A numerical study of steady and unsteady viscoelastic flow past bounded cylinders

We consider two-dimensional, inertia-free, flow of a constant-viscosity viscoelastic fluid obeying the FENE-CR equation past a cylinder placed symmetrically in a channel, with a blockage ratio of 0.5. Through numerical simulations we show that the flow becomes unsteady when the Deborah number (using the usual definition) is greater than De ≈ 1.3, for an extensibility parameter of the model of L² = 144. The transition from steady to unsteady flow is characterised by a small pulsating recirculation zone of size approximately equal to 0.15 cylinder radius attached to the downstream face of the cylinder. There is also a rise in drag coefficient, which shows a sinusoidal variation with time. The results suggest a possible triggering mechanism leading to the steady three-dimensional Gortler-type vortical structures, which have been observed in experiments of the flow of a viscoelastic fluid around cylinders. The results reveal that the reason for failure of the search for steady numerical solutions at relatively high Deborah numbers is that the two-dimensional flow separates and eventually becomes unsteady. For a lower extensibility parameter, L² = 100, a similar recirculation is formed given rise to a small standing eddy behind the cylinder which becomes unsteady and pulsates in time for Deborah numbers larger than De ≈ 4.0–4.5.     

Simple constitutive modelling of nonlinear viscoelasticity under general extension

We study the behaviour of a single integral constitutive equation, capable of providing analytic expressions for the viscoelastic stress in extensional flows of a variety of deformation histories and geometries, ranging from uniaxial to equibiaxial. It is based on the use of a stress damping function, with a power-law dependence on the elongation, λ: h(λ) = 1/λⁿ. The parameter n (0 ≤ n ≤ 2) signifies the nonlinear viscoelastic character of the material and, therefore, is an inverse measure of network connectivity strength of the underlying microstructure. This renders the constitutive approach applicable to incompressible polymers of a variable degree of branching, strain hardening and stress thinning behavior. Methods of connecting n with the macromolecular architecture and the alignment strength of the flow are also explored.     

Turbulent channel flow of dilute polymeric solutions: Drag reduction scaling and an eddy viscosity model

Direct numerical simulation of viscoelastic turbulent channel flows up to the maximum drag reduction (MDR) limit has been performed. The simulation results in turn have been used to develop relationships between the flow and fluid rheological parameters, i.e. maximum chain extensibility, Reynolds number, Re_τ, and Weissenberg number, We_τ and percent drag reduction (%DR) as well as the slope increment of the mean velocity profile. Moreover, based on the trends observed in the mean velocity profile and the overall momentum balance three different regimes of drag reduction (DR), namely, low drag reduction (LDR; 0 ≤ %DR ≤ 20), high drag reduction (HDR; 20 ≤ %DR ≤ 52) and MDR (52 ≤ %DR ≤ 74) have been identified and mathematical expressions for the eddy viscosity in these regimes are presented. It is found that both in LDR and HDR regimes the eddy viscosity varies with the distance from the channel wall. However, in the MDR regime the ratio of the eddy viscosity to the Newtonian one tends to a very small value around 0.1 within the channel. Based on these expressions a procedure that relies on the DNS predictions of the budgets of momentum and viscoelastic shear stress is developed for evaluating the mean velocity profile.     

Integral-based constitutive equation for rubber at high strain rates

High-speed experiments were conducted to characterize the deformation and failure of Styrene Butadiene Rubber at impact rates. Dynamic tensile stress–strain curves of uniaxial strip specimens and force–extension curves of thin sheets were obtained from a Charpy tensile impact apparatus. Results from the uniaxial tension tests indicated that although the rubber became stiffer with increasing strain rates, the stress–strain curves remained virtually the same above 280 s⁻¹. Above this critical strain rate, strength, fracture strain and toughness decreased with increasing strain rates. When strain rates were below 180 s⁻¹, the initial modulus, tensile strength and breaking extension increased as the strain rate increased. Between strain rates of 180 and 280 s⁻¹, the initial modulus and tensile strength increased with increasing strain rates but the extension at break decreased with increasing strain rates. A hyper-viscoelastic constitutive relation of integral form was used to describe the rate-dependent material behavior of the rubber. Two characteristic relaxation times, 5 ms and 0.25 ms, were needed to fit the proposed constitutive equation to the data. The proposed constitutive equation was implemented in ABAQUS Explicit via a user-defined subroutine and used to predict the dynamic response of the rubber sheets in the experiments. Numerical predictions for the transient deformation and failure of the rubber sheet were within 10% of experimental results.     

