Exponential Shear Research Articles

Group theoretical analysis and similarity solutions for stress boundary layers in viscoelastic flows

Lie group theory is used to obtain point symmetries of the boundary layer equations derived in the literature for the high Weissenberg number flow of upper convected Maxwell (UCM) and Phan-Tien-Tanner (PTT) type of viscoelastic fluids. The equations are reduced to ordinary differential equation systems with the use of scaling and spiral transformation groups. Similarity solutions are obtained and discussed for different cases such as flow around corners, flow over moving and stretching walls, and exponential shear flows.

Comparison of Various Shear Deformation Theories for Bending, Buckling, and Vibration of Rectangular Symmetric Cross-ply Plate with Simply Supported Edges

In the present study, unified shear deformation theory which was proposed by Soldatos, K.P. and Timarci, T. (1993). A Unified Formulation of Laminated Composite, Shear Deformable Five-degrees-of-freedom Cylindrical Shells on the Basis of a Unified Shear Deformable Shell Theory, Compos. Struct., 25(1–4): 165–171 is used to analyze simply supported symmetric cross-ply rectangular plates for deflections, stresses, natural frequencies, and buckling loads. This theory enables the selection of different in-plane displacement components to represent shear deformation. Exponential shear deformation theory, which is proposed by Karama et al., 40(6) (2003). Mechanical Behavior of Laminated Composite Beam by New Multi-layered Laminated Composite Structures Model with Transverse Shear Stress Continuity, Int J Solids Struct., 40: 1525–1546, is used for the first time to analyze the problem considered. Results which are found by exponential theory are compared with those obtained by using the parabolic shear deformation theory of Reddy, J.N., 51(4) (1984). A Simple Higher-order Theory for Laminated Composite Plates, J Appl Mech., 51: 745–752, the trigonometric shear deformation theory of Touratier, M., 29(8) (1991). An Efficient Standard Plate Theory, Int. Jnl. Engng. Sci., 29(8): 901–916, the hyperbolic shear deformation theory of Soldatos, K.P., 94(3–4) (1992). A Transverse Shear Deformation Theory for Homogeneous Monoclinic Plates, Acta Mech., 94: 1995–220 and with the available three-dimensional elasticity solutions. The study shows that while the transverse displacement and the stresses are best predicted by the exponential shear deformation theory, the parabolic shear deformation and the hyperbolic shear deformation theories yield more accurate predictions for the natural frequencies and the buckling loads.

The response of 2D foams to continuous applied shear in a Couette rheometer

The continuum model that has reproduced the experimental observation of exponential shear localisation for straight-edge boundary conditions is adapted to the case of circular geometry. Essentially the same effect is found. However, the scenario of possible velocity profiles is much richer. Our calculations elucidate many recent experiments qualitatively and suggest further extensions of them. Various limits are analysed. In particular, the localisation length vanishes as the inner-boundary velocity tends to zero.

Analysis of Shear Effects on a Partially Cavitating Wing by Using a Lifting-Line Theory for an Exponential Shear Flow

Analysis of Shear Effects on a Partially Cavitating Wing by Using a Lifting-Line Theory for an Exponential Shear Flow

Exponential shear flow of branched polyethylenes in rotational parallel-plate geometry

