Drag reduction change of polyethyleneoxide solutions in pipe flow

Change of drag reduction (DR) along a tube (D=2 mm, L=4 m) was experimentally investigated. To attain turbulent flow with Re=8 × 10⁴, a tank operated under high pressure up to 16 MPa. Solutions of different brands of polyethyleneoxide (PEO) with concentrations from 1 ppm to 100 ppm were tested. The results indicate that DR is not a constant value but depends on the time and intensity of interaction between the polymer and the turbulent flow. There are three regions with different behaviors of DR: growth, maximum, and slope down. Maximum DR coincides with the Virk ultimate DR and can be described by the suggested simple formula . A decrease in the DR maximum has not been found even for high shear stresses τ _p < 800 Pa. DR dynamics for four brands of PEO with different molecular weight was studied. Direct experimentally determined DR may be greater than the Virk ultimate value if the change in velocity profile is not taken into account. The corrected DR never exceeds the ultimate DR. Received: 10 April 2000/Accepted: 24 May 2001     

Axial annular flow of plastic fluids: dead zones and plug-free flow

In axial annular flow, the shear stress decreases from its value τ(κR) at the inner cylinder to 0 at r = λR and increases from then on to τ(R) at the outer cylinder. For plastic fluids with a yield stress τ _c, λ will be such that flow commences when τ(κR) = τ(R) = τ _c. For fluids with position-dependent yield stresses (electro- and magnetorheological fluids are examples), the situation is more complex. While it is possible that yielding and flow occur everywhere, it is also possible that flow occurs only in parts of the fluid-filled space, and a dead zone (region in which the fluid is at rest) close to one of the walls exists. In that case, the fluid will flow no matter how small the applied pressure difference is. If P is large enough, the dead zone ceases to exist and flow without any plug is possible. The fluid flows as if no yield stress exists.

Three-dimensional viscoelastic simulation of coextrusion process: comparison with experimental data

 Three-dimensional numerical simulation of viscoelastic coextrusion process has been performed and numerical results were compared with the experimental data of Karagiannis et al. (1990). By varying the magnitude of the second normal stress difference and its ratio of Fluid I and Fluid II, we were able to control the interface profile and the degree of encapsulation along the downstream direction. By increasing the parameter α (α_Fluid I=α_Fluid II) from 0.1 to 0.4 in the Giesekus model and increasing the α ratio (α_Fluid Iα_Fluid II) between Fluid I and Fluid II from 2.0 to 4.0 in the permissible range of realistic polymeric systems, the interface profile and the degree of encapsulation along the downstream direction were fitted with the experimental results. There was little difference between the numerical results and the experimental data in the interface profile and the degree of encapsulation along the downstream direction when the α ratio was set to 3.0 (0.3:0.1). Fluid I with larger magnitude of the second normal stress difference protrudes into Fluid II with smaller magnitude of the second normal stress difference around the symmetric plane, while Fluid II wraps around Fluid I near the side walls. As the ξ ₁ ratio (ξ _{1 ,Fluid I}ξ _{1 ,Fluid II}) increases from 1.0 to 3.0 for the two-mode Phan-Thien and Tanner model, it was found that the curvature of the interface profile increased, and the difference between the numerical results and the experimental data in the interface profile and the degree of encapsulation along the downstream direction was almost negligible when the ξ ₁ ratio was set to 3.0 (0.54:0.18). Although the parameters of viscoelastic models were fitted by using the shear viscosity data only, quantitative agreements between the numerical results and the experimental coextrusion data were quite satisfactory. Received: 24 April 2001 Accepted: 5 June 2001     

The role of the mathematical mean value theorem (MVT) in rheometry: an easy way to convert apparent flow curves into correct ones

