Elastic Normal Stresses Research Articles

How Fluid Pseudoplasticity and Elasticity Affect Propeller Flows in Biogas Fermenters

AbstractMixing in biogas fermenters is complex due to the non‐Newtonian rheology of biogenic substrates, which exhibit both pseudoplasticity and elasticity. It is yet unclear how these non‐Newtonian properties affect propeller flows and the mixing behavior in fermenters. Therefore, propeller flows in Newtonian as well as shear‐thinning inelastic and elastic fluids are compared numerically and validated against particle image velocity (PIV) data. Elastic normal stresses lead to an increase of pumping rates in the laminar regime and a suppression of the formation of a propeller jet in the transitional regime. Thus, flow rates are severely overestimated by the inelastic, shear‐thinning model in this regime. The results indicate that elasticity is critical for an accurate modeling of flows of biogenic substrates.

A PIV study to quantify the relationships between viscoelasticity and flow behavior in stirred tanks

Quantitative data on fluid dynamics in stirred tanks with viscoelastic fluids are scarcely available. Therefore, a particle image velocimetry study on flow phenomena occurring when agitating viscoelastic liquids with radial and axial impellers is presented. Different flow regimes exist, which result from interactions between elastic normal stresses and inertial forces: While for all agitators an axial pumping effect is found in the elastic regime, (semi-)radial flow patterns result in inertia-dominated flows. In the transition regime, locally limited zones exist dominated by either elasticity or inertia. This leads to a flow compartmentalization, a related pseudo-stagnation flow and reduced circulation rates. Which of the flow regimes is present can be predicted by the use of the elasticity number. With it, circulation flow numbers and flow distributions are standardized: In elastic flow conditions, these are constant. Thus, a scaling of the log-normally distributed velocities to the tip speed in this regime is possible.

An indirect method to determine nonlinearly elastic shear stress-strain constitutive relationships for nonlinear torsional vibration of nonlinearly elastic shafts

PurposeIn this paper, the authors aim to propose an effective method to indirectly determine nonlinear elastic shear stress-strain constitutive relationships for nonlinear elasticity materials, and then study the nonlinear free torsional vibration of Al–1%Si shaft.Design/methodology/approachIn this study the authors use BoxLucas1 model to fit the determined-experimentally nonlinear elastic normal stress–strain constitutive relationship curve of Al–1%Si, a typical case of isotropic nonlinear elasticity materials, and then derive its nonlinear shear stress-strain constitutive relationships based on the fitting constitutive relationships and general equations of plane-stress and plane-strain transformation. Hamilton’s principle is utilized to gain nonlinear governing equation and boundary conditions for free torsional vibration of Al–1%Si shaft. Differential quadrature method and an iterative algorithm are employed to numerically solve the gained equations of motion.Findings The effect of four variables, namely dimensionless fundamental vibration amplitude , radius and length , and nonlinear-elasticity intensity factor , on frequencies and mode shapes of the shafts is obtained. Numerical results are in good agreement with reference solutions, and show that compared with linearly elastic shear stress-strain constitutive relationships of the shafts made of the nonlinear elasticity materials, its actual nonlinearly elastic shear stress-strain constitutive relationships have smaller torsion frequencies. In addition, but having opposite hardening effect, the rest of the four variables have softening effect on nonlinearly elastic torsion frequencies. Eventually, taking into account nonlinearly elastic shear stress-strain constitutive relationships, changes of the four factors, i.e. , , and , cause inflation and deflation behaviors of mode shapes in nonlinear free torsional vibration.Originality/valueThe study could provide a reference for indirectly determining nonlinear elastic shear stress-strain constitutive relationships for nonlinear elasticity materials and for structure design of torsional shaft made of nonlinear elasticity materials.

