On the no-slip boundary conditions for squeeze flow of viscous fluids

A typical class of boundary conditions for squeeze flow problems in lubrication approximation is the one in which the squeezing rate and the width between the squeezing plates are constant. This hypothesis is justified by claiming that the plates moves so slowly that they can be considered static. In this short note we prove that this assumption leads to a contradiction and hence cannot be used.     

Yield stress of structured fluids measured by squeeze flow

Various structured fluids were placed between the parallel circular plates of a squeeze-flow rheometer and squeezed by a force F until the fluid thickness h was stationary. Fluid thickness down to a few microns could be measured. Most fluids showed two kinds of dependence of f on h according to an experimentally-determined thickness h ^*. If h > h ^* then F varied in proportion to h ⁻¹ as predicted by Scott (1931) for a fluid with a shear yield stress τ₀. The magnitude of τ₀ from squeeze-flow data in this region was compared with the yield stress measured by the vane method. For some fluids τ₀ measured by squeeze flow was less than the vane yield stress, suggesting that the yield stress of fluid in contact with the plates was less than the bulk yield stress. If h < h ^* then F varied approximately as h ^−5/2 and the squeeze-flow data in this region analysed with Scott's relationship gave a yield stress which increased as the fluid thickness decreased. This previously unreported effect may result from unconnected regions of large yield stress in the fluid of size similar to h ^* which are not sensed by the vane and which become effective in squeeze flow only when h < h ^*. Received: 13 December 1999/Accepted: 4 January 2000     

An investigation of squeeze flow as a viable technique for determining the yield stress

A systematic study of squeeze flow (SF) was performed on different concentrations of Carbopol with varying yield stresses. A sample of constant volume was placed between two parallel plates and a series of constant force steps applied, following the plate separation as a function of time. Precise rheological measurements of the model yield stress fluids were performed in addition to the well-controlled SF tests. These rheological measurements were used in conjunction with the SF equations to determine the time-dependent plate separation, allowing a direct comparison of theory and experiment throughout the entire test. The limiting height achieved during constant force SF reveals information about the yield stress of the fluid as predicted by the theory. It appears that by carefully controlling the experimental conditions of the squeeze test one can obtain yield stress values that agree with the rheological measurements within 10%. Additionally, the validity of the lubricational theory was tested; not only for the determination of the yield stress but throughout the flow as well.     

The yield surface of viscoelastic and plastic fluids in a vane viscometer

Vane viscometers are often used to investigate the low shear rate properties of plastic fluids. The shear stress is determined by assuming that the material is held in the space between the vane blades so that it behaves like a rigid cylinder. Experimental evidence supports this assumption and the aim of the present study is to model numerically the yield process in a vane rheometer using viscoelastic and plastic fluids. The finite element method has been used to model the behavior of Herschel-Bulkley (Bingham), Casson and viscoelastic (Maxwell type) fluids. The penalty function approach for the pressure approximation and a rotating reference frame are used together with fine meshes containing more than 1300 elements. The results show that for Herschel-Bulkley (Bingham), and Casson fluids a rotating rigid cylinder of fluid is trapped inside the periphery of the vane, the shear stress is uniformly distributed over the surface of the cylinder. Finally a modified second order fluid is used to simulate the viscoelastic behaviour, anticipated to be an intermediate between the elastic deformation and the plastic flow, to provide a more realistic simulation of the yield process about a vane. In this case, contrast with the concentration of the elastic strain rate at the blade tips, a nearly uniform distribution of the plastic shear rate is still found. This implies that the plastic shear always distributes uniformly during the entire yielding process. Evidently the assumption of uniform shear on a rotating cylinder of material occluded in the blades of a vane is a valid and useful model for many types of fluid possessing a yield stress.     

The constant flow rate problem for fluids with increasing yield stress in a pipe

Experiments show that the degradation effect observed during both stirring and pipelining tests of some coal-water slurries is mainly to be ascribed to the increase of the yield stress. Regardless of the particular mathematical model adopted to investigate the dynamics of these fluids, engineering applications force us to consider the problem of how long a constant flow rate can be maintained during the pipelining process. We choose a Bingham model where the yield stress is assumed to increase with the dissipated energy as in [5]. It is first shown that the constant flow rate problem is equivalent to solving a nonlinear functional equation in the unknown pressure gradient that generalizes the classical algebraic Buckingham equation for the same problem with constant rheological parameters. By means of a fixed-point argument we also prove that the functional equation has one and only one solution which is local in time. We finally find an estimate from below of the interval of the interval of existence. Numerical results are rather good and agree with those expected from the engineering point of view.     

