The viscous squeeze-film behaviour of shear-thinning fluids

A finite element technique is proposed to predict the purely viscous squeeze-film behaviour of an arbitrary shear-thinning fluid confined between parallel discs and subjected to a constant load. The technique requires establishment of the distribution of viscosity in the gap. The variable viscosity is modelled by a discrete number of Newtonian fluids, with each fluid lying in a region bounded by lines of constant shear rate. Each of these Newtonian regions is further divided into regions which appear as “finite element” rectangles in the r-z plane. The equations governing squeeze-film flow are applied to this finite element network and an ordinary differential equation is ultimately derived which governs the gap decrease with time. Solving this equation is not simple because the coefficients of two terms change as the gap decreases. When the number of Newtonian fluids is sufficient, the technique predicts the squeeze-film time of a power-law fluid to within a fraction of a percent. Application of the technique to synovial fluid viscosity prevents the cartilage surfaces from touching for only a fraction of a second.     

Effects of bubble deformation on the viscosity of dilute suspensions

The relative viscosity (μ_rel=suspension viscosity/suspending fluid viscosity) of low Reynolds number, dilute and surfactant-free bubble suspensions in simple shear is studied with a rotating cylinder, Couette rheometer. The conditions of the experiments correspond to capillary numbers (Ca) of order 1 and bridge previous experimental, theoretical and numerical results that focused on either Ca⪡1 or Ca⪢1. The suspensions are shear thinning with μ_rel>1 for small Ca. At large Ca, μ_rel approaches a constant that is less than 1. These results are explained by a scaling analysis that considers how regions of viscous dissipation in and around bubbles change as bubbles are deformed by the flow.     

An investigation of non-newtonian flow past a sphere

Any experimental work on the flow of a polymer solution or any theoretical analysis on the basis of a visoelastic constitutive equation does not always bring out viscoelastic effects but may be showing a non-Newtonian viscosity effect. Therefore, in order to obtain a clear understanding about viscoelastic effects, it is desirable to have a sufficient knowledge of the non-Newtonian viscosity effect. To facilitate this, finite-difference numerical solutions of non-Newtonian flow were carried out using a non-Newtonian viscous model for the Reynolds numbers of 0.1, 1.0, 20 and 60.Drag force measurements and flow visualization experiments were also performed over a wide range of experimental conditions using polymer solutions. The present work appears to support the following idea: When compared with the Newtonian case on the basis of DV_∞P/η₀, where η₀ is the zero shear viscosity, it is on account of the non-Newtonian viscosity that the friction and pressure drags decrease, that the separating vortices behind the sphere become larger, and that no shift occurs in the streamlines. On the other hand, it is due to viscoelasticity that the normal force drag increases, that the separating vortices behind the sphere become smaller, and that an upstream shift occurs in the streamlines.     

Elastico-viscous squeeze films: Part 2. superimposed rotation

The first part of this paper contains a reconsideration of the conventional squeeze-film situation. It is shown that the Part 1 concentration on the half-time t_{$\text{1}\text{2}$} sometimes hides interesting elastico-viscous effects, since these are most pronounced at short times after the load is applied.The bulk of the paper is concerned with the more general situation in which a rotational flow is superimposed on the basic squeezing flow. This is brought about by rotating the bottom plate. An approximate theoretical analysis is shown to lead to a satisfactory prediction of observed behaviour under conditions of light loading.The experiments show substantial transient stress-overshoot effects under some conditions and there is also a possible indication of long-range memory effects in some of the experiments.     

Stability of a liquid jet into incompressible gases and liquids: Part 2. Effects of the irrotational viscous pressure

In this paper we investigate the effects of an irrotational, viscous pressure on the stability of a liquid jet into gases and liquids. The analysis extends our earlier work (part 1) in which the stability of the viscous jet was studied assuming that the motion and pressure are irrotational and the viscosity enters through the jump in the viscous normal stress in the normal stress balance at the interface. The liquid jet is always unstable; at high Weber numbers the instability is dominated by capillary instability; at low W the instability is dominated by Kelvin–Helmholtz (KH) waves generated by pressures driven by the discontinuous velocity. In the irrotational analysis the viscosity is important but the effects of shear are neglected. In fact a discontinuous velocity is not compatible with the continuity of the tangential components of velocity and shear stress so that KH instability is not properly posed for exact study using the no-slip condition but some of the effects of viscosity can be ascertained using viscous potential flow. The theory is called viscous potential flow (VPF). Here we develop another irrotational theory in which the discontinuities in the irrotational tangential velocity and shear stress are eliminated in the global energy balance by selecting viscous contributions to the irrotational pressure. These pressures generate a hierarchy of potential flows in powers of the viscosity, but only the first one, linear in viscosity, in the irrotational viscous stress, is thought to have physical significance. The tangential velocity and shear stress in an irrotational study cannot be made continuous, but the effects of the discontinuous velocity and stress in the mechanical energy balance can be removed “in the mean.” This theory with the additional viscous pressure is called VCVPF, viscous correction of VPF. VCVPF is VPF with the additional pressures. The theory here cannot be compared with an exact solution, which would not allow the discontinuous velocity and stress. In other problems, like capillary instability, in which VCVPF can be compared with an exact solution, the agreements are uniformly excellent in the wave number when one of the fluids is gas and in good but not uniform, agreement when both fluids are liquids.     

