with increasing CMC mass concentration, which led to the increase in both the terminal rise velocity and average acceleration of the trailing bubble, consequently, the minimum in-line coalescence height of bubbles decreased. An empirical correlation for estimating the minimum in-line bubble coalescence height was proposed, the calculating values accords well with experimental data with a mean relative deviation only 7.6%.     

Interaction between Taylor bubbles rising in stagnant non-Newtonian fluids

The interaction between Taylor bubbles rising in stagnant non-Newtonian solutions was studied. Aqueous solutions of carboxymethylcellulose (CMC) and polyacrylamide (PAA) polymers were used to study the effect of different rheological properties: shear viscosity and viscoelasticity. The solutions studied covered a range of Reynolds numbers between 10 and 714, and Deborah numbers up to 14. The study was performed with pairs of Taylor bubbles rising in a vertical column (0.032 m internal diameter) filled with stagnant liquid. The velocities of the leading and trailing bubbles were measured by sets of laser diodes/photocells placed along the column. The velocity of the trailing bubble was analysed together with the liquid velocity profile in the wake of a single rising bubble (Particle Image Velocimetry data obtained from the literature). For the less concentrated CMC solutions, with moderate shear viscosity and low viscoelasticity, the interaction between Taylor bubbles was similar to that found in Newtonian fluids. For the most concentrated CMC solution, which has high shear viscosity and moderate viscoelasticity, a negative wake forms behind the Taylor bubbles, inhibiting coalescence since the bubbles maintain a minimum distance of about 1D between them. For the PAA solutions, with moderate shear viscosity but higher viscoelasticity than the CMC solutions, longer wake lengths are seen, which are responsible for trailing bubble acceleration at greater distances from the leading bubble. Also in the PAA solutions, the long time needed for the fluid to recover its initial shear viscosity after the passage of the first bubble makes the fluid less resistant to the trailing bubble flow. Hence, the trailing bubble can travel at a higher velocity than the leading bubble, even at distances above 90D.     

Rising behavior of single bubble in infinite stagnant non-Newtonian liquids

This work is an experimental study of the rising behavior of single air bubbles in infinite stagnant non-Newtonian liquids. Aqueous solutions of carboxymethyl cellulose (CMC) are selected to study the effect of rheological properties. The high speed photography is employed to record the bubble motion in CMC solutions. The bubble size, rising trajectory, bubble shape and velocities are determined by digital image processing technique. As expected, the rheological properties have great influence on the rising behavior of single bubble. In the less concentrated CMC solutions, the bubble rising process can be divided into three stages according to spatial evolution of bubble shape. The deformation changes the trajectories of rising bubbles and bubble hydrodynamics. As the solution concentration increases, the transitional stage gradually disappears. In the most concentrated CMC solution, the first continuous shape flattening stage is directly followed by a rising process with bubble shape basically constant, the rectilinear path and constant rising velocity. Dimensional analysis is performed to formulate a general dimensionless correlation for the deformation and motion of bubbles in infinite liquids by considering the rheological properties.     

On the dynamics of bubbles in polymer aqueous solutions

By analysing experimentally the rheological behaviors of polyethylene oxide solutions, polyacrylamide solutions, carboxymethylcellulose solutions, and hydroxyethylcellulose solution, a rheological model is proposed. The effect of various polymer additives on an air-liquid surface tension is made clear. The equation of motion for a bubble and the pressure equation are derived by using the new rheological expression, and then the effects of a kind of polymer and a polymer concentration on the bubble radius-time history and impulse pressure are numerically examined.     

A rising bubble in a polymer solution

We propose a boundary integral method to study the shape of a bubble rising under gravity in a dilute polymer solution. Constitutive properties are modelled using a FENE model [M.D. Chilcott, J.M. Rallison, J. Non-Newtonian Fluid Mech. 29 (1988) 381] with a pure surface tension interface. We employ a birefringent strand representation [O.G. Harlen, J.M. Rallison, M.D. Chilcott, High-Deborah-number flows of dilute polymer, J. Non-Newtonian Fluid Mech.34 (1990) 319–349] of the wake to simulate the shape and the time-dependent motion of the bubble. Steady and non-steady solutions reproduce qualitatively the bubble deformation seen in experiment with a small region of very high curvature near the rear stagnation point of the bubble. We find a limit point for steady axisymmetric solutions if the polymer concentration is increased or the surface tension is decreased. Rise speed jump discontinuities were not found.     

