Non-oscillation Criteria for Hypoelastic Models under Simple Shear Deformation

Ten objective rates, spinning or non-spinning, are critically examined from the viewpoint of Sturm's theorems in ordinary differential equations. Upon developing implication relations of oscillatory, non-oscillatory, and disconjugate behavior, we establish oscillation and non-oscillation criteria which pick out the objective stress rates that lead to oscillatory and non-oscillatory responses in simple shear deformation, respectively. Among the hypoelastic equations associated with the spinning objective rates examined, the Jaumann equation is an oscillatory minorant, the homogeneous Xiao–Bruhns–Meyers equation is a non-oscillatory majorant, and the homogeneous Green–Naghdi equation is a disconjugate majorant. If (Sturm comparable) non-spinning objective rates are also taken into consideration, then the Durban–Baruch equation becomes an oscillatory minorant, but the other two equations remain to play the same roles. The Jaumann equation is a Sturm majorant for all the other nine homogeneous hypoelastic equations, and the homogeneous Szabó–Balla-2 equation is a Sturm minorant for all the other nine homogeneous hypoelastic equations. Most of the solutions of the zeroth-grade hypoelastic equations at simple shear have already been published, except for those of Szabó and Balla, to which the closed-form solutions are derived here. Moreover, all solutions are extended to include the effect of initial stresses. This revised version was published online in August 2006 with corrections to the Cover Date.     

Anomalous predictions of pressure drop and heat transfer in ducts of arbitrary cross-section with modified power law fluids

 The apparent viscosities of purely viscous non-Newtonian fluids are shear rate dependent. At low shear rates, many of such fluids exhibit Newtonian behaviour while at higher shear rates non-Newtonian, power law characteristics exist. Between these two ranges, the fluid's viscous properties are neither Newtonian or power law. Utilizing an apparent viscosity constitutive equation called the “Modified Power Law” which accounts for the above behavior, solutions have been obtained for forced convection flows. A shear rate similarity parameter is identified which specifies both the shear rate range for a given fluid and set of operating conditions and the appropriate solution for that range. The results of numerical solutions for the friction factor–Reynolds number product and for the Nusselt number as a function of a dimensionless shear rate parameter have been presented for forced fully developed laminer duct flows of different cross-sections with modified power law fluids. Experimental data is also presented showing the suitability of the “Modified Power Law” constitutive equation to represent the apparent viscosity of various polymer solutions. Received on 21 August 2000     

Elongational and shear rheology of carbon nanotube suspensions

Rheological behavior of concentrated suspensions of chemical vapor deposition carbon nanotubes in uniaxial elongation and simple shear is studied experimentally and theoretically. Nanotubes are suspended in viscous host liquids—castor oil or its blends with n-decane. The elongational measurements are performed by analyzing self-thinning (due to surface tension effect) liquid threads of nanotube suspensions. A quasi-one-dimensional model is used to describe the self-thinning process, whereas corrections accounting for thread nonuniformity and necking are introduced a posteriori. The effects of nanotube concentration and aspect ratio, viscosity of the suspending liquid, and initial diameter of the self-thinning thread in uniaxial elongation are elucidated. The results for uniaxial elongation are compared with those for simple shear. The correspondence in the results of the shear and elongational measurements is addressed and interpreted. The results conform to the Herschel–Bulkley rheological constitutive equation (i.e., power law fluids with yield stress). However, the yield stress in elongation is about 40% higher than in simple shear flow, which suggests that the original Herschel–Bulkley model need modification with the yield stress being a function of the second invariant of the deviatoric stress tensor. The present effort is the first to study capillary self-thinning of Herschel–Bulkley liquids, which are exemplified here by suspensions of carbon nanotubes.     

An unsymmetric constitutive equation for anisotropic viscoelastic fluid

A continuum constitutive theory of corotational derivative type is developed for the anisotropic viscoelastic fluid–liquid crystalline (LC) polymers. A concept of anisotropic viscoelastic simple fluid is introduced. The stress tensor instead of the velocity gradient tensor D in the classic Leslie–Ericksen theory is described by the first Rivlin–Ericksen tensor A and a spin tensor W measured with respect to a co-rotational coordinate system. A model LCP-H on this theory is proposed and the characteristic unsymmetric behaviour of the shear stress is predicted for LC polymer liquids. Two shear stresses thereby in shear flow of LC polymer liquids lead to internal vortex flow and rotational flow. The conclusion could be of theoretical meaning for the modern liquid crystalline display technology. By using the equation, extrusion–extensional flows of the fluid are studied for fiber spinning of LC polymer melts, the elongational viscosity vs. extension rate with variation of shear rate is given in figures. A considerable increase of elongational viscosity and bifurcation behaviour are observed when the orientational motion of the director vector is considered. The contraction of extrudate of LC polymer melts is caused by the high elongational viscosity. For anisotropic viscoelastic fluids, an important advance has been made in the investigation on the constitutive equation on the basis of which a series of new anisotropic non-Newtonian fluid problems can be addressed. The project supported by the National Natural Science Foundation of China (10372100, 19832050) (Key project). The English text was polished by Yunming Chen.     