An experimental study of non-isothermal miscible displacements in a Hele-Shaw cell

Non-isothermal miscible displacements in a radial Hele-Shaw cell were experimentally investigated using a scheme in which room temperature liquids of relatively high viscosity were displaced by high-temperature (80 °C), less-viscous liquids. Fundamental characteristics have been presented regarding how the effect of a non-isothermal field on miscible displacement patterns varies in terms of factors such as the viscosity ratio of the more- and less-viscous liquids at 20 °C, M₂₀, the rate of an increase in the pattern’s area, R, and the gap width of the cell, b. The concept of area density was used to quantitatively evaluate the effect of the non-isothermal fields on the patterns. We have found that the effect of the non-isothermal field on the patterns does not monotonically vary with M₂₀ and b. In contrast, it increases with R in the present experimental condition. The experimental results can be explained by introducing an assumption in which heat is transferred mainly to the plates of the cell, in other words, the temperature of the more-viscous liquid remains constant, whereas that of the less-viscous liquid spatiotemporally decreases and the viscosity of it increases along with the temperature decrease. Visualization of non-isothermal field in the cell has been done by means of a thermo sheet and the results support the assumption mentioned above.     

Can extensional viscosity be measured with opposed-nozzle devices?

Opposed-nozzle devices are widely used to try to measure the extensional viscosity of low-viscosity liquids. A thorough literature survey shows that there are still several unanswered questions on the relationship between the quantity measured in opposed-nozzle devices and the true extensional viscosity of the liquids. In addition to extensional stresses, opposed nozzle measurements are influenced by dynamic pressure, shear on the nozzles, and liquid inertia. Therefore the ratio of the apparent extensional viscosity that is measured to the shear viscosity that is independently measured is greater than three even for Newtonian liquids. The effect of inertia on the extensional measurements is analyzed by computer-aided solution of the Navier-Stokes system, and by experiments on low-viscosity Newtonian liquids(1 mPa sS 800 mPa s). The effect of nozzle separation-to-diameter ratio on the average residence time of the liquid is analyzed under the assumption of simple extensional flow kinematics. The average residence time of the liquid is independent of this ratio unless the radial inflow section of the extensional flow volume is related to the nozzle separation. Experiments indicate that in some cases widening the gap lowers the apparent extensional viscosity that is measured, whereas in other cases the opposite is true. In the light of these theoretical considerations and experimental observations, the use of systematic corrections to extensional viscosity measurements on non-Newtonian liquids is not recommended. Thus opposed nozzle devices should be considered as useful indexers rather than rheometers. Finally, measurements on a series of semi-dilute solutions of high molecular weight poly(ethylene oxide) in. water are also reported.Dedicated to the memory of Anastasios C. Papanastasiou     

Instantaneous Squeeze-Film Force Between a Heat Exchanger Tube and a Support Plate for Arbitrary Tube Motion

The instantaneous squeeze-film force between a heat exchanger tube and a support plate is studied. Based on a two-dimensional rectangular plate model, a short-sleeve squeeze-film model for arbitrary tube motion is developed. The instantaneous squeeze-film force is expressed in normal and tangential directions. The normal squeeze-film force consists of four nonlinear terms, the viscous, unsteady inertia, convective inertia and centripetal inertia terms. Three nonlinear terms, the viscous, unsteady inertia and Coriolis inertia terms, make up the tangential squeeze-film force. An experimental apparatus was developed in order to evaluate the theoretical models against measurements of a finite length squeeze film. A modified model based on the experimental data is obtained where the viscous terms for both directions are multiplied by the instantaneous Reynolds number. All the inertia terms are multiplied by constant coefficients. The modified model is in good agreement with most experimental cases for unsymmetrical linear motion, approximate circular motion and elliptical motion. The form of the modified model is suitable for predicting instantaneous squeeze-film forces in the simulation of heat exchanger tube vibration. Further work using different sized components and fluid properties is required in order to finalize coefficient values.     