Exponential shear flow, as a strong flow with the potential to generate a high degree of molecular stretching, has attracted considerable interest in recent years. So far, exponential shear flow has been realized by either sliding-plate or cone-and-plate (CP) geometry. Both geometries guarantee homogeneous shear flow. Here, we present experimental data on exponential shear flow of several long-chain branched polyethylene melts with different degrees of strain hardening obtained by using parallel-plate (PP) geometry in a rotational rheometer. This type of geometry, which is standard in linear-viscoelastic characterization of polymer materials, produces inhomogeneous shear flow. A comparison of exponential shear flow data obtained by PP and CP geometry is made. Additionally, the experimental data are compared to predictions of the rubber-like liquid (RLL) and the molecular stress function (MSF) theories. For this purpose, the relaxation spectra of the polymer melts considered were obtained by standard linear-viscoelastic characterization. In addition, two irrotational parameters and one rotational parameter are required by the MSF theory. While the irrotational parameters were obtained from fitting to elongational viscosity data, the value of the rotational parameter was used as given in the literature. It can be concluded that viable experimental data in exponential shear flow can be obtained by PP geometry. For finite linear-viscoelasticity (RLL theory), predictions of reduced shear stress for CP and PP geometry coincide, but nonlinear material behavior (as modeled by the MSF theory) leads to small differences between both geometries. Furthermore, it is shown that the MSF predictions are in excellent agreement with the experimental data in exponential shear flow and that this type of flow leads to much less chain stretching than elongational flow.

Internal friction spectroscopy in Li 2O–2SiO 2 partially crystallised glasses

Oscillatory torsion deformation experiments were performed on partially crystallised Li 2O–2SiO 2 glasses in the temperature range 350–480 °C and with frequencies between 20 and 0.002 Hz. The experiments were carried out in a torsion deformation apparatus exerting a small strain on cylindrical samples. Data obtained at varying temperatures and frequency were reduced to master plots using a normalised frequency. The frequency shift factor has been taken as a function of temperature in an Arrhenian form, yielding an activation energy of a background Q −1 close to the activation energy of oxygen defect diffusion (=120 kJ/mol). The master curves of real and imaginary components of shear modulus and internal friction indicate a stretched exponential shear stress relaxation with a an exponent of ≈0.45, characteristic of a broadened relaxation spectrum. The dynamic viscosity was estimated at temperatures of 470 and 480 °C. The extrapolation of dynamic viscosity to zero frequency allowed estimation of the relaxed shear viscosity. The presence of crystals increases the relaxed shear viscosity by ≈ 0.2 log( Pa s) /10 vol.% of crystallinity. Dependence of the relative shear viscosity of partially crystallised lithium disilicate melts on crystal content is discussed.

Characterisation of an Industrial Polymer Melt Through Either Uniaxial Extension or Exponential Shear Data: An Application of the Pom-Pom Model

Abstract We present new non-linear data in extension and two different shear histories. These data are used to compare the effectiveness of using exponential shear data and uniaxial extension data to characterise the non-linear response of an industrial LDPE melt with the pom-pom molecular model. We conclude that extension and exponential shear both allow good predictions to be made in simple shear. However, the characterisation spectrum obtained from exponential shear data fails to predict the correct degree of strain hardening at low extension rates. From this study we are able to suggest circumstances under which exponential shear provides a useful characterisation of branched polymer melts.

An experimental and simulation study of dilute polymer solutions in exponential shear flow: Comparison to uniaxial and planar extensional flows

The rheological properties of dilute polymer solutions in exponential shear, uniaxial, and planar extensional flows are compared using Brownian dynamics simulations of both freely draining, flexible bead-rod and bead-spring chains. We introduce a novel stress function for exponential shear which uses the extinction angle χ to take into account the orientation of the chain as it aligns in the flow. Comparing this new stress function during startup and relaxation in exponential shear with τ11−τ22 in planar extensional flow and τ11−12(τ22+τ33) in uniaxial extensional flow, we find that for both models there is a quantitative agreement among the three different flows over a large range of Wi, strain, and chain length. Furthermore, the distributions of maximum extension show a microstructural equivalence between ensembles of chains in all three flows up to strains of 3 or 4 at all values of Wi simulated. Finally, we show three comparisons between experiment and simulation of the various flows: (a) simulations ...

