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She flapped her legs like a wind-up doll the entire time. I tugged her shirt over her head. She screamed. I tried to yank her jacket on, and she went no-bones, melting onto the floor. One of her flailing legs connected with my face. The pain was raw and white and I snapped. This is where I need to take a break and tell you that my parents, for all their foibles and deep dysfunction, had never so much as yelled at me, forget spanking or hitting. I was raised to be an organic granola pacifist, someone whose go-to in times of conflict and stress has always been research followed by earnest communication.
The idea of striking a child was as foreign and abhorrent to me as cutting off my own finger. Hand still in the air, I fled. I jumped into my car. I started it.
I raced out of that driveway, the snowdrifts a sun-blocking wall of white on each side. My eyes were dry. It took just past the end of the driveway for my prefrontal lobe to calm the animal in me. My daughter was three years old and alone in our house. She was frightened of the dark and the entire basement, would grab my hand with her chubby fingers when strangers talked to her, was as defenseless as a newborn fawn.
My fear bowed to nausea. I tried to turn the car around, but the snow was too high, only a single lane plowed on my back country road. I had to drive two more icy miles before there was enough space to change direction, and by then, I was sobbing so hard that I was choking.
It had been ringed with terror. I pulled into the driveway and leapt out of the car without turning it off. Potty-trained for well over a year, she had wet herself in fear. The dark stain flowered on the front of her elastic-waisted jeans. A puddle had formed underneath her.
She was staring at the ceiling, shuddering. I picked her up. I held her until she stopped shaking and the sobbing came, that heaving gale of the shattered child. I apologized, but I knew there would never be enough sorries. I cleaned her up, me up. I expected him to take her away from me, for daycare to call the authorities. They would have been well within their rights. I know I'm not alone. There are many of us who need to reprocess our garbage, but who can't bear the idea of writing memoir, whether it's because we are too close to the trauma , don't want to hurt or be hurt by those we're writing about, or simply prefer the vehicle of fiction.
I kept up writing May Day , rubbing it like a worrystone, afraid to relapse into that gaping darkness where I was the monster. The answer depends very much on the particular field of interest. Here we shall discuss what we would anticipate for a number of industrial situations, but the same would be true for academic studies in the same general area: Areas of interest Personal products and cosmetics Paints, coatings and inks Polymers Lubricants Detergents Pharmaceuticals Food and biotechnology Minerals Types of liquid creams, lotions, pastes suspensions solutions, melts oils, greases suspensions, pastes dispersions, lotions, creams dispersions, emulsions, pastes slurries, suspensions This is not an exhaustive list for which see Whorlow's book  , but liquids found in any other areas fall somewhere within those examples found in this list.
Obviously in each area, certain specific items of test equipment will be used, such as small extruders, paint rigs, Stevens texture testers and penetrometers, etc. However, here we are interested in those pieces of rheological equipment that we should also find. The following important features are generally measured in rheological studies for typical liquids listed above, with some specific examples: 51 A Handbook of Elementary Rheology Product Property Examples general 'thickness' - consumer-perceived flow properties suspendability - long-term physical stability technical attributes - slow draining microstructural characteristic - state of flocculation Processing Examples machineability - 'fly' or misting in ink or coatings machines process engineering - heat transfer, pump specification, mixing efficiency 8.
The following are some general classes of rheometer: 8. Exercise: Find the Internet websites of the most prominent viscometer and rheometer manufacturers, viz. Table 1: Commercial rheometers available today and their available functions. References  Whorlow, R. When we do this over a wide-enough range of either shear rate or shear stress, we generally see the kind of behaviour shown in figures , when, as described in chapter 4, we plot the results on logarithmic axes.
At low-enough shear rates or shear stresses, the viscosity is constant with a value ijo , but at some point it begins to decrease, and usually enters a straight-line region on a logarithmic plot, which indicates power-law behaviour, see figure 9. This decrease of viscosity with shear rate is called shear thinning and must be distinguished from a decrease of viscosity with time of shearing which is called thixotropy, see later. Figure 9: Definition diagram of the various models and the ranges that they cover.
This is particularly true for polymer melts and shampoos. On the other hand, if the shear-rate range is too high, then the 56 Chapter 9: Shear- thinning liquids lower shear-rate behaviour is not seen, then the kind of behaviour shown in figures 14 and 15 is observed. This is often seen for dispersions and emulsions measured in typical laboratory viscometers. Lastly, the situation sometimes arises where both the lower and higher shear rate behaviour is difficult to see, and then only the power-law region is seen. This is shown to be the approximate behaviour seen in figure This is fairly typical of most situations where the power-law is forced to fit the data, when careful scrutiny— try looking along the curve from the side— will show that there is always some curvature.