Multiple-time scale approach to stress growth in suspensions of rodlike macromolecules

The rheologial properties of a dilute suspension of rigid rodlike macromolecules at the inception of a steady homogeneous flow are examined within the framework of the kinetic theory of polymeric liquids. The cases of simple shear and elongational flows are considered under the assumption of small non-dimensional shear rates, ϵ = κ₀/6D, where D is the rotation diffusion of the particles. The method of multiple-time scales is used to solve the problem and the resulting transport coefficients are shown to reveal new features as compared with the results of the straightforward perturbation technique. In particular, the onset of stress oscillations leading to an undershoot following the overshoot of the shear viscosity and the overshoot of the normal stress difference are detected as ϵ approaches unity.     

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.     

A novel geometry for rheological characterization of viscoelastic materials

A novel geometry for generating a viscometric flow presents advantages of both cone and plate and parallel plate geometries, regarding uniform shear field and adjustable range of measurement. Kinematics and dynamics of the generated flow have been described mathematically utilizing an orthogonal curvilinear coordinate system based on the shapes of the shearing surfaces which are similar to the surface that generates the flow. Simple equations that allow the calculation of quantities of experimental interest in the rheological characterization of liquid materials, namely, shear rate, shear stress and two normal stress differences, have also been derived.The geometry, called pseudosphere, was tested with two types of fluids (Newtonian and pseudoplastic). Results show that the geometry can be used with low viscosity liquids (Newtonian liquids) by only adjusting the gapH. The behavior of pseudoplastic fluids for both low and moderately high viscosity could also be studied with this geometry. Very reproducible results were obtained when compared with those obtained with cone and plate geometry. Regions of lower shear rate could be studied using only the pseudosphere geometry.     

Dynamics of a Taylor bubble in steady and pulsatile co-current flow of Newtonian and shear-thinning liquids in a vertical tube

A computational analysis is carried out to ascertain the effects of steady and pulsatile co-current flow, on the dynamics of an air bubble rising in a vertical tube containing water or a solution of Carboxymethylcellulose (CMC) in water. The mass fraction (m_f) of CMC in the solution is varied in the range 0.1% ⩽ m_f ⩽ 1% to accommodate zero-shear dynamic viscosities in the range 0.009–2.99 Pa-s. It was found that the transient and time-averaged velocities of Taylor bubbles are independent of the bubble size under both steady as well as pulsatile co-current flows. The lengths of the Taylor bubbles under the Newtonian conditions are found to be consistently greater than the corresponding shear-thinning non-Newtonian conditions for any given zero-shear dynamic viscosity of the liquid. In contrast to observations in stagnant liquid columns, an increase in the dynamic viscosity of the liquid (under Newtonian conditions) results in a concomitant increase in the bubble velocity, for any given co-current liquid velocity. In shear-thinning liquids, the change in the bubble velocity with an increase in m_f is found to be relatively greater at higher co-current liquid velocities. During pulsatile shear-thinning flows, distinct ripples are observed to occur on the bubble surface at higher values of m_f, the locations of which remain stationary with reference to the tube for any given pulsatile flow frequency, while the bubble propagated upwards. In such a pulsatile shear-thinning flow, a localised increase in dynamic viscosity is accompanied near each ripple, which results in a localised re-circulation region inside the bubble, unlike a single re-circulation region that occurs in Newtonian liquids, or shear-thinning liquids with low values of m_f. It is also seen that as compared to frequency, the amplitude of pulsatile flow has a greater influence on the oscillating characteristics of the rising Taylor bubble. The amplitude of oscillation in the bubble velocity increases with an increase in the CMC mass fraction, for any given value of pulsatile flow amplitude.     

On stagnation pressure increases in calorically perfect,ideal gases

When stagnation pressure rises in a natural or numerically simulated flow it is frequently a cause for concern, as one usually expects viscosity and turbulence to cause stagnation pressure to decrease. In fact, if stagnation pressure increases, one may suspect measurement or numerical errors. However, this need not be the case, as the laws of nature do not require that stagnation pressure continually decreases. In order to help clarify matters, the objective of this work is to understand the conditions under which stagnation pressure will rise in the unsteady/steady flows of compressible, viscous, calorically perfect, ideal gases. Furthermore, at a more practical level, the goal is to understand the conditions under which stagnation pressure will increase in flows simulated with the Reynolds averaged Navier–Stokes equations and eddy-viscosity turbulence models. In order to provide an improved understanding of increases in stagnation pressure for both these scenarios, transport equations are derived that govern its behavior in the unaveraged and Reynolds averaged settings. These equations are utilized to precisely determine the relationship between changes in stagnation pressure and zeroth, first, and second derivatives of fundamental flow quantities. Furthermore, these equations are utilized to demonstrate the relationship between changes in stagnation pressure and fundamental non-dimensional quantities that govern the conductivity, viscosity, and compressibility of the flow. In addition, based on an analysis of the Reynolds averaged equation (for eddy-viscosity turbulence models), it is shown that stagnation pressure is particularly likely to experience a spurious rise at the outer edges of shear layers that are undergoing convex curvature. Thereafter, numerical experiments are performed which confirm the primary aspects of the theoretical analysis.     

