A self-aligning parallel plate (SAPP) fixture for tribology and high shear rheometry

This paper introduces a novel self-aligning parallel plate fixture for rotational rheometers. The ring-shaped shearing surface of this fixture is held on a low friction single contact point bearing and uses hydrodynamic lubrication forces in order to maintain the parallelism of the freely tilting surfaces over a full rotation. The optimized parallelism of the plates enables to conduct tribological measurements of low frictional stress between the shearing surface materials and a fluid at normal loads down to 1.3 kPa. Limited only by the degree of non-flatness of the surfaces, the new fixture can determine boundary lubrication sliding frictions within 10 % and down to angular velocities of 400 μrad/s. In a controlled gap mode, this setup reaches a gap error of 3.4 μm which enables to reliably conduct rheological measurements down to absolute gaps of the parallel plates of 10 μm and to reach high shear rates up to $10^{5}\;{\rm s}^{-1}$ .     

Obtaining the steady shear rheological properties and apparent wall slip velocity data of a water-in-oil emulsion from gap-dependent parallel plate viscometry data

A two-stage Tikhonov regularisation procedure has been used to obtain rheological properties for a high internal phase emulsion from gap-dependent steady-state parallel plate shear data. This method is beneficial in that it can convert the steady shear data into rheological property functions. The built-in regularisation parameters of the method are able to keep noise amplification under control. The two-stage method is able to obtain not only the shear stress–shear rate function but also the apparent slip velocity as a function of wall shear stress. The method is such that it obtains the rheological functions over the maximum range of shear rate covered by the data. The results obtained using the new method are compared to those obtained using the vane geometry with good agreement being observed.     

Buckling of a flush-mounted plate in simple shear flow

The buckling of an elastic plate with arbitrary shape flush-mounted on a rigid wall and deforming under the action of a uniform tangential load due to an overpassing simple shear flow is considered. Working under the auspices of the theory of elastic instability of plates governed by the linear von Kármán equation, an eigenvalue problem is formulated for the buckled state resulting in a fourth-order partial differential equation with position-dependent coefficients parameterized by the Poisson ratio. The governing equation also describes the deformation of a plate clamped around the edges on a vertical wall and buckling under the action of its own weight. Solutions are computed analytically for a circular plate by applying a Fourier series expansion to derive an infinite system of coupled ordinary differential equations and then implementing orthogonal collocation, and numerically for elliptical and rectangular plates by using a finite-element method. The eigenvalues of the resulting generalized algebraic eigenvalue problem are bifurcation points in the solution space, physically representing critical thresholds of the uniform tangential load above which the plate buckles and wrinkles due to the partially compressive developing stresses. The associated eigenfunctions representing possible modes of deformation are illustrated, and the effect of the Poisson ratio and plate shape is discussed.     

Transient freezing of liquids in forced laminar flow inside a parallel plate channel

A simple analytical approximative solution was given for calculating the time dependent development of the ice-layers at the cooled walls inside a parallel plate channel. By ignoring the effect of acceleration, resulting from converging ice-layers in the axial direction, an analytical solution for the variation of the ice-layer thickness with time and axial position could be obtained. The approximative solution was checked by numerical calculations and good agreement was found.Es wurde ein analytisches Näherungsverfahren entwickelt, das es ermöglicht, die zeitliche Entwicklung der Erstarrungsfronten im gekühlten, ebenen Kanal zu bestimmen. Die Methode liefert unter Vernachlässigung der Beschleunigungsterme durch die konvergenten Eisschichten eine exakte Lösung der Phasengrenzbeziehung. Das Näherungsverfahren wurde mittels numerischer Berechnungen überprüft und stimmt bis zu Wandunterkühlungsverhältnissen vonB=10 sehr gut mit der numerischen Lösung überein.     

Two-dimensional study of drop deformation under simple shear for Oldroyd-B liquids

The effect of viscoelasticity on the deformation of a circular drop suspended in a second liquid in shear is investigated with direct numerical simulations. A numerical algorithm based on the volume-of-fluid method for interface tracking is implemented in two dimensions with the Oldroyd-B constitutive model for viscoelastic liquids. The code is verified against a normal mode analysis for the stability of two-layer flow in a channel; theoretical growth rates are reproduced for the interface height, velocity and stress components. Drop simulations are performed for drop and matrix liquids of different viscosities and elasticities. A new feature is found for the case of equal viscosity, when the matrix liquid is highly elastic and surface tension is low; hook-like structures form at the drop tips. This is due to the growth of first normal stress differences that occur slightly above the front tip and below the back tip as the matrix elasticity increases above a threshold value.     