Laminar flow of power-law fluids past a rotating cylinder

In this work, the continuity and momentum equations have been solved numerically to investigate the flow of power-law fluids over a rotating cylinder. In particular, consideration has been given to the prediction of drag and lift coefficients as functions of the pertinent governing dimensionless parameters, namely, power-law index (1 ≥ n ≥ 0.2), dimensionless rotational velocity (0 ≤ α ≤ 6) and the Reynolds number (0.1 ≤ Re ≤ 40). Over the range of Reynolds number, the flow is known to be steady. Detailed streamline and vorticity contours adjacent to the rotating cylinder and surface pressure profiles provide further insights into the nature of flow. Finally, the paper is concluded by comparing the present numerical results with the scant experimental data on velocity profiles in the vicinity of a rotating cylinder available in the literature. The correspondence is seen to be excellent for Newtonian and inelastic fluids.     

Criteria for the appearance of recirculating and non-stationary regimes behind a cylinder in a viscoplastic fluid

The appearance of a recirculation zone and the formation of non-stationary vortices behind a cylinder in the unconfined flow of a Herschel–Bulkley fluid have been studied by numerical simulation. The Herschel–Bulkley constitutive equation was regularised by using the Papanastasiou model. Special attention was paid to determining the numerical parameters and comparing them to existing results. The influence of the Oldroyd number and power-law index on flow morphology and, in particular, on the unyielded zones was studied over a wide spectrum (0 ≤ Od ≤ 10) and (0.3 ≤ n ≤ 1.8). It was seen that the greater the Oldroyd number, the greater the critical Reynolds numbers and Strouhal number for the two flow regimes. The influence of the power-law index is more complex.     

Conjugate heat transfer in the unbounded flow of a viscoelastic fluid past a sphere

This work addresses the conjugate heat transfer of a simplified PTT fluid flowing past an unbounded sphere in the Stokes regime (Re = 0.01). The problem is numerically solved with the finite-volume method assuming axisymmetry, absence of natural convection and constant physical properties. The sphere generates heat at a constant and uniform rate, and the analysis is conducted for a range of Deborah (0 ≤ De ≤ 100), Prandtl (10⁰ ≤ Pr ≤ 10⁵) and Brinkman (0 ≤ Br ≤ 100) numbers, in the presence or absence of thermal contact resistance at the solid–fluid interface and for different conductivity ratios (0.1 ≤ κ ≤ 10). The drag coefficient shows a monotonic decrease with De, whereas the normalized stresses on the sphere surface and in the wake first increase and then decrease with De. A negative wake was observed for the two solvent viscosity ratios tested (β = 0.1 and 0.5), being more intense for the more elastic fluid. In the absence of viscous dissipation, the average Nusselt number starts to decrease with De after an initial increase. Heat transfer enhancement relative to an equivalent Newtonian fluid was observed for the whole range of conditions tested. The dimensionless temperature of the sphere decreases and becomes more homogeneous when its thermal conductivity increases in relation to the conductivity of the fluid, although small changes are observed in the Nusselt number. The thermal contact resistance at the interface increases the average temperature of the sphere, without affecting significantly the shape of the temperature profiles inside the sphere. When viscous dissipation is considered, significant changes are observed in the heat transfer process as Br increases. Overall, a simplified PTT fluid can moderately enhance heat transfer compared to a Newtonian fluid, but increasing De does not necessarily improve heat exchange.     

Comment on “Stress boundary conditions for plate bending” by F.Y.M. Wan [Int. J. Solids Struct. 40 (2003) 4107–4123]

By using the reciprocal theorem of elasticity, the author obtained the appropriate stress boundary conditions for the Levy solution for plate bending accurate to all order for plates of general edge geometry and loading. Two special cases of k = 0 (axisymmetric deformation of a circular plate) and k ⩾ 2 (unsymmetric deformation of a circular plate) were discussed in detail in the paper.     

An experimental investigation of negative wakes behind spheres settling in a shear-thinning viscoelastic fluid

We present detailed experimental results examining “negative wakes” behind spheres settling along the centerline of a tube containing a viscoelastic aqueous polyacrylamide solution. Negative wakes are found for all Deborah numbers (2.43≤De(˙γ)≤8.75) and sphere-to-tube aspect ratios (0.060≤a/R≤0.396) examined. The wake structures are investigated using laser-Doppler velocimetry (LDV) to examine the centerline fluid velocity around the sphere and digital particle image velocimetry (DPIV) for full-field velocity profiles. For a fixed aspect ratio, the magnitude of the most negative velocity, U _min , in the wake is seen to increase with increasing De. Additionally, as the Deborah number becomes larger, the location of this minimum velocity shifts farther downstream. When normalized with the sphere radius and the steady state velocity of the sphere, the axial velocity profiles become self-similar to the point of the minimum velocity. Beyond this point, the wake structure varies weakly with aspect ratio and De, and it extends more than 20 radii downstream. Inertial effects at high Reynolds numbers are observed to shift the entire negative wake farther downstream. Using DPIV to investigate the transient kinematic response of the fluid to the initial acceleration of the sphere from rest, it is seen that the wake develops from the nonlinear fluid response at large strains. Measurements of the transient uniaxial extensional viscosity of this weakly strain-hardening fluid using a filament stretching rheometer show that the existence of a negative wake is consistent with theoretical arguments based on the opposing roles of extensional stresses and shearing stresses in the wake of the sphere. Received: 10 November 1997 Accepted: 1 May 1998     