Calculation of stress in FGM beams

Content of the paper is oriented to calculation of elastic normal stress in the Functionally Graded Beams (FGM). Spatial variation of material properties is considered in the lateral, transversal and longitudinal direction of the straight beam. The displacements and internal forces are calculated using our new FGM finite beam element. Heterogeneous material properties are homogenized by extended mixture rules, laminate theory and reference volume element (RVE). Obtained results by our approach are evaluated and compared with the ones obtained by the 3D solid finite elements.

Stable core-annular flows of viscoelastic fluids using the visco-plastic lubrication technique

We give an experimental demonstration that stable core-annular flows can be achieved when lubricating a viscoelastic core fluid with a yield stress fluid. We use Carbopol as the lubricating yield stress fluid and Polyethylene Oxide (PEO) as the core fluid. The yield stress in the lubricating fluid preserves a ring of unyielded material around the interface, restricting the growth of instabilities. When the inlet radius is smaller than that of the established flow the core fluid stream expands, resulting in a net relaxation of the elastic normal stresses as the flow becomes fully developed. At low flow rates ( Q 1) of the core fluid this relaxation does not break the surrounding plug. At larger Q 1 secondary flows and then instabilities are observed. The secondary flows are interesting in that the elastic instability is frozen into the yield stress fluid at the interface, as the stresses drop below the yield stress in the developing flow. At still larger Q 1 the surrounding plug is broken and the interface may deform. In this case elasticity appears to retard the degree of interfacial mixing, in comparison to Newtonian core fluids. When the inlet radius is larger than that of the established flow a stabilizing effect is observed. The study shows that the visco-plastic lubrication technique is feasible for multi-layer flows with elasticity.

Experimental investigation of purely elastic instabilities in periodic flows

Purely elastic instabilities in the flows of polyisobutylene-based Boger fluids in the periodic channel (PC) and past a (linear) periodic array of closely spaced cylinders (PAC) in a channel have been experimentally examined. The test geometries have the same wavelength and amplitude associated with the periodic variation in the cross-sectional area. Pressure measurements show temporal fluctuations that appear when the Weissenberg number exceeds approximately 0.7 and 1.1 for the PC and PAC geometries, respectively. Flow visualization using digital particle imaging velocimetry (DPIV) shows that below the critical Weissenberg number, flows are two-dimensional and steady consisting of slow vortex motion in the wake region for the flow past the PAC flow and in the near wall region for the PC flow. Inception of flow instability results in the breakdown of the symmetric vortices in the flow past the PAC flow and in time-dependent vortex motion in the PC flow. The onset of flow instability is also accompanied by enhanced flow resistance. The power spectral composition of the pressure fluctuations is uniform along the flow direction for both geometries. This implies that in time-dependent simulations a unit cell with periodic boundary conditions can faithfully represent these geometries. The onset Weissenberg number values, when scaled with the maximum curvature of the streamlines, are very close to each other indicating that the instability arises from the coupling between streamline curvature and elastic normal stresses.

The axisymmetric contraction–expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop

The flow of a polystyrene Boger fluid through axisymmetric contraction–expansions having various contraction ratios (2≤ β≤8) and varying degrees of re-entrant corner curvatures are studied experimentally over a large range of Deborah numbers. The ideal elastic fluid is dilute, monodisperse and well characterized in both shear and transient uniaxial extension. A large enhanced pressure drop above that of a Newtonian fluid is observed independent of contraction ratio and re-entrant corner curvature. Streak images, laser Doppler velocimetry (LDV) and digital particle image velocimetry (DPIV) are used to investigate the flow kinematics upstream of the contraction plane. LDV is used to measure velocity fluctuation in the mean flow field and to characterize a global elastic flow instability which occurs at large Deborah numbers. For a contraction ratio of β=2, a steady elastic lip vortex is observed while for contraction ratios of 4≤ β≤8, no lip vortex is observed and a corner vortex is seen. Rounding the re-entrant corner leads to shifts in the onset of the flow transitions at larger Deborah numbers, but does not qualitatively change the overall structure of the flow field. We describe a simple rescaling of the deformation rate which incorporates the effects of lip curvature and allows measurements of vortex size, enhanced pressure drop and critical Deborah number for the onset of elastic instability to be collapsed onto master curves. Transient extensional rheology measurements are utilized to explain the significant differences in vortex growth pathways (i.e. elastic corner vortex versus lip vortex growth) observed between the polystyrene Boger fluids used in this research and polyisobutylene and polyacrylamide Boger fluids used in previous contraction flow experiments. We show that the role of contraction ratio on vortex growth dynamics can be rationalized by considering the dimensionless ratio of the elastic normal stress difference in steady shear flow to those in transient uniaxial extension. It appears that the differences in this normal stress ratio for different fluids at a given Deborah number arise from variations in solvent quality or excluded volume effects.