Shear and squeeze rheometry of suspensions of magnetic polymerized chains

We present the first experimental results on the magnetorheology of suspensions of non-Brownian magnetic ellipsoidal particles. These particles are made of spherical iron particles linked by polymers and are called polymerized chains. Steady shear, oscillatory shear, and oscillatory squeeze rheological tests have been performed. The rheological properties of the suspension of polymerized chains have been compared with those of the suspension of spherical iron particles. In shear flow, both suspensions develop nearly the same yield stress, while in squeeze flow, the yield stress is several times higher for the suspension of polymerized chains. We show that the squeezing force of a suspension of spherical particles is an increasing function of the magnetic field intensity at low magnetic fields but decreases dramatically at higher fields. Surprisingly, this phenomenon, attributed to cavitation or air entrainment, does not occur in the suspension of polymerized chains.     

Potential flow of viscous fluids: Historical notes

In this essay I will attempt to identify the main events in the history of thought about irrotational flow of viscous fluids. I am of the opinion that when considering irrotational solutions of the Navier–Stokes equations it is never necessary and typically not useful to put the viscosity to zero. This observation runs counter to the idea frequently expressed that potential flow is a topic which is useful only for inviscid fluids; many people think that the notion of a viscous potential flow is an oxymoron. Incorrect statements like “… irrotational flow implies inviscid flow but not the other way around” can be found in popular textbooks.     

Non-Newtonian fluids with a yield stress

A modified Herschel–Bulkley model [E. Mitsoulis, S.S. Abdali, Flow simulation of Herschel–Bulkley fluids through extrusion dies, Can. J. Chem. Eng. 71 (1993) 147–160] predicts an infinite apparent viscosity at vanishing shear rate. Furthermore, the dimensions of one parameter depend on another parameter. In this contribution, we propose a generalized model based on earlier work by De Kee and Turcotte [D. De Kee, G. Turcotte, Viscosity of biomaterials. Chem. Eng. Commun. 6 (1980) 273–282] and on the work of Papanastasiou [T.C. Papanastasiou, Flows of materials with yield, J. Rheol. 31 (1987) 385–404] to solve the problems associated with the modified Herschel–Bulkley model. Compared to the responses of the Papanastasiou model and the modified Herschel–Bulkley model, the proposed generalized model provides the expected improvements and is capable of predicting successfully the rheological behavior (viscosity and yield stress) of Carbopol 980 dispersions.     

Discrete element study of viscous flow in magnetorheological fluids

Using discrete element simulations, we gain insight into the structure of a magnetorheological fluid (MRF) under shear. In simulations with flat walls, the particles arrange in chains, sheet-like structures, or columns along the magnetic field lines, depending on the strength of the applied external magnetic field. Corresponding to the structure formation, three different types of failure mechanisms can be identified. For the characterization of the different regimes, specific particle coordination numbers are introduced. The three structural regimes can be distinguished and described by means of these coordination numbers. To analyze the contact between the MRF particles and the walls of the shear cell, additional simulations with rough walls have been conducted. The resulting structure formation could be successfully classified by the introduced coordination numbers. Based on the analysis of the shear stress transmission both in the case of flat and rough walls, possibilities for shear stress enhancement for technological applications are discussed.     

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.     

Static yield stress of ferrofluid-based magnetorheological fluids

In this work, the yield stress of ferrofluid-based magnetorheological fluids (F-MRF) was investigated. The fluids are composed of a ferrofluid as the liquid carrier and micro-sized iron particles as magnetic particles. The physical and magnetorheological properties of the F-MRF have been investigated and compared with a commercial mineral oil-based MR fluid. With the addition of a ferrofluid, the anti-sedimentation property of the commercial MR fluids could be significantly improved. The static yield stress of the F-MRF samples with four different weight fractions (ϕ) of micro-sized iron particles were measured using three different testing modes under various magnetic fields. The effects of weight fraction, magnetic strength, and test mode on the yielding stress have been systematically studied. Finally, a scaling relation, , was proposed for the yield stress modeling of the F-MRF system.     

A new yield stress scaling function for electrorheological fluids

A new scaling function capable of modeling the yield stress behavior of electrorheological (ER) fluids through the full range of electric fields is proposed. In spite of its simple form, a comparison of the model predictions with experimental data for both ac and dc fields and the polarization model shows that the proposed model correctly predicts the yield stress behavior both quantitatively as well as qualitatively.     