Molecular approach in linear polymer dynamics: Theory and numerical experiment

A constitutive equation for polymer solutions and melts is obtained on the basis of the dynamics of noninteracting dumbbells moving in a nonlinear anisotropic fluid. The equation obtained is used to describe nonlinear effects under conditions of simple shear and steady-state flow in a circular tube and for the numerical investigation of a flow in a finite cylinder with a rotating end face. Barnaul. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–13, January–February, 2000.     

Transformation of Long Nonlinear Waves in a Two-Layer Viscous Fluid Between a Gently Sloping Lid and Bottom

Dynamics of three-dimensional disturbances of the interface between two fluid layers of different densities is considered analytically and numerically. An evolutionary integrodifferential equation is derived, which takes into account long-wave contributions of inertia of the layers and surface tension, small but finite amplitude of disturbances of the interface between two incompressible immiscible fluids, gentle slopes of the lid and bottom, and nonstationary shear stresses at all boundaries. Numerical solutions of this model equation for several (most typical) nonlinear problems of transformation of two- and three-dimensional waves are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 45–57, November–December, 2005.     

Evolution of long nonlinear waves on the interface of a stratified viscous fluid flow in a channel

The dynamics of disturbances of the interface between two layers of incompressible immiscible fluids of different densities in the presence of a steady flow between the horizontal bottom and lid is studied analytically and numerically. A model integrodifferential equation is derived, which takes into account long-wave contributions of inertial layers and surface tension of the fluids, small but finite amplitude of disturbances, and unsteady shear stresses on all boundaries. Numerical solutions of this equation are given for the most typical nonlinear problems of transformation of both plane waves of different lengths and solitary waves. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 49–61, July–August, 2007.     

The use of the capillary rheometer for the rheological evaluation of extrudable cement-based materials

In this paper, the rheological properties of an extrudable cement-based paste are investigated by means of an original ram extrusion apparatus (capillary rheometer). The experimental results indicate that a careful measurement of the die pressure is necessary to obtain a realistic viscosity vs shear rate curve, as required in extrusion technology. In particular, it is shown that the optimal test configuration is when the pressure measurement is made directly inside the rheometer die. By applying this rheological methodology in steady-state conditions, it has been observed that the extrudable cement-based material here evaluated obeys to a simple power–law equation, in the range of shear rates investigated, which are suitable for an industrial extrusion process. This paper was presented at the third Annual European Rheology Conference (AERC) held in Hersonissos, Crete, Greece, April 24–27, 2006.     

Effect of a hydrotrope on the viscoelastic properties of polymer-like micellar solutions

Low-viscosity micellar aqueous solutions of cetyltrimethylammonium bromide (CTAB) undergo a major change in the presence of the hydrotrope, potassium 1-phenylmethylsulfate (KPhMS), producing a highly viscoelastic entanglement network of polymer-like micelles. The system studied here shows typical shear banding flow behavior, which tends to disappear with increasing the hydrotrope-to-surfactant concentration ratio (C _H / C _S). The linear rheological response was analyzed with the model of Granek–Cates, whereas the nonlinear behavior was reproduced with the Bautista–Manero–Puig (BMP) model. Both models introduce a kinetic equation to account for the breaking and reformation of the micelles, and they predict the linear and nonlinear rheological data very well. This paper was presented at Annual European Rheology Conference (AERC) held in Hersonisos, Crete, Greece, April 27–29, 2006.     

The interchain pressure effect in shear rheology

Recently, the tube diameter relaxation time in the evolution equation of the molecular stress function (MSF) model (Wagner et al., J Rheol 49: 1317–1327, 2005) with the interchain pressure effect (Marrucci and Ianniruberto, Macromolecules 37:3934–3942, 2004) included was shown to be equal to three times the Rouse time in the limit of small chain stretch. From this result, an advanced version of the MSF model was proposed, allowing modeling of the transient and steady-state elongational viscosity data of monodisperse polystyrene melts without using any nonlinear parameter, i.e., solely based on the linear viscoelastic characterization of the melts (Wagner and Rolón-Garrido 2009a, b). In this work, the same approach is extended to model experimental data in shear flow. The shear viscosity of two polybutadiene solutions (Ravindranath and Wang, J Rheol 52(3):681–695, 2008), of four styrene-butadiene random copolymer melts (Boukany et al., J Rheol 53(3):617–629, 2009), and of four polyisoprene melts (Auhl et al., J Rheol 52(3):801–835, 2008) as well as the shear viscosity and the first and second normal stress differences of a polystyrene melt (Schweizer et al., J Rheol 48(6):1345–1363, 2004), are analyzed. The capability of the MSF model with the interchain pressure effect included in the evolution equation of the chain stretch to model shear rheology on the basis of linear viscoelastic data alone is confirmed.     