On the motion of long bubbles in vertical tubes

Potential flow theory has been applied to study the shape and speed of an infinitely long bubble rising through flowing liquid in a vertical tube. In particular, the combined effects of surface tension and externally forced liquid motion are examined. An analytical formula for the bubble rise velocity in stagnant liquid is proposed, and shown to be in good agreement with experimental data for all values of surface tension. Numerical solutions for the bubble velocity in upward flowing liquid are obtained for laminar and turbulent velocity profiles. Approximate expressions for the bubble velocity, where the effects of liquid motion and surface tension are incorporated through the Reynolds and inverse Eötwos, are proposed and compared with experimental data. The predicted changes in bubble shape have, to a large extent, been confirmed through comparisons with photographic evidence for a wide range of parameters.     

Nitrogen/non-Newtonian fluid two-phase upward flow in non-circular microchannels

This paper presents experimental investigations on nitrogen/non-Newtonian fluid two-phase flow in vertical noncircular microchannels, which have square or triangular cross-section with the hydraulic diameters being D_h = 2.5, 2.886 and 0.866 mm, respectively, by visualization method. Three non-Newtonian aqueous solutions with typical rheological properties, i.e., 0.4% carboxymethyl cellulose (CMC), 0.2% polyacrylamide (PAM) and 0.2% xanthan gum (XG) are chosen as the working fluids. The common flow patterns are identified as slug flow, churn flow and annular flow. The dispersed bubble flow is only found in the case with nitrogen/CMC solution two-phase flow in the largest channel. A new flow pattern of nitrogen/PAM solution two-phase flow, named chained bubble/slug flow, is observed in all the test channels. The flow regime maps are also developed and the results show that the rheological properties of the non-Newtonian fluid have remarkable influence on the flow pattern transitions. The geometrical factors of the microchannel such as the cross-section shape and hydraulic diameter of the channel can also affect the flow regime map. Finally, the results obtained in this work are compared with the available flow pattern transitions.     

Fringe element reconstruction for front tracking for three‐dimensional incompressible flow analysis

Fringe element reconstruction technique for tracking the free surface in three‐dimensional incompressible flow analysis was developed. The flow field was calculated by the mixed formulation based on a four‐node tetrahedral element with a bubble function at the centroid (P1+/P1). Since an Eulerian approach was employed in this study, the flow front interface was advected by the flow through a fixed mesh. For accurate modelling of interfacial movement, a fringe element reconstruction method developed can provide not only an accurate treatment of material discontinuity but also surface tension across the interface. The effect of surface tension was modelled by imposing tensile stress directly on the constructed surface elements at the flow front interface. To verify the numerical approach developed, the developed algorithm was applied to two examples whose solutions are available in references. Good agreement was obtained between the simulation results and these solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

2D lattice Boltzmann investigation of saturated pool boiling using a tunable surface tension model: Prandtl number effects on film boiling

In this paper, the saturated pool boiling is investigated using lattice Boltzmann method. The written FORTRAN code is validated in two aspects: For flow, the thermodynamic consistency test and Laplace law are applied and for heat transfer, the space- and time- averaged Nusselt number is compared with Berenson analytical solution in film boiling regime. In addition, the results of bubble generation and departure are compared with some well-known analytical solutions to show the accuracy of the code. It is confirmed that bubble departure diameter and the departure frequency are related to the gravity acceleration with powers of ? 0.505 and 0.709, respectively, which is in a very good agreement with the existing analytical expressions. The present model has the ability to tune different surface tensions independent of liquid/vapor density ratio, which was unreachable using other existing numerical models of boiling. Thus, the sole effects of surface tension on boiling can also be taken into consideration using the present model. It is also shown that the departure diameter is related to the surface tension with a power of 0.485, which is in good agreement with the analytical expressions. Temperature contours are shown together with flow lines to have a better viewpoint for studying the bubble’s behavior. An intensive temperature gradient is observed in the necking area at the departure time. All the four boiling regimes in the boiling curve are simulated under constant temperature boundary condition. The Prandtl number effects on vapor bubble dynamics in the film boiling regime are investigated using the improved Shan and Chen model for the first time. Results revealed that bubbles are more resistant to depart from the vapor blanket with increasing the Prandtl number.     