Using the pom-pom equations to analyze polymer melts in exponential shear

Exponential shear flows of polymer melts are investigated. These flows are of interest because they share some properties of both planar extension and simple, constant-rate, shear. In particular, embedded points separate exponentially in time, like extensional flows, yet the flow direction and velocity gradient are perpendicular, as for all shear deformations. A comparison between the predictions of the “pom-pom” molecular model of McLeish and Larson [J. Rheol. 42, 81–110 (1998)] in exponential shear and experimental data is presented. The solutions in exponential shear are used to contrast this flow with simple shear and extensional flows and the physical processes driving the solutions are analyzed in each of the three geometries. Accurate quantitative predictions are made using the multimode approach, and the possibility of using exponential shear to obtain the multimode nonlinear spectrum of a melt is explored. It is concluded that although exponential shear is less effective than extensional flows at exposing the influence of the nonlinear parameters of a melt, there are still situations where it is a useful flow for obtaining these parameters.

309 無限指数せん断流中における翼の揚力線理論(流体工学II)

309 無限指数せん断流中における翼の揚力線理論(流体工学II)

A Lifting-Line Theory of a Wing in Exponential Shear Flow and Application to Supercavitating Hydrofoil.

A lifting line theory of a wing in exponential shear flow is presented. The governing equation due to Chen et al. is reduced to two ordinary differential equations by separation of variables, and a general solution is obtained by linearly combining the fundamental solutions. Induced attack angle is derived from the general solution and the condition of far downstream, and relation of lift force to effective attack angle is described by the Taylor series expansion in the vicinity of geometrical attack angle. From the induced attack angle and the lift force, a lifting-line equation is derived, and solved by Gaussian elimination method. By applying the theory to supercavitating hydrofiol, the effects of shear parameter, cavitation number and aspect ratio are clarified through some numerical calculations.

Exponential shear flow of linear, entangled polymeric liquids

A previously proposed reptation model is used to interpret exponential shear flow data taken on an entangled polystyrene solution. Both shear and normal stress measurements are made during exponential shear using mechanical means. The model is capable of explaining all trends seen in the data, and suggests a novel analysis of the data. This analysis demonstrates that exponential shearing flow is no more capable of stretching polymer chains than is inception of steady shear at comparable instantaneous shear rates. In fact, all exponential shear flow stresses measured are bounded quantitatively by stress measurements taken during inception of steady shear. Information taken from the model about chain stretching suggests that normal stress measurements are strong indications of stretching, whereas shear stress measurements are indicative of both chain stretching and segment orientation.

Dynamic slip and nonlinear viscoelasticity

AbstractSeveral significant problems arise when film is fabricated on a large scale. One of these is the appearance of irregularities on the extrudate surfaces when the polymer melt is extruded at high rates. These irregularities vary in intensity and form and are generally known as sharkskin melt fracture. This phenomenon, which occurs when the wall shear stress exceeds a critical value, is a limiting factor for production rates in many industrial extrusion operations such as film blowing of polyethylene. We used a sliding plate rheometer incorporating a shear stress transducer to study slip in both steady and unsteady flows. By combining a dynamic slip model with a nonlinear viscoelastic constitutive model, we determined the slip model parameters for LLDPE film resin with and without a fluoropolymer sharkskin suppressant. The models give good prediction of our slip data in steady shear but show insufficient gap dependence in exponential shear. Our own film blowing studies demonstrated the efficiency of the sharkskin suppressant; it has more than doubled the throughput in our laboratory setup. The fluoropolymer additive was found to profoundly affect both the steady and dynamic slip parameters. Hence, the sharkskin suppressant alters how the LLDPE remembers its past slipping motions.

Exponential shear flow of branched polymer melts

The behavior of a low-density polyethylene melt in exponential shear strain histories is examined and compared to its behavior in constant rate planar elongation. A new set of shear stress and first normal stress difference data in exponential shear are presented and used in several different material functions that have been previously proposed. Viscosities composed of principal stress differences for the two flows showed no correspondence suggesting that, contrary to previous assertions, exponential shear and constant rate planar elongation flows are fundamentally different. It is further suggested that the presence of vorticity makes exponential shear a weak, rather than strong, flow.

