Steady and transient shear flows of polystyrene solutions I: Concentration and molecular weight dependence of non-dimensional viscometric functions

Start up from rest and relaxation from steady shear flow experiments have been performed on monodisperse polystyrene solutions with molecular weight ranging from 1.3 × 10⁵ to 1.6 × 10⁶ and concentration c ranging from 5% to 40%. A method of reduced variables based on the use of a characteristic time τ_w is proposed. τ_w is defined as the product of zero shear viscosity with the steady state elastic compliance.Reduced steady and transient viscometric functions so obtained depend on the ratio M/M_e (where M_e is the entanglement molecular weight). Limiting forms are obtained when M/M_e ? 18. In steady flow, a simple correlation is found between shear and normal stresses.In stress relaxation experiments, independent of shear rate, the long-time behaviour can be characterised by a single relaxation time τ₁, which is identical for shear and normal stresses. τ₁ can be simply related to the zero shear rate viscosity and the limiting elastic compliance.     

On the similarity solutions of wave propagation for a general class of non-linear dissipative materials

Deductive similarity analysis is employed to study one-dimensional wave propagation in rate dependent materials whose constitutive laws are special cases of Maxwellian materials (σ_t = φ(ε, σ)ε_t + ψ(ε, σ), ε = strain, σ = stress). The general problem is shown not to have a similar solution although many special cases have the independent similar variable (x ? c)/(t ? d)^e. These cases are studied and tabulated. Analytic similar solutions are presented for several cases and a discussion of permissable boundary conditions is given.     

Mechanical Properties of Hard W-C Coating on Steel Substrate Deduced from Nanoindentation and Finite Element Modeling

Mechanical properties of a hard and stiff W-C coating on steel substrate have been investigated using nanoindentation combined with finite element modeling (FEM) and extended FEM (XFEM). The significant pile-up observed around the indents in steel substrate caused an overestimation of hardness and indentation modulus. A simple geometrical model, considering the additional contact surfaces due to pile-up, has been proposed to reduce this overestimation. The presence of W-C coating suppressed the pile-up in the steel substrate and a transition to sink-in behavior occurred. The FEM simulations adequately reproduced the surface topography of the indents in the substrate and coating/substrate systems as well. The maximum principal stresses of the indented W-C/steel coated system were tensile; they were always located in the coating and evolved in 3 stages. Cohesive cracking occurred during loading in the sink-in zone (stage III) when the ultimate tensile strength (σ _max ) of the coating was reached. The obtained hardness (H _c ), indentation modulus (E _c ), yield stress (Y) and strength (σ _max ) of the W-C coating were H _c ? =?20 GPa, E _c ? =?250 GPa, Y?=?9.0 GPa and σ _max ? =?9.35 GPa, respectively. XFEM resulted in fracture energy of the W-C coating of G?=?38.1 J?·?m^-2 and fracture toughness of K _IC ? =?3.5 MPa?·?m^0.5.     

Power-law creep and residual stresses in a carbopol gel

We report on the interplay between creep and residual stresses in a carbopol microgel. When a constant shear stress σ is applied below the yield stress σ _y, the strain is shown to increase as a power law of time, γ(t) = γ ₀ + (t/τ) ^α , with an exponent α = 0.39 ± 0.04 that is strongly reminiscent of Andrade creep in hard solids. For applied shear stresses lower than some typical value σ _c ? 0.2σ _y, the microgel experiences a more complex, anomalous creep behavior, characterized by an initial decrease of the strain, that we attribute to the existence of residual stresses of the order of σ _c that persist after a rest time under a zero shear rate following preshear. The influence of gel concentration on creep and residual stresses are investigated as well as possible aging effects. We discuss our results in light of previous works on colloidal glasses and other soft glassy systems.     

Linear viscoelasticity of ZrO 2 nanoparticle dispersions with associative polymers

ZrO₂ nanoparticle dispersions containing associative polymers exhibit two relaxation modes: Maxwellian behavior at high frequency imparted by the associating polymers and a power law spectrum at low frequency generated by the particle dynamics. The timescales and volume fraction dependence of the dispersions reflect weak attractions between particles with adsorbed polymer layers dispersed in a percolated network of associative polymer. The Baxter stickiness parameter extracted from the high frequency viscosity data indicates strong attractions, whereas the high frequency modulus reveals three sources of elasticity: micelle–micelle associations in the solution, rigidity of the particles and adsorbed layer, and adsorbed layer–adsorbed layer interactions. The sol-gel transition of the dispersions occurs around 12–14% particle loading. Comparison with latex dispersions suggests a slower relaxation mode and greater rigidity with the ZrO₂ particles.     