Critical concentrations of polymers in solutions according to measurements of the viscosity and specific surface area of aerogels resulting after sublimation of the solvent

Conclusion Measurement of the viscosities of polystyrene and cyclolinear polyphenylsiloxane depending on their concentrations has revealed critical concentrations of these polymers to exist corresponding to the formation of a fluctuating continuous supermolecular network in the solutions. It has been shown on the example of polystyrene solutions that changing from a good to a poor solvent has little effect on the value of the critical concentration. Calculation of the critical concentration with the use of the parameters employed in the free volume theory after the Pezzin method shows good agreement with the critical concentrations determined from the shape of the viscosity vs. concentration dependences of polystyrene solutions.A method is described for producing polymer aerogels by sublimation of the frozen solvent from polymer solutions. It is shown that to obtain aerogels with very high specific surface areas from crystallized solutions is necessary to fulfil at least two conditions: 1. to freeze them quickly; 2. to sublimate the solvent in vacuo at a sufficiently low temperature. Special attention is drawn to the fact that the effectivity of sublimation drying should be estimated by the specific surface area of the preparations obtained as a result of drying.Mesurement of the specific surface areas of aerogels obtained under appropriate conditions from polymer solutions of different concentrations shows that with a poor solvent the aerogels have specific surface areas one decimal order lower. Hence it is concluded that solutions of polymers in poor solvents should yield stronger and less permeable polymeric systems. In all the cases studied the dependence of the specific surface area of the aerogel on the concentration of the polymer in the solution has a distinct maximum which corresponds to the critical concentration determined viscometrically. This is evidence of the correspondence between the structure of the aerogels and the structure of the initial solutions, at least, at concentrations in the region of their critical and above-critical values. Sublimation of the solvent from frozen polymer solutions results in contraction of the samples, which is the most considerable at polymer concentrations below critical. This is also connected with the relatively low specific surface areas of aerogels obtained from solutions of low concentrations.     

Analytic theory for the determination of velocity and stability of bubbles in a Hele-Shaw cell

An asymptotic theory is presented for the determination of velocity and linear stability of a steady symmetric bubble in a Hele-Shaw cell for small surface tension. In the first part, the bubble velocity U relative to the fluid velocity at infinity is determined for small surface tension T by determining a transcendentally small correction to the asymptotic series solution. It is found that for any relative bubble velocity U in the interval (U _c , 2), solutions exist at a countably infinite set of values of T (which has zero as its limit point) corresponding to the different branches of bubble solutions. The value of U _c decreases monotonically from 2 to 1 as the bubble area increases from 0 to . However, for a bubble of an arbitrarily given size, as T 0, a solution exists on any given branch with the relative bubble velocity U satisfying the relation 2–U=cT ^2/3, where c depends on the branch but is independent of the bubble area. The analytical evidence further suggests that there are no solutions for U>2. These results are in agreement with earlier analytical results for a finger.In Part II an analytic theory is presented for the determination of the linear stability of the bubble in the limit of zero surface tension. Only the solution branch corresponding to the largest possible U for given surface tension is found to be stable, while all the others are unstable, in accordance with earlier numerical results.This research has been supported by National Science Foundation Grant DMS-8713246. Partial support was also provided by the NASA Langley Research Center (NAS1-18605) while the author was in residence at the Institute of Computer Applications in Science and Engineering.     

Drag reduction by coupled systems: microbubble injection with homogeneous polymer and surfactant solutions

 The influence of homogeneous surfactant and homogeneous polymer solutions on the performance of microbubble skin friction reduction was investigated on an axisymmetric body. Carbon dioxide was injected into water, homogeneous surfactant (Aerosol OT) solutions, and homogeneous dilute polymer (Polyethylene oxide) solutions. Integrated skin friction measurements were obtained at two freestream velocities as a function of gas injection rate and polyethylene-oxide concentration. A moderate (50%) decrease in surface tension had little to no effect on the drag reducing characteristics of microbubble injection. At similar gas injection rates, microbubble injection exhibited more drag reduction in the polymer solutions than obtained with microbubble injection into water. However, the increased drag reduction obtained with polymer additives was no more than a multiplicative factor related to the baseline levels of drag reduction achieved by the individual methods, and suggests the mechanism for microbubble skin friction reduction acts independently of the polymer drag reduction. Received: 17 April 1998 / Accepted: 12 October 1998     

Surface tension driven oscillations of a bubble in a viscoelastic liquid

Small amplitude surface tension driven oscillations of a spherical bubble in a dilute polymer solution are considered. The rheological properties of the liquid are modelled by using a 3-constant constitutive equation of the Oldroyd type. The Laplace transform of the solution of the initial value problem is inverted numerically. As in the Newtonian fluid case, both a discrete and a continuous spectrum occurs. In addition to the non-dimensional parameters in the corresponding problem for a Newtonian fluid, the results depend on two other parameters: the ratio of the relaxation time of the polymer solution and the time scale of the flow (the Deborah number) and the product of the polymer concentration and the intrinsic viscosity. For small bubbles in an aqueous solution having a small relaxation time, significant additional damping is found even for dilute solutions.     