Non-isothermal modification of purely elastic flow instabilities in torsional flows of polymeric fluids

Previous experimental measurements and linear stability analyses of curvilinear shearing flows of viscoelastic fluids have shown that the combination of streamwise curvature and elastic normal stresses can lead to flow destabilization. Torsional shear flows of highly elastic fluids with closed streamlines can also accumulate heat from viscous dissipation resulting in nonuniformity in the temperature profile within the flow and nonlinearity in the viscometric properties of the fluid. Recently, it has been shown by Al-Mubaiyedh et al. [Phys. Fluids 11, 3217 (1999)] that the inclusion of energetics in the linear stability analysis of viscoelastic Taylor–Couette flow can change the dominant mode of the purely elastic instability from a nonaxisymmetric and time-dependent secondary flow to an axisymmetric stationary Taylor-type toroidal vortex that more closely agrees with the stability characteristics observed experimentally. In this work, we present a detailed experimental study of the effect of viscous heating on the torsional steady shearing of elastic fluids between a rotating cone and plate and between two rotating coaxial parallel plates. Elastic effects in the flow are characterized by the Deborah number, De, while the magnitude of the viscous heating is characterized by the Nahme–Griffith number, Na. We show that the relative importance of these two competing effects can be quantified by a new dimensionless thermoelastic parameter, Θ=Na1/2/De, which is a material property of a given viscoelastic fluid independent of the rate of deformation. By utilizing this thermoelastic number, experimental observations of viscoelastic flow stability in three different fluids and two different geometries over a range of temperatures can be rationalized and the critical conditions unified into a single flow stability diagram. The thermoelastic number is a function of the molecular weight of the polymer, the flow geometry, and the temperature of the test fluid. The experiments presented here were performed using test fluids consisting of three different high molecular weight monodisperse polystyrene solutions in various flow geometries and over a large range of temperatures. By systematically varying the temperature of the test fluid or the configuration of the test geometry, the thermoelastic number can be adjusted appreciably. When the characteristic time scale for viscous heating is much longer than the relaxation time of the test fluid (Θ≪1) the critical conditions for the onset of the elastic instability are in good agreement with the predictions of isothermal linear stability analyses. As the thermoelastic number approaches a critical value, the strong temperature gradients induced by viscous heating reduce the elasticity of the test fluid and delay the onset of the instability. At even larger values of the thermoelastic parameter, viscous heating stabilizes the flow completely.

A Numerical Investigation of the Detachment of the Trailing Particle From a Chain Sedimenting in Newtonian and Viscoelastic Fluids

Particles sedimenting in a viscoelastic fluid form chains. It has been observed in experiments that, sometimes, the last particle in the chain gets detached. In this paper, we investigate this phenomenon. It is known that a long chain falls faster than a single particle in fluids. This long body effect tends to detach the last particle from the chain. The wake effect and the normal stress effect are the mechanisms that work against the long body effect. The last particle is not detached if the inertial wake effects are strong enough to cause substantial drag reduction on it. The detachment is also restricted by the elastic normal stress of the viscoelastic fluid. [S0098-2202(00)01003-8]