Lattice orientation effects on void growth and coalescence in fcc single crystals

Void growth and coalescence in fcc single crystals were studied using crystal plasticity under uniaxial and biaxial loading conditions and various orientations of the crystalline lattice. A 2D plane strain unit cell with one and two cylindrical voids was employed using three-dimensional 12 potentially active slip systems. The results were compared to five representative orientations of the tensile axis on the stereographic triangle. For uniaxial tension conditions, the void volume fraction increase under the applied load is strongly dependent on the crystallographic orientation with respect to the tensile axis. For some orientations of the tensile axis, such as [1 0 0] or [1 1 0], the voids exhibited a growth rate twice as fast compared with other orientations ([1 0 0], [2 1 1]). Void growth and coalescence simulations under uniaxial loading indicated that during deformation along some orientations with asymmetry of the slip systems, the voids experienced rotation and shape distortion, due mainly to lattice reorientation. Coalescence effects are shown to diminish the influence of lattice orientation on the void volume fraction increase, but noteworthy differences are still present. Under biaxial loading conditions, practically all differences in the void volume fraction for different orientations of the tensile axes during void growth vanish. These results lead to the conclusion that at microstructural length scales in regions under intense biaxiality/triaxiality conditions, such as crack tip or notched regions, the plastic anisotropy due to the initial lattice orientation has only a minor role in influencing the void growth rate. In such situations, void growth and coalescence are mainly determined by the stress triaxiality, the magnitude of accumulated strain, and the spatial localization of such plastic strains.     

New contributions on laminar flow of inelastic non-Newtonian fluid in the two-dimensional symmetric expansion: Creeping and slowly moving flow conditions

The steady flow of generalized Newtonian fluid in a two-dimensional 1:3 sudden expansion was studied numerically. Finite volume method was applied to solve the momentum equations along with the continuity equation and the Power law rheological model within the laminar flow regime for a range of Reynolds number and Power law index values. The values of generalized Reynolds number, based on physical and rheological properties, upstream channel height and bulk velocity, were varied between 0.0001 ≤ Re_gen ≤ 10, while the Power law index values mapped the 0.60 ≤ n ≤ 1.40 range, allowing for the investigation of both shear-thinning and shear-thickening effects at creeping as well as slowly moving fluid flow conditions. We report accurate results of a systematic study with a focus on most important characteristics of recirculating fluid flow in the downstream section of sudden expansion geometry. It is shown that for the creeping flow regime there exist finite sized redevelopment length, extra pressure drop (Couette correction) and recirculation zones (also called as Moffatt vortices) that are influenced by the non-Newtonian viscous behaviour.     

Numerical simulation of the fountain flow instability in injection molding

For the first time, the viscoelastic flow front instability is studied in the full non-linear regime by numerical simulation. A two-component viscoelastic numerical model is developed which can predict fountain flow behavior in a two-dimensional cavity. The eXtended Pom-Pom (XPP) viscoelastic model is used. The levelset method is used for modeling the two-component flow of polymer and gas. The difficulties arising from the three-phase contact point modeling are addressed, and solved by treating the wall as an interface and the gas as a compressible fluid with a low viscosity. The resulting set of equations is solved in a decoupled way using a finite element formulation. Since the model for the polymer does not contain a solvent viscosity, the time discretized evolution equation for the conformation tensor is substituted into the momentum balance in order to obtain a Stokes like equation for computing the velocity and pressure at the new time level. Weissenberg numbers range from 0.1 to 10. The simulations reveal a symmetric fountain flow for Wi = 0.1–5. For Wi = 10 however, an oscillating motion of the fountain flow is found with a spatial period of three times the channel height, which corresponds to experimental observations.     

