Time-dependent fluid behaviour

Time-dependent fluid behaviour may be further sub-divided into two categories thixotropy and rheopexy or negative thixotropy. 

Figure 1.11 Schematic shear stress-shear rate behaviour for time-dependent fluid behaviour...

The above brief discussion of time-dependent fluid behaviour provides an introduction to the topic, but Mewis [1979] and Barnes [1997] have given detailed accounts of recent developments in the field. Govier and Aziz [1982], moreover, have focused on the practical aspects of the flow of time-dependent fluids in pipes. 

At high particle concentrations, slurries are often non-Newtonian. For non-Newtonian fluids, the relationship between the shear stress and shear rate, which describes the rheology of the slurry, is not linear and/or a certain minimum stress is required before flow begins. The power-law, Bingham plastic and Herschel-Bulkley models are various models used to describe the flow behaviour of slurries in which these other types of relationships between the shear stress and shear rate exist. Although less common, some slurries also display time-dependent flow behaviour. In these cases, the shear stress can decrease with time when the shear rate is maintained constant (thixotropic fluid) or can increase with time when the shear rate is maintained constant (rheopectic fluid). Milk is an example of a non-settling slurry which behaves as a thixotropic liquid. 

Analytic teclmiques often use a time-dependent generalization of Landau-Ginzburg ffee-energy fiinctionals. The different universal dynamic behaviours have been classified by Hohenberg and Halperin [94]. In the simple example of a binary fluid (model B) the concentration difference can be used as an order parameter m.. A gradient in the local chemical potential p(r) = 8F/ m(r) gives rise to a current j... 

When the apparent viscosity is a function of the shear rate, the behaviour is said to he shear-dependenf, when it is a function of the duration of shearing at a particular rate, it is referred to as time-dependent. Any shear-dependent fluid must to some extent be time-dependent because, if the shear rate is suddenly changed, the apparent viscosity does not alter instantaneously, but gradually moves towards its new value. In many eases, however, the time-scale for the flow process may be sufficiently long for the effects of time-dependence to be negligible. 

For a Newtonian fluid, the shear stress is proportional to the shear rate, the constant of proportionality being the coefficient of viscosity. The viscosity is a property of the material and, at a given temperature and pressure, is constant. Non-Newtonian fluids exhibit departures from this type of behaviour. The relationship between the shear stress and the shear rate can be determined using a viscometer as described in Chapter 3. There are three main categories of departure from Newtonian behaviour behaviour that is independent of time but the fluid exhibits an apparent viscosity that varies as the shear rate is changed behaviour in which the apparent viscosity changes with time even if the shear rate is kept constant and a type of behaviour that is intermediate between purely liquid-like and purely solid-like. These are known as time-independent, time-dependent, and viscoelastic behaviour respectively. Many materials display a combination of these types of behaviour. 

The second category, time-dependent behaviour, is common but difficult to deal with. The best known type is the thixotropic fluid, the characteristic of which is that when sheared at a constant rate (or at a constant shear stress) the apparent viscosity decreases with the duration of shearing. Figure 1.21 shows the type of flow curve that is found. The apparent viscosity continues to fall during shearing so that if measurements are made for a series of increasing shear rates and then the series is reversed, a hysteresis loop is observed. On repeating the measurements, similar behaviour is seen but at lower values of shear stress because the apparent viscosity continues to fall. 

In order to proceed with the evaluation of the time-dependent Poisson ratio v(0, both sets of relaxation behaviour are required. Now from Chapter 2 we know the Poisson ratio is the ratio of the contractile to the tensile strain and that for an incompressible fluid the Poisson ratio v = 0.5. Suppose we were able to apply a step deformation as we did for a shear stress relaxation experiment. The derivation then follows the same course as that to Equation (4.69) ... 

To complete this chapter, we would like to mention that recent monographs have reviewed the use of in-situ spectroscopies for monitoring heterogeneously catalysed reaction under supercritical conditions, although very few studies in this field has been devoted to the study of the fluid-solid interface.182 The use of a multi-technique approach in order to maximise information under real, in-situ conditions has also been reviewed recently.183 The combined use of powerful spectroscopies with simultaneous on-line analysis of the catalytic activity of the sample will become more widespread in application allowing an interpretation of catalytic behaviour in terms of the physico-chemical properties of the solid. The next frontier in spectroscopic characterisation of metal catalysts will consist of time-dependent analysis of the gas/liquid-solid interface, particularly with a view to analyse short-lived intermediates during catalysed reactions and with the aim to determine the response of the catalyst surface and relate these responses to the physico-chemical properties of the solid. 

Abstract Chalk is the constituent material of numerous oil reservoirs in North Sea. The mechanical behaviour of a saturated chalk has been largely studied. However, different aspects of its behaviour are not yet well understood material characteristics depend on the saturating fluids and chalk response is time-dependent. This paper proposes the PASACHALK numerical model an elasto-plastic constitutive law is presented, which reproduces the different plastic mechanisms of the chalk (pore collapse and shear failure) and the influence of pore fluids. The water sensitivity of this soft rock is explained by the existence of suction effects in chalk. Finally, a simulation of a hypothetical reservoir is proposed to show the response of the elasto-plastic model during depletion phase and water injection phase. 