The mean value theorem of integral calculus guarantees that the apparent viscosity η _a can easily be converted into the correct viscosity η. For ordinary liquids there is a direct identity between η _a and η but the apparent shear rate (or apparent shear stress) has to be shifted to the representative shear rate γ˙^ (or representative shear stress τ^). A model free approximation scheme is introduced which implies a constant shift factor. The corresponding approximation for η is acceptable for liquids most commonly encountered. For plastic fluids the relation between η and η _a is more complex since it involves a function depending upon α; the yield stress relative to the maximum stress within the viscometer. Using the same approximation scheme as before the shift factor will involve α as well. The corresponding approximation of η is shown to be acceptable for the whole range of α. Received: 7 February 2000/Accepted: 15 February 2000     

Three views of viscoelasticity for Cox–Merz materials

A slight rearrangement of the classical Cox and Merz rule suggests that the shear stress value of steady shear flow, , and complex modulus value of small amplitude oscillatory shear, G^∗(ω) = (G^′2 + G^″2)^1/2, are equivalent in many respects. Small changes of material structure, which express themselves most sensitively in the steady shear stress, τ, show equally pronounced in linear viscoelastic data when plotting these with G^∗ as one of the variables. An example is given to demonstrate this phenomenon: viscosity data that cover about three decades in frequency get stretched out over about nine decades in G^∗ while maintaining steep gradients in a transition region. This suggests a more effective way of exploiting the Cox–Merz rule when it is valid and exploring reasons for lack of validity when it is not. The τ −G^∗ equivalence could also further the understanding of the steady shear normal stress function as proposed by Laun.     

Effect of an Exit-Wedge Angle on Pinch-off and Mass Entrainment of Vortex Rings in Air

The effects of exit-wedge angle on evolution, formation, pinch-off, propagation and diffusive mass entrainment of vortex rings in air were studied using digital particle image velocimetry. Vortex rings were generated by passing a solenoid-valve-controlled air jet through a cylindrical nozzle. Experiments were performed over a wide range of exit-wedge angles (10° ≤ α ≤ 90°) of the cylindrical nozzle, initial Reynolds numbers (450 ≤ Re ≤ 4,580) and length-to-diameter ratios (0.9 ≤ L/D ≤ 11) of the air jet. For sharp edges (α ≤ 10°), a secondary ring may emerge at high Reynolds numbers, which tended to distort the vortex ring if ingested into it. For blunt edges (α ≥ 45°), by contrast, stable vortex rings were produced. The formation phase of a vortex ring was found to be closely related to its evolution pattern. An exit-wedge angle of 45° was found to be optimal for rapid pinch-off and faster propagation and better stability of a vortex ring. Diffusive mass entrainment was found to be between 35% and 40% in the early stages of a vortex ring propagation and it gradually increased throughout the course of vortex ring propagation. Entrainment fraction was found to be sensitive to the L/D ratio of the initial jet and decreases when the L/D ratio is increased.     

Influence of flow path modification on oscillation of cavity shear layer

This study investigates the influence on the oscillating characteristics of a cavity shear layer by introducing either a sloped bottom or a flow path modifier at the bottom of the cavity. All the experiments are performed in a recirculating water channel. The laser Doppler velocimetry system and the laser sheet technique are employed to perform the quantitative velocity measurements and the qualitative flow visualization, respectively. The Reynolds number, based on the momentum thickness at the upstream edge of the cavity, is kept at about Re _{θ 0}=194 ± 3.4. It is found that, in addition to the feedback effect, the upstream moving part of the recirculating flow inside the cavity also plays an important role in changing the oscillating characteristics of the unstable shear layer. As the bottom of the cavity is either negatively or positively sloped, the oscillating characteristics of the cavity shear layer are modified to different extents. Significant reduction of the oscillating amplitude within the cavity is found while the bottom slope increases up to d/L=± 2/5. As the bottom slope further increases up to d/L=± 1/2, the self-excited oscillation is completely suppressed. In addition, the ability to suppress the self-excited oscillation by the negative bottom slopes is superior to that in the case of a positive bottom slope. Depending upon the fence locations, the upstream moving part of the recirculating flow will perturb the unstable shear layer at different x/L locations, leading to different oscillating amplitudes. The ability to promote the enlarged oscillating amplitude of the unstable shear layer is better for a fence inclined at a positive angle than for one at a negative angle. Received: 31 May 2000/Accepted: 11 January 2001     