The following are some examples of just some of the simpler forms of equations which fit different parts of the flow curve, see figure 17 for an overall picture. Figure Definition diagram of parts of the flow curve. When this model is used to describe non-Newtonian liquids, the degree of shear thinning is dictated by the value of m, with m tending to zero describes more Newtonian liquids, while the most shear-thinning liquids have a value of m tending to unity. If we make various simplifying assumptions, it is not difficult to show that the Cross equation can be reduced to Sisko, power-law and Newtonian behaviour, see below.
There is another Cross-like model which uses the stress rather than the shear rate as the independent variable, it has been called the Ellis or sometimes the Meter model, and for some specific values of the exponent, it has been gives other names: for an exponent of unity it has been called the Williamson or Dougherty and Krieger model, while for an exponent of two it has been called the Philippoff model, etc. The Carreau model is very similar to the Cross model, but with the whole of the bottom line within brackets, i. Figure 18 shows the typical form of the results that would normally have been measured and plotted graphically over the last 50 years.
Over a reasonable range of shear rates, the shear stress seemed to be a linear function of shear rate, but now displaced upwards by a constant value, which is called the yield stress. This was found by extrapolation to where the shear rate was zero. This seemed to show that there would be no flow at all unless the stress was higher than this critical value. Shear rate, y Figure Definition diagram of the Bingham model. In the power-law model, the consistency k has the strange units of Pa.
This law alone is sufficient to describe many non-Newtonian flows, and the use of it is described in chapter 10 with many examples, which are analogous the Newtonian examples shown above. The power-law description does well for most structured liquids from shear rates around 1 to 10 3 s 1 or so, but at about 10 3 s 1 , there is usually some flattening. However, for engineering calculations, power-law predictions are quite reasonable if limited to the medium range. This is called the Sisko equation, and it is very good at describing the flow behaviour of most emulsions and suspensions in the practical everyday shear rate range of 0.
Exercise: In figure 19, which models would you attempt to fit to each the flow curves shown? First we plot the data on a linear basis to see if it fits the Bingham equation. Then the values of cr 0 and rj p can be used to predict flows of the liquid in other geometries, see chapter If the linear plot shows curvature, the data should be plotted on a logarithmic basis. From this plot we can see if we have a reasonable straight line, or a straight-line and some curvature. If a fair straight line is seen, then the data can be submitted for power-law regression and the k and n extracted and used for prediction.
Last, if there is considerable curvature, then the Sisko model, or if necessary the complete Cross equation is indicated. However, here we have a problem, because the equation is non- linear and simple regression analysis is inadequate. However, most modern viscometers have suitable software for this purpose, otherwise other specialised commercial mathematical software.
Chapter 10 will show how these simple non-Newtonian equations — Bingham, power-law and Sisko — can be used to describe the flow of liquids in various geometries. However a note of warning must be sounded — the equations should only be used to predict behaviour within the range of the measurements taken to produce the various fitting parameters. Then the range of shear rates in both situations — the viscometer and the pipe— should be approximately in the same range of decades, say 10 - s 1.
These are generally power-law equations, but some other flow laws are also included. For most engineering applications the power law description is adequate if the parameters for the flow law relate either to the local point where the calculation is being made, or generally over the range being considered, see chapter 8, see also [1- 3].
If thevalueof n only changes slowly over a range of flow rates, then this approach can be used to deal piecewise with thewholeflow-curve, even though it does not strictly follow power-law behaviour over a wide range of shear rates. It is very good for describing the flow of most emulsions and dispersions through pipes. Workers who handle molten chocolate favour the Casson model; blood is also bel i eved to conform to thi s model. In most practical cases where slip layers are observed, they are usually less than one mi cron in thickness, but because the viscosity is so low, they can still dominate the flow.
We can invert these expressions so that given a known shear stress and a measured viscosity at that stress, the velocity can be calculated. This approach assumes that the emptyi ng i s si ow enough to I eave no materi al on the wal I s of the vessel. F,V Figure 1 : The parallel-plate plastometer. The maximum stress a m in that situation is given by 87i 3. Kotomin, Moscow, personal communication, Sept. References  Bird, R. It was usually thought that at stresses below the yield stress no flow takes place, and only elastic behaviour is seen.
This is in fact only half the story, since indeed although there are such materials which appear to show this kind of behaviour, in reality there is as much happening in terms of flow below as above the 'yield stress'. Figures see  for details show a number of examples of such liquids, where the viscosity falls many orders of magnitude over a narrow range of shear stress, and indeed when approaching this critical stress region from regions of high stress it appears that the viscosity goes to infinity at a certain minimum stress.