Large strain deformation of polycrystalline metals at low homologous temperatures

Large strain compression data (true strains to about ?3.0) are presented for polycrystalline α U and α Fe at room temperature. The results, together with other published data at low homologous temperatures (≈0.2 T_m), where T_m is the absolute melting temperature, suggest that a steady-state flow stress σ_s is approached after extensive strain-hardening, α U exhibits a very high strain-hardening rate, with σ_s ≈ 2900 MPa (420 ksi) indicating that cold-working is a very potent method of strengthening this metal. All the data evaluated can be fit by the stress-strain relation σ = σ_s? exp (?(Nε)^p)(σ_s? σ_y), where σ_y is the yield stess, p is a constant equal to a for the metals analyzed, N is a constant associated with the strain-hardening characteristics of a material, σ is true stress, and ε is true strain.     

Experimental investigation of the efficiency of gas screens at a burning-out surface with the blowing of a foreign gas

The article gives the results of a systematic experimental investigation of the efficiency of a gas screen at a burning-out graphite surface organized in different manners and with the blowing of a foreign gas into the boundary layer. In the experiments, the Reynolds number was varied within the limits R_Δ ≈ 1.2·10⁵ = 1.6·10⁶, R* ≈ 100–2000, R_Z = 2.3·10²–1.5·10⁴; the enthalpy factor of the nonisothermicity in the initial cross section varied from i_s/i₀ = 1 with the tangential blowing of nitrogen to i_s/i₀ ≈ 34 with the blowing of helium through the heated porous section.     

Drag of a strongly heated sphere at small Reynolds numbers

Using an asymptotic small-perturbation method, the flow around a strongly heated sphere at small Reynolds numbers Re ≪ 1 is considered with account for thermal stresses in the gas in the higher-order approximations, beyond the Stokes one. It is assumed that the value of the Prandtl number Pr is arbitrary and the temperature dependence of the viscosity is described by a power law with an arbitrary exponent. In the O(Re²) and O(Re³ ln(Re)) approximations, the drag force of a heated sphere is found over a wide range of the ratios of sphere’s temperature to the gas free-stream temperature T _W /T _∞. The limits of applicability of the first (in Re) approximation are investigated, including the negative-drag effect, attributable to the action of the thermal stresses. The results are compared with numerical calculations of the flow around a hot sphere. The limits of applicability of the approximations found are examined. Similar results are obtained for the standard Navier-Stokes equations in which the thermal stresses are neglected.     

Shock structure in a viscoelastic medium with a nonlinear dependence of the Maxwellian viscosity on the parameters of the material

The shock-profile structure in a viscoelastic medium with a nonlinear dependence of the Maxwellian viscosity (a quantity inverse to the tangential stress-relaxation time ) on the substance parameters is investigated in this paper on the basis of a model proposed in [1]. The presence of such a dependence of the relaxation time permitted the extraction of sections with an abrupt change in the quantities on the profile, called plastic waves, by using additional relationships. The model of the isotropic medium used is characterized by an equation of state in the form of a dependence of the density of the internal energy E per unit mass on the strain-tensor invariants and the entropy S.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 101–108, September–October, 1974.     

Viscosity and dynamics of nanorod (carbon nanotubes, cellulose whiskers, stiff polymers and polymer fibers) suspensions

The zero shear viscosity and the dynamic behaviors of different nanorod dispersions (carbon nanotubes (CNTs), cellulose whiskers, polymer nanofibers, crosslinked polymer nanofibers, and stiff polymers such as poly(γ-benzyl-α-l-glutamate) (PBLG)) were compared and discussed from literature data. Their Brownian dynamic behaviors have always been discussed in the frame of the Doi–Edwards theory. In agreement with this theory, the straight rigid rods (CNTs, cellulose whisker, polymer nanofibers) obey a master curve in the reduced viscosity (or rotary diffusivity) c power laws on viscosity (η ₀ ∝ φ ³) and diffusivity (D _r ∝ ? ^?2). On the contrary, stiff polymer chains and crosslinked polymer fibers at temperature above T _g exhibit different and two distinct dynamic behaviors. Despite their deviation from the ideal rigidity, surprisingly it can be noted that stiff polymers such as PBLG have been extremely used in the literature to verify the Doi–Edwards theory. Finally, flexible crosslinked chains at T > T _g , do not obey the Doi–Edwards theory, and their dynamics are close to the physics of polymer solutions in terms of power laws.     