Turbulence Modeling Effects on the Prediction of Equilibrium States of Buoyant Shear Flows

The effects of turbulence modeling on the prediction of equilibrium states of turbulent buoyant shear flows were investigated. The velocity field models used include a two-equation closure, a Reynolds-stress closure assuming two different pressure-strain models and three different dissipation rate tensor models. As for the thermal field closure models, two different pressure-scrambling models and nine different temperature variance dissipation rate ɛ_τ) equations were considered. The emphasis of this paper is focused on the effects of the ɛ_τ-equation, of the dissipation rate models, of the pressure-strain models and of the pressure-scrambling models on the prediction of the approach to equilibrium turbulence. Equilibrium turbulence is defined by the time rate of change of the scaled Reynolds stress anisotropic tensor and heat flux vector becoming zero. These conditions lead to the equilibrium state parameters, given by /ɛ, _τ/ɛ_τ, , Sk/ɛ and G/ɛ, becoming constant. Here, and _τ are the production of turbulent kinetic energy k and temperature variance , respectively, ɛ and ɛ_τ are their respective dissipation rates, R is the mixed time scale ratio, G is the buoyant production of k and S is the mean shear gradient. Calculations show that the ɛ_τ-equation has a significant effect on the prediction of the approach to equilibrium turbulence. For a particular ɛ_τ-equation, all velocity closure models considered give an equilibrium state if anisotropic dissipation is accounted for in one form or another in the dissipation rate tensor or in the ɛ-equation. It is further found that the models considered for the pressure-strain tensor and the pressure-scrambling vector have little or no effect on the prediction of the approach to equilibrium turbulence. Received 21 April 2000 and accepted 21 February 2001     

Stability of an Initially, Stably Stratified Fluid Subjected to a Step Change in Temperature

The onset of convective instability in an initially quiescent, stably stratified fluid layer between two horizontal plates is analyzed with linear theory. The bottom boundary is heated suddenly from below, subjected to a step change in surface temperature. The critical time t _c to mark the onset of Rayleigh-Bénard convection is predicted by propagation theory. This theory uses the length scaled by , where α denotes thermal diffusivity. Under the normal mode analysis the dimensionless disturbance equations are obtained as a function of τ(=αt/d ²) and ζ(=Z/), where d is the fluid layer depth and Z is the vertical distance. The resulting equations are transformed to self-similar ones by using scaling and finally fixing τ as τ_c under the frame of coordinates τ and ζ. For a given γ, Pr and τ_c, the minimum value of Ra is obtained from the marginal stability curve. Here γ denotes the temperature ratio to represent the degree of stabilizing effect, Pr is the Prandtl number and Ra is the Rayleigh number. With γ=0, the minimum Ra value approaches the well-known value of 1708 as τ_c increases. However, it is inversely proportional to τ_c ^3/2 as τ_c decreases. With increasing γ, the system becomes more stable. It is interesting that in the present system, propagation theory produces the stability criteria to bound the available experimental data over the whole domain of time. Received 5 November 2001 and accepted 29 March 2002 Published online: 2 October 2002 RID="*" ID="*" This work has been supported by both SK Chemicals Co. Ltd. and LG Chemical Ltd., Seoul under the Brain Korea 21 Project of the Ministry of Education. Communicated by H.J.S. Fernando     

Rheological and mechanical properties of silica colloids: from Newtonian liquid to brittle behaviour

