Inelastic Time-Dependent Fluids

Viscoelasticity was introduced in Section 11.5. A polymer example may be useful by way of reeapitulalion. Imagine a polymer melt or solution confined in the aperture between two parallel plates to which it adheres. One plate is rotated at a constant rate, while the other is held stationary. Figure 11-3la shows the time dependence of the shear stress after the rotation has been stopped, r decays immediately to zero for an inelastic fluid but the decrease in stress is much more gradual if the material is viscoelastic. In some cases, the residual stresses may... 

In fluids with time-dependent behavior, the effects of time can be either reversible or irreversible. If the time effects are reversible, the fluids are either thixotropic or rheopectic. Thixotropy is the continuous decrease of apparent viscosity with time under shear and the subsequent recovery of viscosity when the flow is discontinued. Rheopexy is the continuous increase of apparent viscosity with time under shear it is also described by the term anti-thixotropy. A good review on thixotropy was given by Mewis [45]. Polymer melts do exhibit some thixotropic effects however, thixotropy can also occur in inelastic fluids. The time scale of thixotropy is not necessarily associated with the time scale for viscoelastic relaxation. 

To the practicing process engineer, the most important question should be whether the more general analysis will be applicable e.g., are all the boundary conditions well known, and will it result in more accurate predictions. For instance, it probably does not make much sense to use a sophisticated non-isothermal melt conveying analysis to predict the melt temperature at the end of the screw if the actual screw temperature in the process is unknown. These practical considerations do not necessariiy appiy to the academic. On the other hand, it would make sense to perform a non-isothermal analysis to predict the effect of barrel temperature fluctuations on meit temperature or conveying rate. Besides the complications already discussed, there is the additional complication that the polymer melt is not a pure, inelastic power iaw fluid and significant time-dependent effects can occur (e.g., [122-128]). 

At this point we need to classify fluids into those that are inelastic, by which we mean that the stress at a given time depends only on the deformation rate at that time, and those that are viscoelastic, by which we mean that the stress at a given... 

Exponential shear is therefore not a flow with constant stress history. The stress in this flow tends to grow without limit, even in an inelastic or linearly elastic fluid, and this makes the presentation of data an important issue. Doshi and Dealy [57] and Dealy [58] have argued that the results of an exponential shear experiment should be reported in terms of a time-dependent exponential viscosity rf that is defined in terms of the instantaneous shear rate ... 

The results of the scant work with visco-elastic fluids are conflicting but suggest that relationship between mixing and circulation times is strongly dependent on the geometrical arrangement and that it is not yet possible to accoimt quantitatively for the effects of visco-elasticity. Visco-elastic fluids appear to be much more difficult to mix than inelastic fluids. Reference should be made to a review paper for fmther details of studies in this mea [Takahashi, 1988]. 

The initial cost of a mixer including its impeller, gear box and motor is closely related to the torque, T, requirement rather than its power. Deduce the relationship between torque and size of the impeller for the same mixing time as a function of geometrical scale, for turbulent conditions in the vessel. Does your answer depend upon whether the fluid is Newtonian or inelastic shear-thinning in behaviour What will be the ratio of torques for a scale-up factor of 2 ... 

CMC bubbles of the same volume regardless of the Reynolds number [26]. In stirred liquids, the impeller disturbs the motion of bubbles and can accelerate or retard them, depending on the existence of recirculation currents. Nevertheless, the average residence time of PAA bubbles is expected to be lower than in inelastic fluids. Simple visual observation allowed us to confirm that the residence time of air bubbles in PAA was longer than in CMC, but not necessarily longer than in XTN... 

The temperature instability of a two-dimensional reactive fluid of N hard disks bounded by heat conducting walls has been studied by molecular dynamics simulation. The collision of two hard disks is either elastic or inelastic (exothermic reaction), depending on whether the relative kinetic energy at impact exceeds a prescribed activation barrier. Heat removal is accomplished by using a wall boundary condition involving diffuse and specular reflection of the incident particles. Critical conditions for ignition have been obtained and the observations compared with continuum theory results. Other quantities which can be studied include temperature profiles, ignition times, and the effects of local fluctuations. 

---