Classification of stress-intensity factors from isochromatic-fringe patterns

The stresses in the local neighborhood of a crack tip have been used to develop a relation between the isochromatic-fringe orderN, its position parametersr and θ and the stress field expressed in terms of stress intensities,K _I ,K _II , and a far-field stress σ _ox . This relation was programmed and a plotting routine was developed to map isochromatic (σ₁ ? σ₂) fields in the neighborhood of the crack tip. The stress intensitiesK _I andK _II and the far-field stress σ _ox were varied and isochromatic fields were constructed for each combination. As bothK _II and σ _ox influence the size, shape and orientation of the isochromatics loops in a systematic manner, the pictorial representation of the isochromatic fields can be used to classify the state of stress (K _I ,K _II and σ _ox ) at the crack tip. Isochromatics which classify six different states of stress have been illustrated and methods used to determineK _I ,K _II , andσ _ox in five of the six states are given.     

Rheology of suspensions of spherical particles in a newtonian and a non-newtonian fluid

Properties of suspensions of spherical glass beads (25–38 μm dia.) in a Newtonian fluid and a non-Newtonian (NBS Fluid 40) fluid were measured at volume fractions, φ, of 0%, 10%, 20% and 30%. Measurements were made using a modified and computerized Weissenberg Rheogoniometer. Properties measured included steady shear viscosity, η($\text{γ}\text{.}$), first normal stress difference, N₁($\text{γ}\text{.}$), linear viscoelastic properties, η′(ω) and G′(ω), shear stress relaxation, σ^? ($\text{γ}\text{.}$, t), and growth, σ⁺($\text{γ}\text{.}$, t) and normal stress relaxation, N₁^?($\text{γ}\text{.}$, t).For a the Newtonian fluid, increasing φ causes both η and η′ to increase, with η′ showing a slight frequency dependence. Both N₁ and G′ are zero and stress relaxation and growth occur essentially instantaneously. For the NBS fluid, both η and η′ increse with φ at all $\text{γ}\text{.}$ and ω, respectively, the increase being greater as $\text{γ}\text{.}$ and ω approach zero. N₁ and G′ are less affected by the presence of the particles than η and η′ with the effect on G′ being more pronounced than on N₁. For fixed $\text{γ}\text{.}$, stress relaxation and growth exhibit greater non-linear effects as φ is increased. A model for predicting a priori the linear viscoelastic properties for suspensions was found to yeild reasonable estimates up to φ = 20%.     

Determination of source parameter in parabolic equations

The authors consider the problem of finding u=u(x, t) and p=p(t) which satisfy u = Lu + p(t) + F(x, t, u, _x, p(t)) in Q _T=Ω×(0, T], u(x, 0)=ø(x), x∈Ω, u(x, t)=g(x, t) on ?Ω×(0, T] and either ∫_G(t) Φ(x,t)u(x,t)dx = E(t), 0 ? t ? T or u(x₀, t)=E(t), 0≤t≤T, where Ω?R ⁿ is a bounded domain with smooth boundary ?Ω, x ₀∈Ω, L is a linear elliptic operator, G(t)?Ω, and F, ø, g, and E are known functions. For each of the two problems stated above, we demonstrate the existence, unicity and continuous dependence upon the data. Some considerations on the numerical solution for these two inverse problems are presented with examples.     

Effect of the turbulent viscosity relaxation time on the modeling of nozzle and jet flows

Certain modifications of three-equation turbulence models are proposed. They are intended for increasing the accuracy of the calculations of turbulent flows in nozzles with boundary layer separation and in supersonic jets with complicated shock wave structures. Basing on the idea of the inclusion of flow prehistory in terms of an additional relaxation equation for nonequilibrium turbulent viscosity we propose three modifications of the k-ω-µ _t model based on the k-ω model and a version of the k- ?-µ _t turbulence model. In these modifications we introduce an additional dependence of the nonequilibrium turbulent viscosity relaxation time on different physical parameters which can be important near the point of boundary layer separation from the nozzle wall, such as viscous effects and effects of large gradients of the mean velocity and the kinetic energy of turbulence (turbulent pressure). The comparison of the results of the calculations with the experimental data shows that all the proposed versions of the three-equation models make it possible to improve the accuracy of the calculations of turbulent flows in nozzles and jets.     