Rheological and mechanical properties of aqueous mono-disperse silica suspensions (Ludox? HS40) are investigated as a function of particle volume fraction (ϕ _p ranging from 0.22 to 0.51) and water content, using shear rate tests, oscillatory methods, indentation and an ultrasonic technique. As the samples are progressively dried, four regimes are identified; they are related to the increasing particle content and the existence and behaviour of the electrical double layer (EDL) around each particle. For 0.22 ≤ ϕ _p ≤ 0.30), the suspensions are stable due to the strong electrostatic repulsion between particles and show Newtonian behaviour (I). As water is removed, the solution pH decreases and the ionic strength increases. The EDL thickness therefore slowly decreases, and screening of the electrostatic repulsion increases. For 0.31 ≤ ϕ _p ≤ 0.35, the suspensions become turbid and exhibit viscoelastic (VE) shear thinning behaviour (II), as they progressively flocculate. For 0.35 ≤ ϕ _p ≤ 0.47, the suspensions turn transparent again and paste-like, with VE shear thinning behaviour and high elastic modulus (III). At higher particle concentration, the suspensions undergo a glass transition and behave as an elastic brittle solid (IV, ϕ _p = 0.51).     

Influence of molecular parameters on the stress dependence of viscous and elastic properties of polypropylene melts in shear

The stress dependencies of the steady-state viscosity η and, particularly, that of the steady-state elastic compliance J _e of various linear isotactic polypropylenes (PP) and one long-chain branched PP are investigated using creep-recovery tests. The creep stresses applied range from 2 to 10,000 Pa. In order to discuss the stress-dependent viscosity η and elastic compliance J _e with respect to the influence of the weight average molar mass M _w and the polydispersity factor M _w/M _n the PP are characterized by SEC–MALLS. For the linear PP, linear steady-state elastic compliances J_e⁰J_{\rm e}^0 in the range of 10^− 5–10^− 3 Pa^− 1 are obtained depending on the molar mass distribution. J_e⁰J_{\rm e}^0 of the LCB-PP is distinctly higher and comes to lie at around 10^− 2 Pa^− 1. J_e⁰J_{\rm e}^0 is found to be independent of M _w but strongly dependent on polydispersity. η and J _e decrease with increasing stress. For the linear PP, J _e as a function of the stress τ is temperature independent. The higher M _w/M _n the stronger is the shear thinning of η and the more pronounced is the stress dependence of J _e. For the LCB-PP, the strongest stress dependence of η and J _e is observed. Furthermore, for all PP J _e reacts more sensitively to an increasing stress than η. A qualitative explanation for the stronger stress dependence of J _e compared to η is given by analyzing the contribution of long relaxation times to the viscosity and elasticity.     

Vortex ring generation due to the coalescence of a water drop at a free surface

 Results are presented of an experimental investigation of vortex ring formation by a fluid drop contacting a free surface with negligible velocity. The pool fluid is mixed with fluorescein dye, and a laser sheet is used to illuminate a plane of the flow. A series of representative images is recorded by a CCD camera and speculation is made regarding specific sources of vorticity flux through the free surface. Two scaling analyses previously presented by other investigators are demonstrated to be equivalent under the assumptions of this experiment, and they provide the motivation for a series of test runs in which the duration of the coalescence process, τ_*, is related to variations in drop diameter L and fluid surface tension σ. Experimental results are in agreement with the analyses, showing τ_*∼σ^-1/2 and τ_*∼L ^3/2. Received: 22 December 1995 / Accepted: 15 October 1996     

Measurements and scaling of wall shear stress fluctuations

Measurements of velocity and wall shear stress fluctuations were made in an external turbulent boundary layer developed over a towed surface-piercing flat plate. An array of eight flush-mounted wall shear stress sensors was used to compute the space-time correlation function. A methodology for in situ calibration of the sensors for ship hydrodynamic applications is presented. The intensity of the wall shear stress fluctuations, τ _rms/τ _avg was measured as 0.25 and 0.36 for R _θ =3,150 and 2,160 respectively. The probability density is shown to exhibit positive skewness, and lack of flow reversals at the wall. Correlations between velocity and wall shear stress fluctuations are shown to collapse with outer boundary layer length and velocity scales, verifying the existence of large-scale coherent structures which convect and decay along the wall at an angle of inclination varying from 10 to 13° over the range of Reynolds numbers investigated. The wall shear stress convection velocity determined from narrow band correlation measurements is shown to scale with outer variables. The space-time correlation of the wall shear is shown to exhibit a well-defined convective ridge, and to decay 80% over approximately 3δ for R _θ =3,150. Published online: 7 November 2002     