A procedure to determine the viscosity function from experimental data of capillary flow

The accurate calculation of the viscosity η as function of the shear rate &γdot; from capillary viscometry is still a matter of debate in the literature. In fact, this problem involves the inversion of an integral equation, which leads to multiple solutions due to the unavoidable noise present in the experimental data. The purpose of this work is to develop an efficient procedure to determine the viscosity function from experimental data of capillary flow without presenting the difficulties inherent in other methods discussed previously in the literature. The system identification procedure is used here to estimate the parameters of a viscosity model, which is appropriately selected for the fluid under study through preliminary calculations involving the apparent shear rate – shear stress data. Once the model is chosen by satisfying criteria for the fit goodness and its parameters are evaluated, a smooth and continuous function η(γdot;) is obtained in the range of experimental shear rates. The procedure proposed is also applicable to fluids in shear flow that present two Newtonian plateaus, as it is typically found in macromolecular dilute solutions. The mean value theorem of continuous functions is used to reduce significantly the computational time. Received: 15 November 1999 Accepted: 7 November 2000     

Analyzing the influence of compressibility on the rapid pressure–strain rate correlation in turbulent shear flows

The influence of compressibility on the rapid pressure–strain rate tensor is investigated using the Green’s function for the wave equation governing pressure fluctuations in compressible homogeneous shear flow. The solution for the Green’s function is obtained as a combination of parabolic cylinder functions; it is oscillatory with monotonically increasing frequency and decreasing amplitude at large times, and anisotropic in wave-vector space. The Green’s function depends explicitly on the turbulent Mach number M _t , given by the root mean square turbulent velocity fluctuations divided by the speed of sound, and the gradient Mach number M _g , which is the mean shear rate times the transverse integral scale of the turbulence divided by the speed of sound. Assuming a form for the temporal decorrelation of velocity fluctuations brought about by the turbulence, the rapid pressure–strain rate tensor is expressed exactly in terms of the energy (or Reynolds stress) spectrum tensor and the time integral of the Green’s function times a decaying exponential. A model for the energy spectrum tensor linear in Reynolds stress anisotropies and in mean shear is assumed for closure. The expression for the rapid pressure–strain correlation is evaluated using parameters applicable to a mixing layer and a boundary layer. It is found that for the same range of M _t there is a large reduction of the pressure–strain correlation in the mixing layer but not in the boundary layer. Implications for compressible turbulence modeling are also explored.      

Coupled nonlinear constitutive models for rarefied and microscale gas flows: subtle interplay of kinematics and dissipation effects

The constitutive relations of gases in a thermal nonequilibrium (rarefied and microscale) can be derived by applying the moment method to the Boltzmann equation. In this work, a model constitutive relation determined on the basis of the moment method is developed and applied to some challenging problems in which classical hydrodynamic theories including the Navier–Stokes–Fourier theory are shown to predict qualitatively wrong results. Analysis of coupled nonlinear constitutive models enables the fundamentals of gas flows in thermal nonequilibrium to be identified: namely, nonlinear, asymmetric, and coupled relations between stresses and the shear rate; and effect of the bulk viscosity. In addition, the new theory explains the central minimum of the temperature profile in a force-driven Poiseuille gas flow, which is a well-known problem that renders the classical hydrodynamic theory a global failure.     

Predictive Capabilities of an Improved Cubic k–ε Model for Inert Steady Flows

Through an improved ε transport equation, a major quality enhancement of the cubic k–ε model, earlier developed in[13], is obtained. The ε-equation of [13],yielding good results for wall-bounded and rotating flows, is combined with the one derived by Shih et al. [20], which produces good results for free shear flows (e.g. the plane jet–round jet anomaly is resolved).Results are presented for the following flows: fully developed stationary and rotating channel and pipe, backward-facing step, sudden pipe expansion, smooth channel expansion and contraction, plane and round jet. Heat transfer predictions in turbulent impinging jets are also discussed. Accurate results are obtained for the mean flow quantities for all test cases, without case dependent model tuning. This revised version was published online in July 2006 with corrections to the Cover Date.     