Ideal elastic fluids of different viscosity and elasticity levels

Four constant viscosity, highly elastic fluids of different viscosity and elasticity levels are presented. The viscosity ranges from 4 × 10^?3 to 5.0 Pa s and the Maxwell relaxation time varies from 0.09 to 4.5 s. The steady and dynamic shear properties are determined. These fluids comply with the requirements of the simple fluid theory except for theG′ andN ₁/2 data where a slight deviation is observed. The results suggest the possibility of preparing a wide range of constant viscosity elastic fluids with specific values of viscosity and relaxation time by manipulating polymer molecular parameters as well as polymer concentration, solvent viscosity and salt addition. The effects of each of these parameters on the rheological behaviour are examined.     

Multiphysics of multifunctional nanofluids based on different oxides nanoparticles and glycerol fluid

Nanofluid (NF) materials consisting of glycerol (Gly) and different inorganic nano oxides (TiO₂, ZnO, Al₂O₃, and SiO₂ for the oxides concentration of 0.01 wt% to the weight of Gly base fluid) were prepared by a two-step method through ultrasonic cavitation process. These nanofluids were investigated by employing an X-ray diffractometer (XRD), ultraviolet–visible (UV–Vis) spectrophotometer, 20 Hz to 1 MHz frequency range dielectric relaxation spectroscopy (DRS), ultrasonic interferometer, and rotational viscometer. The multiphysics of these nanofluids includes structural and optical properties, dielectric permittivity, electrical conductivity, conductivity relaxation, ultrasound velocity, adiabatic compressibility, acoustic impedance, viscosity, density, thermal conductivity, and viscoacoustic relaxation were characterized. The XRD patterns identified monodispersed and stable suspensions of these different characteristic nanoparticles in the hydrogen-bonded 3D supramolecular structure of ultra-high viscous glycerol fluid which were supported by their UV–Vis absorbance analyses. The energy band gap values of the TiO₂ and ZnO containing nanofluids were found primarily ruled by the characteristic optical properties of these oxides nanomaterials. The complex dielectric and various electrical functions studied at 25 °C revealed that the suspension of different oxide nanoparticles in the glycerol fluid increased the static permittivity whereas reduced the direct current electrical conductivity which showed strong conductivity relaxation process dependence. The rheological measurements of the formulated nanofluids were performed over a shear rate range of 0.4–40 s⁻¹ at temperatures of 25–55 °C. The linear relationship between shear rate and shear stress and also the shear rate-independent viscosity revealed the Newtonian behaviour of these nanofluids. The shear viscosity non-linearly decreased with the increase of temperature and exhibited the Arrhenius behaviour for all different oxides containing Gly-based nanofluids. The acoustic parameters of the nanofluids were altered unevenly with types of nano oxides and inferred some structure-property correlations. The promising technologically useable properties of these nanofluids were expected to impact their potential applications in optoelectronics, UV-blocking, sensing, nanodielectrics, energy storing and electric insulation, heat transfer systems, and also in materials processing for the development of innovative soft condensed devices.     

Predictions of linear viscoelastic properties for polydisperse entangled polymers

Different blending laws have been proposed in the literature to describe the polydispersity effect on the rheological behavior of polymer melts. In this paper predictions of linear viscoelastic properties of entangled polydisperse polymers have been derived from the double reptation mixing rule. The results in terms of the relaxation modulus, the zero shear-rate viscosity, η₀, and the steady-state compliance, J _e ⁰, have been obtained using three different relaxation functions for the monodisperse fractions, namely the Tuminello step function, the single exponential function and the BSW function. Both discrete and continuous molecular weight distributions (MWDs) have been investigated. The Generalized Exponential Function (GEX) has been considered in the continuous case. The results showed that, in systems with a large number of components, the predictions of linear viscoelastic properties mainly depend on the double reptation mixing rule assumption, while the choice of the relaxation function is not crucial. In particular, the mathematical simplicity of the Tuminello step relaxation function has allowed analytical computation of the linear viscoelastic properties in closed form. Indeed, the analytical results indicated a dependence of η₀ on the MWD that could be expressed in terms of (M _z/M _w)^0.8, in agreement with experimental results reported in the literature. In the case of J _e ⁰, the analytical model defines a dependence on (M _z/M _w)^5.5, i.e. as expected a strong dependence on the MWD is predicted for the steady-state compliance. Finally, dynamic moduli have been computed from the relaxation modulus and their predictions have been favorably compared with experimental results from the literature. Received: 19 July 1999/Accepted: 24 November 1999     