Finite-Amplitude Elastic Instability of Plane-Poiseuille Flow of Viscoelastic Fluids

The purely elastic stability and bifurcation of the one-dimensional plane Poiseuille flow is determined for a large class of Oldroyd fluids with added viscosity, which typically represent polymer solutions composed of a Newtonian solvent and a polymeric solute. The problem is reduced to a nonlinear dynamical system using the Galerkin projection method. It is shown that elastic normal stress effects can be solely responsible for the destabilization of the base (Poiseuille) flow. It is found that the stability and bifurcation picture is dramatically influenced by the solvent-to-solute viscosity ratio, ε. As the flow deviates from the Newtonian limit and ε decreases below a critical value, the base flow loses its stability. Two static bifurcations emerge at two critical Weissenberg numbers, forming a closed diagram that widens as the level of elasticity increases. [S0021-8936(00)00703-0]

Effect of wall slip on the stability of viscoelastic plane shear flow

We show that for shear flow of an upper convected Maxwell fluid with small but nonzero slip velocity, an increasing dependence of the slip velocity on the elastic normal stress in the flow direction leads to short wavelength flow instability at sufficiently high Weissenberg number (≳10). Pressure-dependent slip can also lead to instability, but only at unrealistically large Weissenberg number. If the slip velocity depends only on shear stress, then the flow is always stable. These analytical results are valid in a specific asymptotic limit, but are independent of the specific form of the model for slip. Numerical results for specific, phenomenological slip models and the Phan-Thien–Tanner bulk constitutive model show that the results are robust in the presence of nonlinear viscoelasticity. The scaling of the critical shear stress for instability with modulus and molecular weight and of the distortion period with polymer relaxation time are qualitatively consistent with experimental observations of the sharkskin instability in linear polyethylenes. The results may also have some relationship to the recent experimental observation of short wavelength instability in plane Couette flow of an entangled solution with wall slip.

Direct simulation of the motion of solid particles in Couette and Poiseuille flows of viscoelastic fluids

This paper reports the results of direct numerical simulation of the motion of a two-dimensional circular cylinder in Couette flow and in Poiseuille flow of an Oldroyd-B fluid. Both neutrally buoyant and non-neutrally buoyant cylinders are considered. The cylinder's motion and the mechanisms which cause the cylinders to migrate are studied. The stable equilibrium position of neutrally buoyant particles varies with inertia, elasticity, shear thinning and the blockage ratio of the channel in both shear flows. Shear thinning promotes the migration of the cylinder to the wall while inertia causes the cylinder to migrate away from the wall. The cylinder moves closer to the wall in a narrower channel. In a Poiseuille flow, the effect of elastic normal stresses is manifested by an attraction toward the nearby wall if the blockage is strong. If the blockage is weak, the normal stresses act through the curvature of the inflow velocity profile and generate a lateral force that points to the centreline. In both cases, the migration of particles is controlled by elastic normal stresses which in the limit of slow flow in two dimensions are compressive and proportional to the square of the shear rate on the body. A slightly buoyant cylinder in Couette flow migrates to an equilibrium position nearer the centreline of the channel in a viscoelastic fluid than in a Newtonian fluid. On the other hand, the same slightly buoyant cylinder in Poiseuille flow moves to a stable position farther away from the centreline of the channel in a viscoelastic fluid than in a Newtonian fluid. Marked effects of shear thinning are documented and discussed.