On the use of a nickel-tube resonator for measuring the complex shear modulus of liquids in the kHz-range

Summary An apparatus for the measurement of liquid-shear impedance in the frequency range 4–200 kHz with the aid of a thin-walled Ni-tube resonator is described. A magnetostrictive mechanism is used for setting the tube into torsional oscillation. Real and imaginary parts of the liquid-shear impedance are found from the change in the 3 dB band-width of the resonance curve and the shift of the resonance frequency, respectively, when the tube is immersed from the air into the liquid. The amount of liquid required is 20 ml. The necessary theory is given and some preliminary results are presented.

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Zusammenfassung Es wird über einen Apparat zur Messung der Scherimpedanz von Flüssigkeiten im Frequenzbereich von 4 bis 200 kHz mit Hilfe eines Resonators berichtet. Der Resonator, ein Nickelrohr mit geringer Wandstärke, wird mittels eines magnetostriktiven Mechanismus in Torsionsschwingungen versetzt. Real- und Imaginärteil der Scherimpedanz der Flüssigkeit werden aus der Änderung der Bandbreite der Resonanzkurve und der Verschiebung der Resonanzfrequenz, wenn das Rohr aus der Luft in die Flüssigkeit eingetaucht wird, berechnet. Die benötigte Flüssigkeitsmenge beträgt 20 ml. Die zugehörige Theorie wird mitgeteilt, und einige vorläufige Meßergebnisse werden vorgestellt.

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a,b outer and inner radii of the tube - 3 dB band-width of resonance curve of the loaded and unloaded tube - c,c ₀ phase velocities of torsional waves in the loaded and unloaded tube - c ^*,c ₀ ^* complex velocities of propagation of torsional waves in the loaded and unloaded tube - c ^* =c ₁ + ic₂ - c _L ^* complex velocity of propagation of shear waves in the fluid - f frequency - f _n, f _n ⁰ resonance frequency ofn-th overtone of the loaded and unloaded tube - G ^* complex shear modulus of the tube material - G _L ^* complex shear modulus of the fluid - i - I ₀ moment of inertia per cm of unloaded tube - I effective moment of inertia per cm of loaded tube - K instrument constant - l length of the tube - m =b/a - M angular momentum - n positive integer - R _L, X_L real and imaginary parts of the characteristic plane shear impedance of the fluid - Z _L characteristic plane shear impedance of the fluid (=R _L + iX_L) - Z _cyl characteristic cylindrical shear impedance of the fluid - , ₀ damping factors of torsional waves in the loaded and unloaded tube - , ₀ phase factors of torsional waves in the loaded and unloaded tube - propagation constant (= + i) - loss angle - tan cos^–1 (/ _max) - viscosity - angular displacement - wave length - displacement amplitude - , ₀ densities of the fluid and the tube material - angular frequency (= 2f) - _n ⁰ = 2f _n ⁰ - _R reduced frequency With 4 figures and 2 tables     

On out-of-plane buckling of anisotropic elastic plate subjected to a simple shear

Out-of-plane buckling of anisotropic elastic plate subjected to a simple shear is investigated. From exact 3-D equilibrium conditions of anisotropic elastic body with a plane of elastic symmetry at critical configuration, the eqution for buckling direction (buckling wave direction) parameter is derived and the shape functions of possible buckling modes are obtained. The traction free boundary conditions which must hold on the upper and lower surfaces of plate lead to a linear eigenvalue problem whose nontrivial solutions are just the possible buckling modes for the plate. The buckling conditions for both flexural and barreling modes are presented. As a particular example of buckling of anisotropic elastic plate, the buckling of an orthotropic elastic plate, which is subjected to simple shear along a direction making an arbitrary angle of θ with respect to an elastic principal axis of materials, is analyzed. The buckling direction varies with θ and the critical amount of shear. The numerical results show that only the flexural mode can indeed exist. Project supported by the National Natural Science Foundation of China (No. 19772032).     