Relating inhomogeneous deformation to local texture in zirconium through grain-scale digital image correlation strain mapping experiments

The effect of local texture on inhomogeneous plastic deformation is studied in zirconium subjected to uniaxial compression. Cross-rolled commercially pure Zr 702 plate that had a strong basal (0 0 0 1) texture through the plate thickness, and a non-basal texture in cross-section, was obtained. At a compressive strain rate of 1 s^?1, samples loaded either in the through-thickness or in-plane directions exhibited significant differences in yield strength, hardening response and failure mechanisms. These macroscopic differences are related to microstructural features by combining information from electron backscattered diffraction with real time in situ imaging and subsequent full-field strain measurements obtained using digital image correlation. Experimental results indicate that the through-thickness loaded zirconium samples, which show a strong basal-texture in the loading direction, do not deform homogeneously – implying the lack of a representative volume element. The detailed surface deformation fields provided by digital image correlation allow for a qualitative and quantitative analysis of the relationship between grain orientation and patterns of deformation bands that form as the precursors to development of an adiabatic shear band in the through-thickness loaded sample. For the in-plane loaded samples, inhomogeneities still exist at the microscale, but the collective behavior of several grains leads to a homogeneous response at the macroscale. It is observed that local texture for hcp polycrystals, which are significantly slip restricted, can directly affect both local and global response, even at low to moderate plastic strains.     

Innovative iteration technique for nonlinear ordinary differential equations of large deflection problem of circular plates

The large deflection problem of a bending circular plate is reduced to a nonlinear ordinary differential equation, and a type of pseudo-linearization is used to obtain the final solution. An innovative iteration technique is suggested. The initial iteration values of some functions for a given loading are adopted from the previous solution for a lower loading. This will significantly extend the range of solution for the non-dimensional loading. Previously, the solution for the non-dimensional loading Q = 100 was obtained. However, in this paper the solution for the non-dimensional loading Q = 1000 is achieved.     

Superelastic shape memory alloy cables: Part I – Isothermal tension experiments

Cables (or wire ropes) made from NiTi shape memory alloy (SMA) wires are relatively new and unexplored structural elements that combine many of the advantages of conventional cables with the adaptive properties of SMAs (shape memory and superelasticity) and have a broad range of potential applications. In this two part series, an extensive set of uniaxial tension experiments was performed on two Nitinol cable constructions, a 7 × 7 right regular lay and a 1 × 27 alternating lay, to characterize their superelastic behavior in room temperature air. Details of the evolution of strain and temperature fields were captured by simultaneous stereo digital image correlation and infrared imaging, respectively. Here in Part I, the nearly isothermal, superelastic responses of the two cable designs are presented and compared. Overall, the 7 × 7 construction has a mechanical response similar to that of straight wires with propagating transformation fronts and distinct stress plateaus during stress-induced transformations. The 1 × 27 construction, however, exhibits a more compliant and stable mechanical response, trading a decreased force for additional elongation, and does not exhibit transformation fronts due to the deeper helix angles of the layers. In Part II that follows, selected subcomponents are dissected from the two cable’s hierarchical constructions to experimentally break down the cable’s responses.     

Comparison of a quasi-Newtonian fluid with a viscoelastic fluid in planar contraction flow

The flow of a 5.0 wt.% solution of polyisobutylene in tetradecane through a planar 4 : 1 contraction exhibiting a shear thinning viscosity is simulated using the flow-type sensitive quasi-Newtonian fluid model. The shear viscosity is fitted by the Giesekus model, which, with the chosen parameters, leads to an extension thickening elongational viscosity. The stress and velocity fields of the numerical simulations are compared with the experimental results of Quinzani et al. [J. Non-Newtonian Fluid Mech. 52 (1994) 1–36] and the numerical results of the viscoelastic simulation using the Giesekus model of Azaiez et al. [J. Non-Newtonian Fluid Mech. 62 (1996) 253–277]. It can be shown that the quasi-Newtonian fluid qualitatively predicts the essential features of the flow in the vicinity of the contraction.     

Influence of single rectangular groove on the flow past a circular cylinder

In the present study, flow control mechanism of single groove on a circular cylinder surface is presented experimentally using Particle image velocimetry (PIV). A square shaped groove is patterned longitudinally on the surface of the cylinder with a diameter of 50 mm. The flow characteristics are studied as a function of angular position of the groove from the forward stagnation point of the cylinder within 0° ≤ θ ≤ 150°. In the current work, instantaneous and time-averaged flow data such as vorticity, ω streamline, Ψ streamwise, u/U_o and transverse, v/U_o velocity components, turbulent kinetic energy, TKE and RMS of streamwise, u_rms and transverse, v_rms velocity components are utilized in order to present the results of quantitative analyses. Furthermore, Strouhal numbers are calculated using Karman vortex shedding frequency, f_k obtained from single point spectral analysis. It is concluded that a critical angular position of the groove, θ = 80° is observed. The flow separation is controlled within 0° ≤ θ < 80°. At θ = 80°, the flow separation starts to occur in the upstream direction. The instability within the shear layer is also induced on grooved side of the cylinder with frequencies different than Karman vortex shedding frequency, f_k.     