One-step,in situ jamming point measurements by immobilization cell rheometry

Dense colloidal suspensions are processed in a wide variety of industries. Challenges for pumping suspensions and slurries at high concentrations include shear thickening and dilation, which can have deleterious consequences. These systems are shear sensitive close to the jamming point, meaning that a significant increase in high shear viscosity can be observed with just a few percent change in volume fractions. Therefore, accurate and rapid determination of the jamming point can greatly aid formulation. Typically, conventional rheometry identifies the jamming point by a time-consuming process, whereby multiple flow curves of suspensions of different volume fraction are measured and extrapolated to the volume fraction where the viscosity diverges. We present an alternative approach for rapid, one-step, experimental determination of the jamming point for aqueous suspensions. The procedure monitors the shear stress under constant shear stress or shear rate as the sample is dewatered using immobilization cell rheometry, until the viscosity diverges. The method is validated by comparing the results of this work with conventional rheometry for a model suspension. Then it is applied to examine the effect of grafting a short-chain polymer to particles, comprising an industrial suspension of silica-coated titania. Polymeric coating of the particles increases the jamming concentration and mitigates shear thickening, qualitatively consistent with predictions from simulations.

A new method is designed to extract the jamming point of a suspension. The procedure monitors the effective viscosity under prescribed shear conditions as the suspension is dewatered using immobilization cell rheometry. The geometry moves down to accommodate solvent evaporation, until the viscosity diverges, and the jamming point is reached.

     

Elastic and inertial effects in pulsatile flow of polymeric liquids in circular tubes

In the earlier theoretical work of Barnes, Townsend, and Walters on pulsatile flow of viscoelastic fluids, it was found that an Oldroyd model predicted that flow enhancement should decrease with increasing frequency of pulsation, whereas their experimental data showed an increase. Their calculations have been repeated for the Goddard—Miller model using two different expressions for the relaxation modulus; it is found that the Goddard—Miller model leads to the same discrepancy between theory and experiment. It is tentatively concluded that secondary-flow effects are playing an important role. The role of the elastic and inertial effects and the role of the shape of the viscosity vs. shear rate curve are discussed.     

Operability windows in viscoelastic slot coating flows using a simplified viscoelastic-capillary model

The viscoelastic-capillary model to predict approximately coating windows for the stable operations of viscoelastic coating liquids is derived using a lubrication approximation in slot coating processes. Pressure distributions and velocity profiles for viscoelastic liquids based on the Oldroyd-B and Phan-Thien and Tanner (PTT) models are solved in the coating bead region considering the Couette-Poiseuille flow feature and the pressure jumps at upstream and downstream menisci. Practical operating limits for the uniform coating of rheologically different liquids that are free from leaking and bead break-up defects are constructed under various conditions, incorporating the position of the upstream meniscus as an important indicator while determining limits. The shift of the uniform operating range shows different patterns for the Oldroyd-B liquid with a constant shear viscosity and the PTT liquid with a shear-thinning nature in comparison with the Newtonian case. The windows predicted by the simplified model are corroborated with experimental observations for one Newtonian and two viscoelastic liquids.     

Shock wave emission during the collapse of cavitation bubbles

Shock wave emission induced by intense laser pulses is investigated experimentally. The present work focuses on the conditions of shock wave emission in glycerine and distilled water during the first bubble collapse. Experimental investigations are carried out in liquids as a function of temperature and viscosity. Comparison is made with the theoretical work of Poritsky (Proc 1st US Natl Congress Appl Mech 813–821, 1952) and Brennen (Cavitation and bubble dynamics, Oxford University Press 1995). To the best knowledge of the authors, this is the first experimental verification of those theories.     

A computational study of some rheological influences on the “splashing experiment”

In various attempts to relate the behaviour of highly-elastic liquids in complex flows to their rheometrical behaviour, obvious candidates for study have been the variation of shear viscosity with shear rate, the two normal stress differences N₁ and N₂, especially N₁, the extensional viscosity, and the dynamic moduli G′ and G″. In this paper, we shall confine attention to ‘constant-viscosity’ Boger fluids, and, accordingly, we shall limit attention to N₁, η_E, G′ and G″.We shall concentrate on the “splashing” problem (particularly that which arises when a liquid drop falls onto the free surface of the same liquid). Modern numerical techniques are employed to provide the theoretical predictions. We show that high η_E can certainly reduce the height of the so-called Worthington jet, thus confirming earlier suggestions, but other rheometrical influences (steady and transient) can also have a role to play and the overall picture may not be as clear as it was once envisaged. We argue that this is due in the main to the fact that splashing is a manifestly unsteady flow. To confirm this proposition, we obtain numerical simulations for the linear Jeffreys model.