High frequency parallel plate probe for the measurement of the complex viscosity of liquids

In this work an instrument is described which measures the complex shear viscosity of liquids in the kHz frequency range. The instrument is driven electromagnetically and operates in resonant mode. The measurement of the primary data, from which the rheological properties of the fluid sample are inferred, does not include any deflection amplitude measuring step and is purely digital. Models allowing the interpretation of the probe primary data in terms of fluid complex viscosity are presented. The theoretically predicted mechanical behaviour of the probe is compared with the measured one and the rheometric ability of the device is discussed.     

Automatic apparatus,based upon a nickel-tube resonator,for measuring the complex shear modulus of liquids in the kHz range

Summary An apparatus for the measurement of liquid-shear impedance in the frequency range 11–170 kHz using a thin-walled nickel-tube resonator is described. The working principle of the method used has been previously published. Compared with the prototype the most important advances in the design concern the suspension of the resonator and complete automatization of the measurements and data handling. At the cost of not measuring at the fundamental frequency accuracy and ease of operation are greatly improved. Results for Newtonian and viscoelastic liquids are presented. Comparison with other types of apparatus is made.

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Zusammenfassung Es wird über einen Apparat zur Messung der Scherimpedanz von Flüssigkeiten im Frequenzbereich von 11 bis 170 kHz berichtet, bei dem ein Nickelrohr mit geringer Wandstärke als Resonator verwendet wird. Das Arbeitsprinzip dieses Apparats wurde schon früher veröffentlicht. Im Vergleich mit dem Prototyp betreffen die wichtigsten Verbesserungen die Aufhängung des Resonators und die völlige Automatisierung der Messungen und der Datenverarbeitung. Die Genauigkeit und die Bequemlichkeit der Handhabung sind erheblich vergrößert, allerdings werden die Messungen jetzt nicht mehr bei der Grundfrequenz durchgeführt. Es werden Ergebnisse für newtonsche und viskoelastische Flüssigkeiten mitgeteilt und ein Vergleich mit anderen Apparaten durchgeführt.

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a outer radius of the tube - A apparatus constant - b distance to screen (in App. B) - B _n 3 dB-bandwith - B _long longitudinal magnetization of the tube - c ₀ velocity of propagation of torsional waves in a tube without losses - C, C ₁,C ₂,C ₃ constants - d _eff thickness of immobilized liquid layer - f frequency - f _n resonance frequency - F =H/(C) - G = G + iG complex shear modulus - H transfer function - i - I moment of inertia per unit length - I _p polar moment per unit length - k 3(2n + 1) - K parameter indicating deviations from ideal resonance - tube length - L inductance of transmitting coil - m ratio of inner and outer radii of the tube - M _el torsion moment caused by elastic deformation of the tube - M _ext torsion moment exerted on the tube by the liquid - n integer - N number of turns in receiving coil - O cross section area of receiving coil - p, p _rest track on screen when tube is in motion and at rest (in App. B) - t time - T _liq moment per unit length exerted by the liquid on the tube - R _L Re{Z _L } - V _in,V _out voltages - X Re{ /6} - X _L Im{Z _L } - Y Im{ /6} - z vertical coordinate - Z variation ofY aroundY ₀ - Z _L characteristic shear impedance - length of transmitting coil - length of receiving coil - rate of shear - propagation constant - loss angle of nickel - _f inertia correction term - _G , _G relative error inG, G - Nederveen transform parameter - , _s viscosity, solvent viscosity - torsion angle - density - angular frequency - ₁ fundamental angular frequency - 0 at resonance without loss terms - n index of used overtones - Ni nickel - A air - N Newtonian calibration liquid With 8 figures and 4 tables     

Relaxational properties of a turbulent shear flow across a cylinder in the presence of a plate

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 61–67, November–December, 1990.     