Linear viscoelastic behavior of perfluorooctyl sulfonate micelles: effects of counter-ions

Linear viscoelastic behavior was investigated for aqueous solutions of perfluorooctyl sulfonate (C₈F₁₇SO⁻ ₃; abbreviated as FOS) micelles having a mixture of tetraethylammonium (N⁺(C₂H₅)₄; TEA) and lithium (Li⁺) ions as the counter-ions. The solutions had the same FOS concentration (0.1 mol l⁻¹) and various Li⁺ fractions in the counter-ions, φ_Li = 0−0.6, and the FOS micelles in these solutions formed threads which further organized into dendritic networks. At T ≤ 15 °C, the terminal relaxation time τ and the viscosity η, governed by thermal scission of the networks, increased with increasing φ_Li up to 0.55. A further increase of φ_Li resulted in decreases of τ and η and in broadening of the relaxation mode distribution. These rheological changes are discussed in relation to the role of TEA ions in thermal scission: Previous NMR studies revealed that only a fraction of TEA ions were tightly bound to the FOS micellar surfaces and these bound ions stabilized the thread/network structures. The concentration of non-bound TEA ions, C_TEA ^*, decreased and finally vanished on increasing φ_Li up to φ_Li ^* ≅ 0.6, and the concentration of the bound TEA ions significantly decreased on a further increase of φ_Li. The non-bound TEA ions appeared to catalyze the thermal scission of the FOS threads, and the observed increases of τ and η for φ_Li < 0.55 were attributed to the decrease of C_TEA ^*. On the other hand, the decreases of τ and η as well as the broadening of the mode distribution, found for φ_Li > 0.55 (where C_TEA ^* ≅ 0), were related to destabilization of the FOS threads/networks due to a shortage of the bound TEA ions and to the existence of concentrated Li⁺ ions. Viscoelastic data of pure FOSTEA and FOSTEA/FOSLi/TEACl solutions lent support to these arguments for the role of TEA ions in the relaxation of FOSTEA/FOSLi solutions. Received: 12 October 1999/Accepted: 1 November 1999     

The interfacial crack between two dissimilar elastic-plastic materials

This paper presents an exact asymptotic analysis on the interfacial crack between two dissimilar elastic-plastic materials. These two materials have identical hardening exponent (n ₁=n ₂) but different hardening coefficient (α₁ ≠ α₂). Two groups of the near-crack-tip fields have been obtained, which not only satisfy the continuity of both tractions (σ_θ, τ_rθ) and displacements (u _r ,u _θ) on the interface, but also meet the traction free conditions on the crack faces. The first group of fields have the mode mixityM ^P quite close toM ^P =1 (MODE I) within the whole range 0 ≤ α₁/α₂ < ∞. As for the second group of fields, which is only obtained within the narrow range 0.9 ≤ α₁/α₂ ≤ 1, it is found that the mode mixity changes sharply with the ratio value α₁/α₂. The project supported by National Natural Science Foundation of China     

Further Study on Pure Shear

It is known that the Cauchy stress tensor T is a pure shear when trT = 0. An elementary derivation is given for a coordinate system such that, when referred to this coordinate system, the diagonal elements of T vanish while the off-diagonal elements τ ₁, τ ₂, τ ₃, are the pure shears. The structure of τ _i (i = 1, 2, 3) depends on one non-dimensional parameter q = 54(detT)² / [tr(T ²)]³, 0 ≤ q ≤ 1. When q = 0, one of the three τ _i vanishes. A coordinate system can be chosen such that the remaining two have the same magnitude or one of the remaining two also vanishes. When q = 1, all three τ _i have the same magnitude. However, there is a one-parameter family of coordinate systems that gives the same three τ _i . For q ≠ 0 or 1, none of the three τ _i vanishes and the three τ _i in general have different magnitudes. Nevertheless, a coordinate system can be chosen such that two of the three τ _i have the same magnitude. ^★Professor Emeritus of University of Illinois at Chicago and Consulting Professor of Stanford University.     