Flow around individual Taylor bubbles rising in stagnant polyacrylamide (PAA) solutions

The flow around single Taylor bubbles rising in stagnant non-Newtonian solutions of polyacrylamide (PAA) polymer was studied using a technique employing simultaneous particle image velocimetry (PIV) and shadowgraphy. Solutions with different weight percentages of polymer, varying from 0.01 to 0.80 wt.%, were used to cover a wide range of flow regimes. The rheological fluid properties and pipe dimension yielded Reynolds numbers between 2 and 1160, and Deborah numbers up to 115. The shape of the bubbles rising in the different solutions was compared and quantified by fitting correlations. The flow around the nose of the bubbles was found to be similar for all conditions studied. Velocity profiles were measured and analysed in the liquid film around the bubbles. A comparison of bubble wake flow patterns was made. For the 0.10 and 0.20 wt.% PAA solutions, long wakes with a recirculation region were observed. Below the wakes, a flow of stretched liquid was found. Negative wakes were also observed for the more concentrated solutions.     

Molecular dynamics study of the liquid-vapor interface of lithium bromide aqueous solutions

The molecular dynamics simulations of the liquid–vapor interface of LiBr aqueous solutions were carried out to investigate the structural and thermophysical properties. As concerns the structural properties, the results of molecular dynamics simulation show that the ions exist in the liquid apart from the surface and this tendency becomes strong as the solute concentration is lowered. This phenomenon is due to the desorption of ion. The calculated values such as density or surface tension agree with experimental ones. As concerns thermophysical properties, the number of water molecules in the bulk gas decreases with an increase of the solute concentration. This result represents the depression of vapor pressure. In addition, in order to investigate the dynamic process of water vapor absorption into LiBr aqueous solution, the molecular dynamics simulation under non-equilibrium condition was carried out. The results show that when the solute concentration is low and the temperature is also low, almost all incident water molecules become trapped at the solution surface and then easily diffuse into the bulk liquid, and when the solute concentration is high and temperature is also high, most incident water molecules become trapped at the solution surface, and the sequent processes are very complicated. Received on 28 September 1998     

Flow of Newtonian and Non-Newtonian Fluids Through Packed Beds: An Experimental Study

The phenomena of flow reduction and flow enhancement was observed in case of viscoelastic and viscoinelastic fluids flowing through packed beds, respectively. In this study, the pressure drop-flow rate behaviors for the flow of Newtonian (water), non-Newtonian viscoinelastic (Carboxy methyl cellulose solution in water, CMC) and viscoelastic (Polyacrylamide solution in water, PAA) fluids have been experimentally studied and pressure drop behavior compared with existing models for viscoinelastic and viscoelastic fluids. Based on the observed data, an appropriate empirical correlation for pressure drop prediction in case of non-Newtonian fluid flowing through packed bed has been proposed. The correlation predicts the data well to within a reasonable accuracy.     

Effect of gas expansion on the velocity of a Taylor bubble: PIV measurements

The effect of gas expansion on the velocity of a Taylor bubble was studied experimentally. The velocity field in the liquid ahead of a Taylor bubble was measured by particle image velocimetry (PIV), and the bubble velocity was measured with two pairs of laser diodes and photocells. The experiments were done in a 7.0 m long vertical tube with a 32 mm internal diameter. Solutions of carboxymethylcellulose (CMC) polymer with weight percentages between 0.01% and 0.1% were used. The expansion of slug gas induces an increase in the bubble velocity and a corresponding displacement of the liquid ahead of the bubble. The velocity of the bubble increases by an amount equal to the maximum velocity in the liquid displaced. For the solutions studied, the induced velocity profile was parabolic and the bubble velocity increase was equal to the liquid velocity at the tube axis, i.e., twice the mean velocity in the liquid displaced. The corrected velocity obtained by subtracting the velocity increase from the value of the bubble velocity is independent of the bubble length.