However, careful and patient measurement below this stress shows that the viscosity is still finite, and eventually levels off to a constant, but very high value at low stress. Figure 1: Flow curve of molten chocolate measured in a vane geometry to eliminate slip. We finally comment on how to make proper use of the mathematical constant that is often called the 'yield stress', but which has no physical reality, see Barnes  for fuller details. Figure 7: Flow curve of a flocculated ink. Figure 8: Flow curve of a toothpaste. A typical dictionary definition of the verb 'to yield' would be 'to give way under the action of force' and this implies an abrupt and extreme change in behaviour to a I ess resi stant state.
The yi el d stress of a sol i d materi al , say a metal I i ke copper, i s the point at which, when the applied stress is increased, it first shows liquid-like behaviour in that it continues to deform for no further increase in stress. Similarly, 72 Chapter Very shear-thinning or 'yield-stress' liquids for a liquid, one way of describing its yield stress is the point at which, when decreasing the applied stress; it first appears to show solid-like behaviour, in that it does not continue to deform.
Anyone testing a very shear-thinning liquid e. However, measurement of the complete flow curve— using the appropriate equipment— shows that no real yield stress exists. When plotted as, say, the logarithm of viscosity against the logarithm of applied stress, the phenomenon is much better illustrated. Within the limited range of measurable shear rates, it looked as though the viscosity of such a 'yield-stress' liquid was increasing asymptotically as the applied shear stress was decreased.
Indeed, for all practical purposes it looked as though there was a definite stress where the viscosity did become infinite. These materials can easily show a million-fold drop in viscosity over a very small region of increasing shear stress, often around or even I ess than a decade, see again figures 1- 10 .
The liquids that appear to have a yield stress are legion. Among them are many original examples noted early in the century such as clay, oil paint, toothpaste, drilling mud, molten chocolate, etc. Nowadays such disparate systems as ceramic pastes, electro- viscous fluids, thixotropic paints, heavy-duty washing liquids, surface-scouring liquids, mayonnaise, yoghurts, purees, liquid pesticides, bio-mass broths, blood, water-coal mixtures, molten liquid-crystalline polymers, plastic explosives, foams, battery and rocket propellant pastes, etc.
Having, for instance, emptied a vat of such a liquid product, we could then calculate, at least in the short-to-medium term when the liquid does not appear to flow, the thickness of the layer left on the wall of the vessel, which has to be removed in any ensuing cleaning operations. This has opened up a new range of previously unobtainable flow behaviour for structured liquids that seemed to have a yield stress.
Measurement in these ultra-low shear-rate regions is now called creep testing, by analogy with the testing of solids under similar low-deformation-rate, long-time conditions; albeit solids creep testing is usually performed in extension rather than in shear. The reason why these types of rheometers lend themselves to measurement of the whole curve is as follows: the complete range of shear rates needed to encompass the whole range of flow can be very large, say 10 10 decades.
However, the concurrent range of shear stress for the same conditi ons can be as I ow as 10 3. The ability to apply this range of stress i n a controlled fashion viathecoupleT applied to a typical geometry is relatively easy compared to generating the equivalent range of shear rates that would need to be applied.
M easuring a wide range of shear rates is much easier than generating them and this measurement is done very accurately these days using optical discs. Figure The vane and the vane -and-basket geometries Barnes  went further than simply using a vanewhen heintroduced a close- fitting wire-mesh cylinder inside the outer containing cylinder to prevent slip there also, which can often occur, but is usually ignored, see figure Of course, use of a vane geometry at very high rotation rates is precluded due to secondary flow developing behind the vanes.
This ensures that slip iscompletely overcome at the rotating member.
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Then, if these equations are used to predict flow in any other situation, where the same general range of shear rates applies, this iscompletely acceptable and un- controversial, and can indeed be very effective. The most popular equations that have been used to describe liquids with yield stresses are the Bingham, Casson and Herschel-Bulkley sometimes called the generalised Bingham models, i.
This behaviour isthen very similar to figures 1 to 10, wherewenotethelargedrop in viscosity for only a moderate increase in stress for all the results shown. However, all solids when stressed, while certainly deforming elastically on a time-scale of less than a second, consequently creep i. If we replace the word 'creep' with 'flow', we have moved smoothly from material science into rheology, and we can happily talk about the viscosity of solids, because, as wehaveseen so far, viscosity determines flow-rate.
Arnstein and Reiner  showed that 'solidified' cements have viscosities of 10 16 - 10 17 Pa. He quoted earlier measurements of the viscosities of various English cheeses in the range 10 10 - 10 11 Pa. H e even measured M adeira cake and came out with viscosities of around 10 9 Pa. Similar creep measurements were made on various kinds of solid soap by Pacor et al, who found values of creep viscosity— depending on formulation and moisture content- in the range 5- x 10 11 Pa.