The effects of viscoelasticity on the transient motion of a sphere in a shear-thinning fluid

We present the first quantitative experimental measurements of the transient motion of a sphere as it accelerates from rest along the centerline of a tube containing a highly elastic, shear-thinning, aqueous polyacrylamide solution. For all shear-rate-dependent Deborah numbers (1.6⩽De(γ̇)⩽4.2) and sphere-to-tube ratios (0.089⩽a/R⩽0.387) investigated, transient oscillations in the velocity of the sphere are observed, often causing the sphere to “rebound,” or reverse directions during the first oscillation. These measurements are in qualitative agreement with the analysis of King and Waters (1972) who presented an analytic solution for the transient motion of a sphere through an unbounded domain of fluid described by the linear Jeffreys model. We also show a similar response in one-dimensional creep experiments which are described quantitatively by a multimode formulation of the upper-convected Maxwell model including a solvent retardation term. Such experiments isolate the shear kinematics from the combined shear and extensional flow around a sphere and indicate that because of the slow quadratic growth of the elastic normal stresses in the fluid at short times and small fluid strains, the initial transient motion of the sphere is governed primarily by a balance of linear viscoelastic stresses and the inertia of the sphere.

Rheological and geometric scaling of purely elastic flow instabilities

We present a new dimensionless criterion that can be used to characterize and unify the critical conditions required for onset of purely elastic instabilities in a wide range of different flow geometries. This scaling incorporates both the presence of non-zero elastic normal stresses in the fluid plus the magnitude of the streamline curvature in the flow, and it can be thought of as the viscoelastic complement of the Görtler number. We present detailed experimental and theoretical evidence that justifies and generalizes the form of the dimensionless criterion. We show how this criterion naturally arises from the linearized stability equations governing the viscoelastic flow and apply it to analytical and experimental results in a number of standard benchmark problems. In geometrically simple flows (e.g. torsional flows such as those in a circular Couette cell or a cone-and-plate rheometer) a characteristic radius of curvature of the streamlines may be readily identified and an analytical solution for the undisturbed base flow can be found. However, in the more complex flows characteristic of those found in commercial polymer processing operations, the base flow must typically be determined numerically and the streamline curvature varies in a complex manner throughout the flow. In the former case, we show how our scaling reduces to well-established results in the literature and for the latter case we present a particularly simple approach for understanding and quantifying the sensitivity of the critical conditions for onset of elastic instability to dimensionless geometric design parameters such as the aspect ratio of the test cell. The generality of the scaling is confirmed by applying it to new experimental measurements in a lid-driven cavity and numerical linear stability calculations for flow past a cylinder in a channel. We also show how the scaling may be generalized to incorporate, at least qualitatively, the variation in the critical conditions with other rheological parameters such as changes in the solvent viscosity, shear-thinning in the viscometric functions, a spectrum of relaxation times and a non-zero second normal stress coefficient. In a number of cases, these modifications and the predicted scaling of the critical onset conditions for purely elastic instabilities in other complex geometries, such as planar contractions or eccentric rotating cylinders, remain to be confirmed by future experiments or calculations.

Extrudate Swelling: Physics, Models, and Computations

Viscous, viscoelastic, or elastic normal stresses are superimposed to pressure within flowing fluids. These stresses act normal to the boundaries of the flow that may deform depending on their modulus or viscosity. At absolutely rigid boundaries of infinite modulus of elasticity any boundary deformation and therefore any fluid expansion or swelling is surpressed (eg, flow in rigid pipes, annuli, channels). Elastic boundaries (eg, flow in veins and arteries, flow by membranes, around inflating/deflating balloons) deform under the action of normal stresses, allowing expansion or swelling of fluid. The same mechanism prevails in lubrication, where pressure and superimposed normal viscoelastic stresses keep surfaces in relative motion apart, with simultaneous increase in load capacity. Viscous boundaries (eg, liquid jet in air or in immiscible liquid, slow extrusion of viscoelastic liquids from dies, expanding/collapsing air-bubbles or liquid-droplets) are displaced by flowing adjacent immiscible fluids, allowing swelling or imposing contraction depending on relative rheological characteristics. Thus, the kind of swelling examined here is independent of density, ie, incompressible, and is due to the action of normal stresses against the boundary that is imposed either by adjacent deformable obstacles or else by surface tension. The resulting swelling is dynamic (ie, it initiates, changes and ceases with the flow) and can be made permanent by solidification, crystallization or glassification. The most profound form of incompressible swelling is the extrude swelling that controls the ultimate shape of extruded parts. Incompressible swelling is enhanced by the ability of macromolecules to deform and recover (eg, viscoelastic) and by the design of flow conduits to impose sharp transitions of deformation modes (eg, singular exit flows). The same swelling is reduced by the ability of molecules (or fibers in fiber-suspensions) to align with the flow streamines, as well as any tendency of solid-like structure formulation (eg, viscoplastic).