A test case for predicting the rheological properties of polymeric liquids: the multiple coupled Maxwell modes model

In this work, the behavior of the multiple coupled Maxwell modes (MCMM) model is examined with regard to the rheological properties of polymer melts in diverse flow fields, including (i) transient, (ii) steady-state shear flow, (iii) small-amplitude oscillatory shear flow, and (iv) transient uniaxial elongational flow. This model is specialized to the case of pair-wise coupling, i.e., each mode interacts with only one other mode. Consequently, each pair of modes acts independently of the others. For a typical polymer melt of industrial interest, a simple optimization technique is developed for determining the number of independent mode pairs, as well as precise values for the corresponding parameters.The goal of this ongoing study is to fit the parameters of various rheological models using a limited number of simple, standard experiments, and then to see whether or not the models can predict data taken from more complicated experiments. In this paper, only the first step is taken in this direction: we examine for one particularly promising rheological model, the MCMM model (but herein restricted to pair-wise coupling), whether or not it can achieve this goal. This restricted model is chosen because it mimics the effect of pair-wise coupling between relaxation modes that is prevalent in current rheological models. Using this model as a test case allows the optimization technique and analytical methodology for achieving the overall goal to be developed. The outcome of this study with regard to developing the methodology was successful, but the particular model chosen, written in terms of coupled Maxwell modes with pair-wise coupling, is not general enough to predict well typical polymer melt rheological behavior.     

Driven torsion pendulum for measuring the complex shear modulus in a steady shear flow

To investigate the viscoelastic behavior of fluid dispersions under steady shear flow conditions, an apparatus for parallel superimposed oscillations has been constructed which consists of a rotating cup containing the liquid under investigation in which a torsional pendulum is immersed. By measuring the resonance frequency and bandwidth of the resonator in both liquid and in air, the frequency and steady-shear-rate-dependent complex shear modulus can be obtained. By exchange of the resonator lumps it is possible to use the instrument at four different frequencies: 85, 284, 740, and 2440 Hz while the steady shear rate can be varied from 1 to 55 s^–1. After treatment of the theoretical background, design, and measuring procedure, the calibration with a number of Newtonian liquids is described and the accuracy of the instrument is discussed.Notation a radius of the lump - A geometrical constant - b inner radius of the sample holder - c constant - C ₁, C ₂ apparatus constants - D damping of the pendulum - e _x , e _y , e _z Cartesian basis - e _r , e , e _z orthonormal cylindrical basis - E geometrical constant - E _t , ⁰ E _t , _t relative strain tensor - f function of shear rate - F _t relative deformation tensor - G (t) memory function - G ^* complex shear modulus - G Re(G ^* ) - G Im(G ^* ) - h distance between plates - H ^* transfer function - , functional - i imaginary unit: i ²= – 1 - I moment of inertia - J _exc excitation current - J ₀ amplitude of J _exc - k ^* = k – ik complex wave number - K torsional constant - K fourth order tensor - l length of the lump - L mutual inductance - M _dr driving torque - M _liq torque exerted by the liquid - ⁰ M _liq, _liq steady state and dynamic part of Mliq - n power of the shear rate - p isotropic pressure - Q quality factor - r radial position - R,R ₀, R _c Re(Z ^*, Z ₀ ^* , Z _c ^* ) - s time - t, t time - T temperature - T, ⁰ T, stress tensor - u velocity - U lock-in output - ₀ velocity - V _det detector output voltage - V _sig, V _cr signal and cross-talk part of V _det - x Cartesian coordinate - X , X ₀, X _c Im(Z ^*, Z ₀ ^* , Z _c ^* ) - y Cartesian coordinate - z Cartesian coordinate, axial position     

Theoretical modeling of microstructured liquids: a simple thermodynamic approach

A new method is presented for accounting for microstructural features of flowing complex fluids at the level of mesoscopic, or coarse-grained, models by ensuring compatibility with macroscopic and continuum thermodynamics and classical transport phenomena. In this method, the microscopic state of the liquid is described by variables that are local expectation values of microscopic features. The hypothesis of local thermodynamic equilibrium is extended to include information on the microscopic state, i.e., the energy of the liquid is assumed to depend on the entropy, specific volume, and microscopic variables. For compatibility with classical transport phenomena, the microscopic variables are taken to be extensive variables (per unit mass or volume), which obey convection-diffusion-generation equations. Restrictions on the constitutive laws of the diffusive fluxes and generation terms are derived by separating dissipation by transport (caused by gradients in the derivatives of the energy with respect to the state variables) and by relaxation (caused by non-equilibrated microscopic processes like polymer chain stretching and orientation), and by applying isotropy. When applied to unentangled, isothermal, non-diffusing polymer solutions, the equations developed according to the new method recover those developed by the Generalized Bracket [J. Non-Newtonian Fluid Mech. 23 (1987) 271; A.N. Beris, B.J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, first ed., Oxford University Press, Oxford, 1994] and by the Matrix Model [J. Rheol. 38 (1994) 769]. Minor differences with published results obtained by the Generalized Bracket are found in the equations describing flow coupled to heat and mass transfer in polymer solutions. The new method is applied to entangled polymer solutions and melts in the general case where the rate of generation of entanglements depends nonlinearly on the rate of strain. A link is drawn between the mesoscopic transport equations of entanglements and conformation and the microscopic equation describing the configurational distribution of polymer segment stretch and orientation. Constraints are derived on the generation terms in the transport equations of entanglements and conformation, and the formula for the elastic stress is generalized to account for reversible formation and destruction of entanglements. A simplified version of the transport equation of conformation is presented which includes many previously published constitutive models, separates flow-induced polymer stretching and orientation, yet is simple enough to be useful for developing large-scale computer codes for modeling coupled fluid flow and transport phenomena in two- and three-dimensional domains with complex shapes and free surfaces.     