Modeling and measuring visco-elastic properties: From collagen molecules to collagen fibrils

Collagen is the main structural protein in vertebrate biology, determining the mechanical behavior of connective tissues such as tendon, bone and skin. Although extensive efforts in the study of the origin of collagen exceptional mechanical properties, a deep knowledge of the relationship between molecular structure and mechanical properties remains elusive, hindered by the complex hierarchical structure of collagen-based tissues. Understanding the viscoelastic behavior of collagenous tissues requires knowledge of the properties at each structural level. Whole tissues have been studied extensively, but less is known about the mechanical behavior at the submicron, fibrillar and molecular level. Hence, we investigate the viscoelastic properties at the molecular level by using an atomistic modeling approach, performing in silico creep tests of a collagen-like peptide. The results are compared with creep and relaxation tests at the level of isolated collagen fibrils performed previously using a micro-electro-mechanical systems platform. Individual collagen molecules present a non-linear viscoelastic behavior, with a Young's modulus increasing from 6 to 16 GPa (for strains up to 20%), a viscosity of 3.84±0.38 Pa s, and a relaxation time in the range of 0.24–0.64 ns. At the fibrils level, stress–strain–time data indicate that isolated fibrils exhibit viscoelastic behavior that could be fitted using the Maxwell–Weichert model. The fibrils showed an elastic modulus of 123±46 MPa. The time-dependent behavior was well fit using the two-time-constant Maxwell–Weichert model with a fast time response of 7±2 s and a slow time response of 102±5 s.     

Creation of very-low-Reynolds-number chaotic fluid motions in microchannels using viscoelastic surfactant solution

Solutions of flexible high-molecular-weight polymers or some kinds of surfactant are viscoelastic fluids. The elastic stress is induced in such viscoelastic fluid flows and grows nonlinearly with the flow-rate resulting in many particular flow phenomena, including purely elastic instability. The purely elastic instability can even result in a kind of chaotic fluid motion, the so-called elastic turbulence, which is a recently discovered flow phenomenon and arises at arbitrarily small Reynolds number. By using viscoelastic surfactant solution, we attempted to create the peculiar chaotic fluid motions in several specially designed microchannels in which flows with curvilinear streamlines can be generated. The viscoelastic working fluids were aqueous solutions of surfactant, CTAC/NaSal (cetyltrimethyl ammonium chloride/sodium salicylate). CTAC solutions with weight concentration of 200 ppm (part per million) and 1000 ppm, respectively, at room temperature were tested. For comparison, water flows in the same microchannels were also visualized. The Reynolds numbers for all the microchannel flows were quite small (for solution flows, the Reynolds numbers were the order of or smaller than one) and the flow should be definitely laminar for Newtonian fluid. It was found that the regular laminar flow patterns for low-Reynolds-number Newtonian fluid flow in different microchannels were strongly deformed in solution flows: either asymmetrical flow structures or time-dependent vortical fluid motions appeared. These chaotic flow phenomena were considered to be induced by the viscoelasticity of the CTAC solutions. Discussions about the potential applications using such kind of chaotic fluid motions were also made.     

The transmission of turbulent boundary layer unsteady pressure and shear stress through a viscoelastic layer

The transmission of unsteady pressure and shear stress, generated by a turbulent boundary layer in water, through a viscoelastic layer backed by a rigid plate is investigated. Analytical models are used to estimate the unsteady pressure and shear stress from 10 to 1000 Hz for a flat plate boundary layer with zero pressure gradient. Additionally, models for the transfer of the unsteady pressures and shear stress through the viscoelastic layer are developed. The models are used to predict the unsteady pressure fluctuations, or flow noise, which would be seen by a finite size sensor embedded under the elastomer layer. The unsteady pressure levels are found to be 20 dB greater than the unsteady shear stress levels across all frequency ranges computed, in agreement with recent measurements. The unsteady pressure transfer functions have a peak at the shear wavenumber and are larger than the shear stress transfer magnitudes from 10 to 50 Hz. The unsteady shear stress transfer functions have a peak at the acoustic wavenumber and are larger than the pressure transfer magnitudes from 50 to 1000 Hz. Over the frequency range examined, the unsteady pressures were found to be the dominant contributor to the sensor flow noise due to the considerably larger magnitude of the unsteady pressures on the top of the viscoelastic layer.