In these systems, weeither have materials in a super-cooled, glassy state, and hence their flow is the same as that of other liquids but is very slow, or else 'solid' polycrystalline materials where flow is usually via the integrated effect of atomic 'jumping' movements al ong grain boundaries. Above the power-law region, other phenomena lead to different kinds of creep behaviour, but among them is a situation where the creep rate often quite large by now tends again towards a linear fundi on of applied stress— the so-called drag-controlled plasticity— i. If we express this response as would normally be done for a strudured liquid, the identical behaviour would be as shown in figures 1- 9 of chapter The creep results of most solid materials, in the customarily observed linear followed by power-law regions, can be fairly well described by the empirical Ellis equation omitting the usual high-shear-stress asymptotic viscosity , viz.
The results can be expressed equally well using the Cross model, which is essentially the same as the Ellis model, but where the stress is replaced by the shear rate. Some further examples of this kind of behaviour are shown in figures 1 and 2. The data has been read off graphs in Frost and Ashby's book  and has been fitted to the Ellis model. Material Temp. Plasticine, which is considered to be a typical soft solid, also shows characteristic non-Newtonian, steady-state liquid behaviour as seen in figure 3, where the results of a creep test are compared to those obtained at the higher deformation rates experienced in extrusion through a short orifice.
Figure 3: Flow curve as viscosity versus shear stress for Plasticine at room temperature.
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One thing worth noting about the degree of 'shear-thinning' seen for sol i ds i s that if we compare the steepness of the shear-thi nni ng regi ons of the very shear-thinning yield-stress liquids shown in chapter 11, they are much 'steeper' than for the solids shown here. This is because their viscosity often arises from structures that are not space-filling as for solids , but are composed of tenuous arrangements such as chains and floes that collapse above a critical shear stress to a low viscosity, whilesolids are always spacefilling.
K; Lelievre,J; Macgibbon, A. H; Taylor, M. H; Papenhuijzen, J. P; Rheol. Acta, 9 3 , ,  Frost, H. J; Ashby, M. When these liquids are deformed, thermo- dynamic forces immediately begin to operate to restore this rest state, just like a stretched spring will always seek to return to its unextended length. Also like a spring, movement from the rest state represents a storage of energy, which manifests itself as an elastic force trying to reproduce the static status quo.
This kind of energy is the origin of elasticity in structured liquids. Initially, the restoring force increases linearly with the distance that any deformation takes the material away from its rest state, but eventually non-linearities will be encountered. The rate of increase of force with deformation then diminishes, until at very large deformations a steady-state situation arises and the elastic force becomes constant.
The Flow of Suspensions (Chapter 15)a Handbook of Elementary Rheology
By the time that these large elastic forces are evident, the microstructure has changed radically, and has become anisotropic. The elastic force manifests itself at small deformations as various elastic moduli, such as the storage modulus, and at steady state i. Alongside these elastic forces are the ever-present viscous forces due to the dissipation, and proportion to the rate not extent of deformation, so that together these produce viscoelastic i.
Most concentrated structured liquids shown strong viscoelastic effects at small deformations, and their measurement is very useful as a physical probe of the microstructure. However at large deformations such as steady-state flow, the manifestation of viscoelastic effects — even from those systems that show a large linear effects — can be quite different.
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Polymer melts show strong non-linear viscoelastic effects see chap. If the stresses and strains in deformations are relatively large, then many time effects are thixotropic in nature. The difference between the two situations — viscoelastic and thixotropic — is that in the linear viscoelastic region the microstructure responds over a certain time scale without changing, while in thixotropy the microstructure does change — by breaking down or building up — and 81 A Handbook of Elementary Rheology such changes take time.
In this chapter we shall deal only with linear viscoelasticity, and the resulting viscoelastic responses shown in various test situations. Before embarking on an investigation of liquids that show obvious elastic effects such as recoil and normal force, it is well to remember that all liquids show elastic effects at a short-enough time or a high-enough frequency.
Put in terms of a Maxwell relaxation time see next section , even what we would think of as simple Newtonian liquids have elastic properties see . Table 1 shows a number of such liquids compared to others that we would normally think of as elastic, giving the modulus that would describe their behaviour at times less than their relaxation time, which is also given.
This is the converse of what we found in chapter 12, where we saw that all solids flow, since here we are saying that all liquids show solid-like properties under appropriate, if extreme, conditions. These consist of combinations of linear elastic and viscous elements, i. They were vibrant. They were excited about life. They were open and available to share all sorts of things.
And so we went to the last program they ever put on, the last retreat they ever put on. We went to it, and my wife, Hannah, and I were kind of blown away. We studied with them for six years.
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