A method for calculating rheological and morphological properties of constant-volume polymer blend models in inhomogeneous shear fields

We show how to formulate two-point boundary-value problems in order to compute fully-developed laminar channel and tube flow profiles for viscoelastic fluid models. The formulation is applied to Couette and pressure-driven flows separately, or a combination of both. The application of this methodology is illustrated analytically for the Upper-Convected Maxwell Model, and it is applied computationally for the Phan-Thien/Tanner and Giesekus Models. Numerical solutions exist for the last two models [J.Y. Yoo, H.C. Choi, On the steady simple shear flows of the one-mode Giesekus fluid, Rheol. Acta 28 (1989) 13–24; P.J. Oliveira, F.T. Pinho, Analytical solution for fully developed channel and pipe flow of Phan-Thien–Tanner fluids, J. Fluid Mech. 387 (1999) 271–280; M.A. Alves, F.T. Pinho, P.J. Oliveira, Study of steady pipe and channel flows of a single-mode Phan-Thien–Tanner fluid, J. Non-Newtonian Fluid Mech. 101 (2001) 55–76], allowing verification of the computational technique. Subsequently, the computational algorithm is applied to the constant-volume polymer blend models of Maffettone and Minale [P.L. Maffettone, M. Minale, Equation of change for ellipsoidal drops in viscous flow, J. Non-Newtonian Fluid Mech. 84 (1999) 105–106 (Erratum), J. Non-Newtonian Fluid Mech. 78 (1998) 227–241] and Dressler and Edwards [M. Dressler, B.J. Edwards, The influence of matrix viscoelasticity on the rheology of polymer blends, Rheol. Acta 43 (2004) 257–282; M. Dressler, B.J. Edwards, Rheology of polymer blends with matrix-phase viscoelasticity and a narrow droplet size distribution, J. Non-Newtonian Fluid Mech. 120 (2004) 189–205]. Rheological and morphological properties of the model blends are thus obtained as functions of the spatial position within the channel, applied pressure drop, and shear rate at the wall.     

Rheometry for large-particulated fluids: analysis of the ball measuring system and comparison to debris flow rheometry

For large-particulated fluids encountered in natural debris flow, building materials, and sewage treatment, only a few rheometers exist that allow the determination of yield stress and viscosity. In the present investigation, we focus on the rheometrical analysis of the ball measuring system as a suitable tool to measure the rheology of particulated fluids up to grain sizes of 10 mm. The ball measuring system consists of a sphere that is dragged through a sample volume of approximately 0.5 l. Implemented in a standard rheometer, torques exerted on the sphere and the corresponding rotational speeds are recorded within a wide measuring range. In the second part of this investigation, six rheometric devices to determine flow curve and yield stress of fluids containing large particles with maximum grain sizes of 1 to 25 mm are compared, considering both rheological data and application in practical use. The large-scale rheometer of Coussot and Piau, the building material learning viscometer of Wallevik and Gjorv, and the ball measuring system were used for the flow curve determination and a capillary rheometer, the inclined plane test, and the slump test were used for the yield stress determination. For different coarse and concentrated sediment–water mixtures, the flow curves and the yield stresses agree well, except for the capillary rheometer, which exhibits much larger yield stress values. Differences are also noted in the measuring range of the different devices, as well as for the required sample volume that is crucial for application.