Stability of Degenerate Stationary Waves for Viscous Gases

This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous gases in the half space. We discuss the following two cases: (1) viscous conservation laws and (2) damped wave equations with nonlinear convection. In each case, we prove that the solution converges to the corresponding degenerate stationary wave at the rate t ^−α/4 as t → ∞, provided that the initial perturbation is in the weighted space L²_a=L²(\mathbb R₊; (1+x)^a dx){L^2_\alpha=L^2({\mathbb R}_+;\,(1+x)^\alpha dx)} . This convergence rate t ^−α/4 is weaker than the one for the non-degenerate case and requires the restriction α < α_*(q), where α_*(q) is the critical value depending only on the degeneracy exponent q. Such a restriction is reasonable because the corresponding linearized operator for viscous conservation laws cannot be dissipative in L²_a{L^2_\alpha} for α > α^*(q) with another critical value α^*(q). Our stability analysis is based on the space–time weighted energy method in which the spatial weight is chosen as a function of the degenerate stationary wave.     

2D and 3D Turbulent Fluctuations in Open Channel Flow with Re τ = 590 Studied by Large Eddy Simulation

Well-resolved 3D Large Eddy Simulations (LES) are presented for open channel flow at a Reynolds number Re _τ  = 590 based on friction velocity u _τ and water depth h. The results are depth-averaged and thereby information is obtained on the 2D horizontal fluctuations in the channel. The total turbulence is decomposed into 2D and 3D fluctuations and the energy content of these as well as their spectral distribution is studied. It is found that only 15% of the fluctuating energy is contained in the 2D fluctuations and that these are mostly of scales larger than the water depth while the 3D fluctuations are restricted by the limited vertical extent of the water body and have scales smaller than the water depth. Information is obtained on the dispersion terms arising from the depth-averaging procedure, and scalar transport due to a vertical line source of tracer is studied thereby investigating the contribution of the 2D and 3D fluctuations to the transverse mixing of the scalar.     

Visualizations of large-scale vortices in flow about a blunt-faced flat plate

A new type of flow visualization method utilizing a smoke-wire, a high-speed camera with high framing rates and a laser light sheet was employed to delineate the unsteady processes of large-scale vortices in the separated shear layer about a blunt-faced flat plate at Re _H  = 560. The sequential images showed that the unsteady behavior of large-scale vortices in the separated shear layer varies as the shedding phases of large-scale vortices alter. Particularly, at a certain phase, a vortex-merging process between the two neighboring large-scale vortices took place. Received: 17 November 1998/Accepted: 1 November 1999     

Effect of Al<Subscript>2</Subscript>O<Subscript>3</Subscript> phases on the enhancement of thermal conductivity and viscosity of nanofluids in engine oil

Thermal conductivity of α-Al₂O₃ was measured using hot wire method. α-Al₂O₃ (20 nm in size) was synthesized by microwave method for which, the results were compared with commercially available γ-Al₂O₃. Thermal conductivity of nanofluids was investigated considering, it is dependency on Al₂O₃ phase. It was observed that by adding 3 wt% of nano γ-Al₂O₃ and α-Al₂O₃ to the engine oil, thermal conductivity increases by 37 and 31%, respectively. The corresponding viscosity increase for the same amount of nano γ-Al₂O₃ and α-Al₂O₃ were 36 and 38%, respectively. It was concluded that the differences in thermal conductivity originate from higher specific surface area of γ-Al₂O₃ compared to the α-Al₂O₃ which is the result of porosity difference, obtained during the synthesis process.