Brownian dynamics simulations of the stress and molecular configuration of polymers in exponential and linearly-ramped shear flow

The rheological and optical properties of dilute polymer solutions during the startup and subsequent relaxation in exponential shear flow are studied using Brownian dynamics simulations of freely draining, flexible bead-rod chains. We find that exponential shear flow is capable of effecting large molecular deformation for finite shear rates and strains, and furthermore, the polymer undergoes an initial stress and index of refraction relaxation that is much faster than a single exponential decay upon cessation of flow. When we compare the first normal stress difference ( τ 11– τ 22) to the corresponding components of the index of refraction tensor, n 11– n 22, we find that there is hysteresis when plotted over the course of a full startup and relaxation experiment. This is very similar to the hysteresis originally discovered by Doyle et al. [J. Non-Newtonian Fluid Mech. 76 (1998) 79–110] for uniaxial extensional flow. Moreover, the shear components of the stress and index of refraction tensors ( τ 12 and n 12) also exhibit hysteretic behavior. However, whereas the hysteresis loop for the 11–22 component traverses up the left branch during startup and relaxes on the right branch, that for the 12 component is in the completely opposite direction, i.e. the right branch corresponds to startup and the left branch relaxation. The presence of stress-index of refraction hysteresis clearly has important implications for the stress-optic rule. We calculate the stress-optic coefficient for the startup process, and find that the normal and shear components give the same value, but that it is strongly dependent on Wi. This result is consistent with that found by Doyle et al. during the startup of steady shear flow [Doyle and Shaqfeh, Dynamic simulation of freely-draining, flexible bead-rod chains: startup of extensional and shear flow, J. Non-Newtonian Fluid Mech. 76 (1998) 43–78]. We also simulated shear flows with a linearly ramped shear rate to compare to our results for exponential shear. We find that for comparable final shear rate and time, both exponential and linear ramping shear produce very similar hysteresis effects in both the shear and normal components suggesting that there may be many unsteady shear flow capable of creating large polymer stretch and stress-birefringence hysteresis. Finally, we compare our bead-rod chain simulations to the FENE dumbbell model, and we find that while the latter can qualitatively capture the normal stress hysteresis, it predicts negligible shear stress hysteresis at equivalent shear rates and shear strains.

On the exact solution of the acoustic wave equation in a parabolic velocity profile in a hard-walled duct

The exact solution of the acoustic wave equation in a parabolic shear flow profile is obtained; the only exact solution given in the literature [Goldstein and Rice (1963); Jones (1977), (1978); Scott (1979); Koutsoyannis (1980)] is for a linear velocity profile, and the solution for the exponential shear flow is given elsewhere [Campos and Serrão (1998)]. The wave equation has three singularities, like the Gaussian hypergeometric equation, but it is of an extended type, since the singularity at infinity is irregular (all three singularities of the Gaussian hypergeometric equation are regular). The other two singularities of the present equation are regular, and one lies at the axis of the duct, and the other at the dicritical layers, where the Doppler-shifted frequency vanishes. The critical layers do not exist (they lie outside the duct) for longitudinal wave vector antiparallel to the mean flow (case I), i.e., propagation upstream in a local frame of reference. In the opposite case of longitudinal wave vector parallel to the mean flow (case II) there are three subcases, depending on whether the Doppler-shifted frequency on the axis of the duct is: (case II A) positive, i.e., critical layers are at imaginary ‘‘distance;’’ (case II B) zero, i.e., the critical layer lies on the axis of the duct: (case II C) negative, i.e., two critical layers exist in the duct. In all cases (I, II A, II B, II C) it is possible to obtain the exact acoustic field over the whole flow region by expanding around the singularities and matching solutions. Since the acoustic wave equation in a shear flow does not lead to a Sturm–Liouville problem, the eigenfunctions need not be orthogonal or complete. There is a single set of natural frequencies and normal modes in cases I, II A, and II B, but not in case II C; in the latter case, where two critical layers lie in the flow, they may separate three sets of eigenvalues/eigenfunctions, in the regions between by the critical layers and the walls.