Porous silicas are usually mesoporous materials and they can be made with a variety of pore dimensions. In particular, silica glasses can be made with well-defined pore diameters, typically in the range 30-250 A, using sol-gel methods. Such a system provides a good model for testing the models of relaxation behaviour of fluids in porous solids. It is normally found that the two-site fast-exchange model for relaxation described above for macroporous systems is still valid. For instance, H and relaxation times have been measured during both adsorption and desorption of water in a porous silica. Despite hysteresis in the observed adsorption isotherms, it was found that the relaxation times depended solely on water content.For deuterated water in some porous silicas, multicomponent relaxation behaviour for T2 and Tip has been observed, and this has been attributed to the fractal nature of the pore structure. 

At the other extreme, in the Newtonian fluid the shearing stress is proportional to the rate of shear, equation (1.1). Many materials show both elastic and viscous effects under appropriate circumstances. In the absence of the time-dependent behaviour mentioned in the preceding section, the material is said to be visco-elastic. Perfectly elastic deformation and perfectly viscous... 

For all fluids, the nature of the flow is governed by the relative importance of the viscous and the inertial forces. For Newtonian fluids, the balance between these forces is characterised by the value of the Reynolds munber. The generally accepted value of the Reynolds number above which stable laminar flow no longer occms is 2100 for Newtonian fluids. For time-independent fluids, the critical value of the Reynolds number depends upon the type and the degree of non-Newtonian behaviour. For power-law fluids (n = n ), the criterion of Ryan and Johnson [1959] can be used. 

There is a further group of fluids which show time-dependent behaviour. The viscosity of these materials is affected by the amount of shearing applied in the recent past to the material. Often the viscosity will decrease with time during shear but recover, sometimes slowly, when the shear stress is removed. This particular behaviour is termed thixotropic and some paints have this property. 

In this sense, the time dependence of the behaviour of the polymeric fluid system is defined as a function of the value of the shear rate used during the experiment. At high shear rates the typical time dependence of the shear stress has the form depicted in Figure 3.251 [714]. 

Note that equation 18.10 in chapter 18 essentially applies to settling at low particle concentrations (below 0.5% by volume) in Newtonian liquids, which have a constant viscosity. In principle, it can be also used in non-Newtonian fluids where viscosity /x then becomes the apparent viscosity but, depending on the type of the non-Newtonian behaviour (= model), its determination may require an iterative procedure. Not only is such behaviour shear-dependent (i.e. the apparent viscosity depends on how fast is the particle settling) but it may also be time-dependent and the model may contain a zero shear viscosity as a parameter. Ref. 4 reviews the state of the art to 1993 and research is still in progress, for example, on particle settling in the Carreau model fluids (e.g. polymeric liquids). ... 

We recall that phase transition is a typical viscoelastic expression 14-17-23-243 of the fluid s compressional behaviour. It is time dependent sample history, sollicitation time and observation time are parameters of importance to describe the apparition of a glassy phase. In our case, the ultrasonic period is less than 1 ps, the observation time longer than 10 s and the sample history corresponds to isothermal compression. 

It is necessary that time-dependent variation of fluid film formation and frictional behaviour in the present knee prosthesis model should be estimated by the analytical study on transient elastohydrodynamic lubrication (3,14), The effect of rolling motion on fluid film formation during walking motion should be also examined. 

For an electrostatically activated micromembrane pump fabricated by bulk micro-machining transient measurements of the pump chamber pressure and the inlet and outlet pressures were made. From these values also the time dependent flow rate can be deduced. A complex dynamic behaviour can be observed, with a low frequency oscillation of the maximum pressure. Measurements can be compared and predicted with the results of an simulation tool PUSI for micropumps connected to a peripheral fluid system. 

In more generalised flows, both the stress and the rate of deformation (strain rate) are tensor quantities, and the constitutive relationship between these may be very complex (Schowalter, 1978 Bird et a/., 1987a). The relationship between stress and shear rate frequently depends on the shear rate (flow rate), as is the case in a simple shear thinning fluid. However, for some fluids the stress may depend on the strain itself as well as on the rate of strain, and such fluids show some elasticity or memory behaviour, in that their stress at a given time depends on the recent strain history such fluids are... 

At higher shear rates, three types of deviations are observable when compared to ideal Newtonian flow (see Fig. 2.1).The first kind of deviation relates to the existence of a flow threshold (yield point). In the case of a Bingham fluid, flow occurs only when the yield stress is exceeded. The second type of deviation is shear thickening, observed where the viscosity increases with shear rate. This is the case for a dilating fluid, behaviour which is seldom apparent in polymers. Last, where viscosity decreases with increase in shear rate, fluxing is observed and such fluids are usually referred to as pseudoplastic fluids. This last phenomenon is a general characteristic of thermoplastic polymers. Flow effects may also be time dependent. Where viscosity does not depend only on the shear rate, but also on the duration of the applied stress, fluids are thixotropic. Polymers in a molten state thus behave as pseudoplastic fluids having thixotropic characteristics. 

Many fluids, including some that are encountered very widely both industrially and domestically, exhibit non-Newtonian behaviour and their apparent viscosities may depend on the rate at which they are sheared and on their previous shear history. At any position and time in the fluid, the apparent viscosity pa which is defined as the ratio of the shear stress to the shear rate at that point is given by ... 

Unfortunately, this group Db depends on the assignment of a single characteristic time to the fluid (perhaps a relaxation time). While this has led to some success, it appears to be inadequate for many viscoelastic materials which show different relaxation behaviour under differing conditions. 

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