Effect of cross slip on crystallographic cracking in anisotropic single crystals

The effect of cross slip on cracking along coplanar slip bands in anisotropic single crystals is examined. By considering the activation of slip due to the stress field at a crack tip, it is demonstrated that: 1. (1) cross slip would not be effective in relaxing elastic normal stresses ahead of a crack propagating along coplanar slip bands, and 2. (2) the unrelaxed elastic normal stresses can be plastically relaxed by out-of-plane noncoplanar slip only. This results in a build-up of large normal stresses ahead of the coplanar crack. The magnitude of the maximum normal stress depends on the size of the noncoplanar secondary slip plastic zone and elastic anisotropy. Based on the presence of relatively high normal stresses on both the coplanar and the cross-slip planes, a rationale for explaining the occurrence of simultaneous cracking on two {111} cross-slip planes and the formation of fracture surface ridges in Mar-M200 single crystal is proposed.

Turbulence suppression in solutions and pure liquids

AbstractExperiments dealing with the onset of turbulent flow in very small capillary tubes are described. The liquids are water, n‐heptane, toluene and four aqueous solutions containing small quantities of a polymeric additive.The transition from laminar to turbulent flow is delayed in a manner dependent on tube diameter. There is an essential similarity between the behavior of all the liquids. Possible reasons for the observed behavior are discussed and it is suggested that the explanation lies in the presence of varying degrees of elasticity in the liquids used.Using normal stress data from a jet thrust apparatus, the transition values of various dimensionless groups are evaluated.The group which gives the most consistent behavior is the inertio‐elastic number equivalent to a Reynolds number in which viscous shear stress is replaced by elastic normal stress.

Flow birefringence and polydispersity

AbstractFlow birefringence data for dilute solutions of various linear and branched polymers of different degrees of polydispersity are discussed. The polymers considered are polystyrene, polypropylene, and polyethylene. By making use of a well‐known stress‐optical relation, reduced elastic normal stresses, which can also be interpreted as coil expansions, are plotted against reduced shear stresses. As elastic normal stresses are rather sensitive to the shape of the molecular weight distribution at high molecular weight, a characteristic, p, related to the skewness of the distribution, is obtained. Experimental values of p are compared with theoretical ones based on special types of distribution functions. Attention is paid to the method of extrapolating experimental results to zero shear stress and concentration.

A new interpretation of isotropic pressure and normal stress in the cone-and-plate, with application to the Oldroyd model

An analysis of the steady cone-and-plate shear of a visco-elastic non-Newtonian liquid is presented to show how the isotropic pressure p is related to the elastic normal stresses and inertia of the sample. Critical to the development is a boundary condition of mass conservation, but no rheological model is used and no undetermined constants are necessary. A consequence is the possibility of finding τ ψψ alone, in addition to the function χ  τ φφ + τ ψψ − 2τ rr always obtained in cone-and plate work. The ratio W  − τ ψψ/χ is evaluated from normal stress data on eight polymer solutions over a shear-rate of 7–1000 sec −1, and found to be approximately 1 6 for highly structure liquids. The parameter W is used to normalize an Oldroyd model, which then has quantitative success in predicting relations between steady-shear and oscillatory-shear stress parameters. This suggest that p  p m may not be the most useful normalizing condition.