Multi-layer channel flows with yield stress fluids

We present results of a computational study of visco-plastically lubricated plane channel multi-layer flows, in which the yield stress fluid layers are unyielded at the interface. We demonstrate that symmetric 3-layer flows may be established for wide ranges of viscosity ratio (m), Bingham number (B) and interface position (y_i), for Reynolds numbers Re ≤ 100. Here an inner Newtonian layer is sandwiched between 2 layers of Bingham fluid. Results are presented illustrating the variation of development length with the main dimensionless parameters and for different inlet sizes. We also show that these flows may be initiated by injecting either fluid into a steady flow of the other fluid. The flows are established quicker when the core fluid is injected into a channel already full of the outer fluid. In situations where the inner fluid flow rate is dominant we observed inertial symmetry breaking in the symmetric start-up flows as Re was increased. Asymmetry is also observed in studying temporal nonlinear stability of these flows, which appear stable up to moderate Re and significant amplitudes. In general the flows destabilize at lower Re and perturbation amplitudes than do the analogous core-annular pipe flows, but 1–1 comparison is hard. When the flow is stable the decay characteristics are very similar to those of the pipe flows. In the final part of the paper we explore more exotic flow effects. We show how flow control could be used to position layers asymmetrically within the flow, and how this effect might be varied transiently. We demonstrate that more complex layered flows can be stably achieved, e.g. a 7-layered flow is established. We also show how a varying inlet position can be used to “write” in the yield stress fluid: complex structures that are advected with the flow and encapsulated within the unyielded fluid.     

A modern look on yield stress fluids

A concept of viscoplasticity advanced exactly one century ago by Bingham appears very fruitful because there are many natural and artificial materials that demonstrate viscoplastic behavior, i.e., they are able to pass from a solid to a liquid state under the influence of applied stress. However, although this transition was originally considered as a jump-like phenomenon occurring at a certain stress—the yield stress—numerous subsequent studies have shown that the real situation is more complicated. A long-term discussion about the possibility of flow at low stresses less than the yield stress came to today’s conclusion denying this possibility as being opposite to the existence of the maximal Newtonian viscosity in viscoelastic polymeric fluids. So, there is a contradiction between the central dogma of rheology which says that “everything flows” and the alleged impossibility for flow at a solid-like state of viscoplastic fluids. Then, the concept of the fragile destruction of an inner structure responsible for a solid-like state at the definite (yield) stress was replaced by an understanding of the yielding as a transition extending over some stress range and occurring in time. So, instead of the yield stress, yielding is characterized by the dependence of durability (or time-to-break) on the applied stress. In this review, experimental facts and the new understanding of yielding as a kinetic process are discussed. Besides, some other alternative methods for measuring the yield stress are considered.     

On the determination flow rates for the compressible and immiscible viscous filtration fluids

The object of this paper is to treat the mathematical problem of the determination of flow rates for compressible and immiscible viscous filtration filtration with a free surface.     

Heat transfer in flow of nonlinear fluids with viscous dissipation

The rotational flow of viscoplastic fluids between concentric cylinders is examined while dissipation due to viscous effects through the energy balance. The viscosity of fluid is simultaneously dependent on shear rate and temperature. Exponential dependence of viscosity on temperature is modeled through Nahme law, and the shear dependency is modeled according to the Carreau equation. Hydrodynamically, stick boundary conditions are applied, and thermally, both constant temperature and constant heat flux on the exterior of cylinders are considered. The governing motion and energy balance equations are coupled adding complexity to the already highly correlated set of differential equations. Introduction of Nahme number has resulted in a nonlinear base flow between the cylinders. As well, the condition of constant heat flux has moved the point of maximum temperature toward the inner cylinder. Taking viscous heating into account, the effects of parameters such as Nahme and Brinkman numbers, material time and pseudoplasticity constant on the stability of the flow are investigated. Moreover, the study shows that the total entropy generation number decreases as the fluid elasticity increases. It, however, increases with increasing Nahme and Brinkman numbers.     

Interaction of viscous drops in a yield stress material

The interaction of Newtonian drops of various volumes moving in yield stress material is investigated experimentally. Tetrachloroethylene drops move in a rectangular reservoir filled with neutralized 0.07% w/w Carbopol gel under the action of gravity. For initially vertically aligned drop pairs, we present time evolution of separation distance, velocities during the interaction and conditions for coalescence of the drops, which depend on the volumes of the drops and the initial separation between them. For the asymmetric interaction of the pairs, we present interaction patterns, which have been used for estimation of the size of the yielded region and its shape around the leading drop.