Vorticity and Regularity for Viscous Incompressible Flows under the Dirichlet Boundary Condition. Results and Related Open Problems

In reference [7] it is proved that the solution of the evolution Navier–Stokes equations in the whole of R ³ must be smooth if the direction of the vorticity is Lipschitz continuous with respect to the space variables. In reference [5] the authors improve the above result by showing that Lipschitz continuity may be replaced by 1/2-H?lder continuity. A central point in the proofs is to estimate the integral of the term (ω · ∇)u · ω, where u is the velocity and ω = ∇ × u is the vorticity. In reference [4] we extend the main estimates on the above integral term to solutions under the slip boundary condition in the half-space R ₊³. This allows an immediate extension to this problem of the 1/2-H?lder sufficient condition. The aim of these notes is to show that under the non-slip boundary condition the above integral term may be estimated as well in a similar, even simpler, way. Nevertheless, without further hypotheses, we are not able now to extend to the non slip (or adherence) boundary condition the 1/2-H?lder sufficient condition. This is not due to the “nonlinear" term (ω · ∇)u · ω but to a boundary integral which is due to the combination of viscosity and adherence to the boundary. On the other hand, by appealing to the properties of Green functions, we are able to consider here a regular, arbitrary open set Ω.      

Molecular interpretation of the behaviour of polyisobutylene in different solvents

The effects of solvent environment on the behaviour of a high molecular weight polyisobutylene dissolved in kerosene and various grades of poly-1-butene solvent mixtures are investigated. The dependence of various molecular parameters such as zero-shear viscosity, intrinsic viscosity, specific viscosity, relaxation time and molecular expansion factor, on the polymer concentration, type of solvent and solvent viscosity is studied in the vicinity of dilute and semidilute regions (near the critical concentrationc ^*). The dependence of these parameters on solvent environment follows qualitatively Zimm's molecular model. The dependence on the polymer concentration deviates from this dilute solution theory. The effects of temperature on the zero-shear viscosity and the Maxwell relaxation time are also presented for two PIB solutions.     

A transition phenomenon of ice formation in water flow between two horizontal parallel plates

Experiments have been performed to investigate the icetransition profiles and heat-transfer characteristics for water flows between two horizontal parallel plates. The experiments are carried out under the condition that upper plate is cooled at uniform temperature kept less than freezing temperature of water, while the lower plate is heated at uniform temperature kept higher than the temperature of water flow. The temperatures of the upper and lower plates range from ?8 to ?14°C and from 10 to 60 °C, respectively, with inlet-water temperature varied from 1.5 to 4.5 °C. The cooling and heating temperature ratios, θ_c and θ_h, are ranging from 1.78 to 9.33 and from 1.22 to 39, respectively. By using three kinds of heightH of 16, 30 and 40 mm between the horizontal parallel plates, the Reynolds and Grashof numbers are varied from 3.2 × 10² to 1.5 × 10⁴ and from 3.4 × 10³ to 8.97 × 10⁶, respectively. As a result of this investigation two ice-transition modes are observed. The first ice-transition mode is due to an interruption of upper and lower thermal boundary layers, while the second mode is due to an instability of laminar boundary layer formed on water-ice interface. In order to determine the kind of ice-transition mode, criterion correlation formulas including the Reynolds numberRe _H , Grashof numberGr _H , and heating temperature ratio θ_h are determined and may be written as follows: For thermal icetransition mode (th.I.T.M.)Re _H /(Gr _H ·θ _h )^0.23<1.6×10^?3 and for hydrodynamical ice-transition mode (hy.I.T.M.)Re _H /(Gr _H ·θ _h )^0.23>2.3×10^?3 By introducing the freezing parameterB _f , correlation equations for local and mean Nusselt numbers along the water-ice interface at steady-state condition are determined. From the current experimental results it is found that the local Nusselt number may be described as the following equation:Nu _x =0.835 Re _H ^0.278 · B _f ^0.834 ·x/H)^?0.139