Pressure-dependent viscosity and free volume of atactic and syndiotactic polystyrene

The effect of pressure on viscosity is an important but often overlooked aspect of the flow properties of polymeric materials. In this work, two polymers (an atactic and a syndiotactic Polystyrene) were characterized to determine the effect of pressure on viscosity. In particular, a device was adopted to increase the exit pressure of a standard capillary rheometer, thus obtaining data of viscosity under high pressure and high shear rates. The Simha-Somcynsky equation of state was applied to the pressure–volume–temperature experimental data of both materials to obtain the dependence of free volume on temperature and pressure. The Doolittle equation was eventually employed to verify the dependence of viscosity on free volume. It was found that, for both materials, a linear relationship holds between the logarithm of zero-shear-rate viscosity (at several temperatures and pressures) and the inverse of free volume.     

Tangential discontinuities of parameters of a polar fluid under shear strain

Possible formation of tangential discontinuities of parameters of a deformable polar fluid is examined by the example of glycerin. It is experimentally established that glycerin under weak shear loads possesses the properties of a non-Newtonian elastoviscoplastic fluid, and formation of tangential discontinuities in viscosity is possible. In the discontinuity region, glycerin has the properties of a low-viscosity fluid, and the structure of the medium is reconstructed after unloading. A rheological equation of the examined fluid is derived, which allows one to analyze the behavior of the medium in different modes of its deformation, including the formation of a local region with reduced viscosity and a tensile stress field. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 41–49, May–June, 2005.     

A parameter investigation of shear-induced coalescence in semidilute PIB–PDMS polymer blends: effects of shear rate, shear stress volume fraction, and viscosity

In this work, drop coalescence of polymer blends under shear flow in a parallel flow apparatus was investigated by optical sectioning microscopy. In each experiment, shear rate was set at values low enough to avoid any break-up phenomena. The time evolution of the drop size distribution was determined by motorized sample scanning and iterative acquisition of stacks of images along sample depth. Drop size and location in the acquired images was found by automated image analysis techniques. A systematic experimental campaign to investigate the effects of shear rate (in the range 0.1–0.5 s⁻¹), volume fraction (2.5–10%), and viscosity of the two phases (3–63 Pa s) at different viscosity ratio (0.1–2.3) was carried out. By comparing data from different experiments, it was found that at any strain value, the average drop size decreases monotonically with the shear stress, calculated as the product of shear rate and matrix viscosity. Furthermore, the coalescence rate slowed down with increasing viscosity ratio. Overall, these results provide an extensive set of data, which can be used as a benchmark for modeling shear-induced coalescence in polymer blends.Paper presented at the Annual Meeting of the European Society of Rheology, Grenoble, April 2005.     

Shear Instability of the Structure of Media Possessing Viscous Fluidity

Instability of the structure of viscous fluids in the Couette flow regime (spontaneous formation of bands, which are tangential discontinuities in terms of viscosity) is experimentally found. This process is demonstrated to be analogous to formation of shear bands in polymethylmethacrylate (Plexiglas) under plastic shear strains. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 70–76, March–April, 2006.     

A comparison of three adsorption equations and sensitivity study of parameter uncertainty effects on adsorption refrigeration thermal performance estimation

This paper presents isosteric-based adsorption equilibrium tests of three activated carbon samples with methanol as an adsorbate. Experimental data was fitted into Langmuir equation, Freundlich equation and Dubinin-Astakov (D–A) equation, respectively. The fitted adsorption equations were compared in terms of agreement with experimental data. Moreover, equation format’s impacts on calculation of the coefficient of performance (COP) and refrigeration capacity of an adsorption refrigeration system was analyzed. In addition, the sensitivity of each parameter in each adsorption equation format to the estimation of cycle’s COP and refrigeration capacity was investigated. It was found that the D–A equation is the best form for presenting the adsorptive property of a carbon-methanol working pair. The D–A equation is recommended for estimating thermal performance of an adsorption refrigeration system because simulation results obtained using the D–A equation are less sensitive to errors of experimentally determined D–A equation’s parameters.     

Simple waves on a shear gas flow in a channel of constant cross section

The plane-parallel unsteady-state shear gas flow in a narrow channel of constant cross section is considered. The existence theorem of solutions in the form of simple waves of a set of equations of motion is proved for a class of isentropic flows with a monotone velocity profile over the channel depth. The exact solution described by incomplete beta-functions is found for a polytropic equation of state in a class of isentropic flows. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 36–43, January–February, 1999.