A kinetic network model for nonlinear flow behavior of molten plastics in both shear and extension

The revised Berkeley kinetic network model is shown to accurately describe experimental data in many types of nonlinear deformation, including large amplitude oscillatory shear (LAOS), exponential shear, planar extension, step biaxial extension, uniaxial extension, and step strain, for the IUPAC X low density polyethylene (LDPE). It is shown that while LAOS is helpful in determining the approximate model parameters, a LAOS test is not sufficient to determine their precise values. Exponential shear, however, is more sensitive to the parameter values, and LAOS and exponential shear together are sufficient for precise parameter determination. Both of these experiments can be performed easily on a sliding plate rheometer incorporating a shear stress transducer.

Can nonlinear deformation amplify subtle differences in linear viscoelasticity?

Large amplitude oscillatory shear (LAOS) and exponential shear can be used to study the nonlinear viscoelastic properties of molten polymers. Using a kinetic network theory, the sensitivity of the nonlinear viscoelasticity of a polymer melt to small changes in the linear spectrum was calculated and compared with the sensitivity of the linear response. In LAOS, the strain amplitude and phase angle of the first harmonic were no more sensitive to changes in the relaxation time than the linear counterparts. The sensitivity of the amplitudes and phase angles of all the harmonics to changes in the linear spectrum decreases with strain amplitude. In exponential shear, the sensitivity of the maximum of the exponential shear viscosity to small changes in the relaxation time was found to decrease with increasing strain rate. However, increasing strain amplitude (in LAOS) and strain rate (in exponential shear) were both found to amplify subtle differences in the kinetic parameters of the model. Thus large shearing deformations do not significantly amplify subtle changes in the relaxation spectrum, but they do amplify subtle differences in the kinetic rate constants.

Matching inviscid/boundary-layer flowfields

New boundary-layer equations are developed and the solutions match all of the boundary-layer properties, except for the normal velocity, exactly with the corresponding inviscid properties. The numerical procedure solves tridiagonal matrices at each marching station. As part of the solution an inviscid transpiration velocity at the surface is calculated from the boundary-layer solution. This transpiration velocity could be used as a boundary condition to calculate a new inviscid solution. If the inviscid/boundary-layer solutions are interacted, then the normal velocity from the boundary-layer solution will match the inviscid values exactly. Solutions are calculated for shear flows over a flat plate. Results from the present method compared well with Navier-Stokes solutions for incompressible constant shear flow and sinusoidal shear flow. Iterations of the inviscid/boundary-layer solutions were not necessary. Compressible flow of an exponential shear at Mach 3 was calculated for cold and hot walls. It was found that inviscid/boundary-layer interactions were significant for hot walls but negligible for cold walls. The present method should yield accurate boundary-layer solutions at Reynolds numbers lower than the traditional boundary-layer equations.

Nonlinear viscoelasticity of concentrated polystyrene solutions: Sliding plate rheometer studies

Because of the degree to which the linearity and polydispersity of polystyrene samples can be controlled in laboratory polymerizations, the rheological properties of melts and solutions of this polymer have been extensively studied. However, in previous work the maximum strain or strain rate was limited to small values due to edge effects in the rotational rheometers used. We used a sliding plate rheometer equipped with a shear stress transducer and a birefringence apparatus to measure simultaneously the shear stress and the third normal stress difference, N3 (≡σ11−σ33), during step strain, start‐up of steady shear, and exponential shear. Depending on the strain history, the maximum strain achieved was between 20 and 80. The steady‐state shear stress in steady shear flow was found to be nearly independent of shear rate over a range of shear rates with a strong suggestion of a maximum, a phenomenon predicted by the Doi–Edwards theory. The relaxation modulus for the shear stress was found to be superposable...