Viscoelastic fluid rheology

As mentioned in the beginning of this review (see Sect. 1), besides the theoretical importance of modelling and experiments in extension of molten polymers, there is an increasing interest in this field of rheology and mechanics of viscoelastic fluids from the technological point of view. This is connected with a wide spectrum of applied problems, the solution of which is based on data on melt extension. Below we shall discuss... 

The behavior of a non-Newtonian viscoelastic fluid can be described by a constitutive equation which takes into account condition (1). Rheological behavior of the fluid is described by an equation derived from White-Metzner-Litvinov model and takes the following form 27,321 ... 

Concentrated emulsions can exhibit viscoelasticity, as can gelled foams and some suspensions. Compared with the previous equations presented, additional coefficients (including primary and secondary normal stress coefficients) are needed to characterize the rheology of viscoelastic fluids [376,382]. 

J. D. Goddard and C. Miller, An Inverse for the Jaumann Derivative and Some Applications to the Rheology of Viscoelastic Fluids, Rheol. Acta, 5, 177-184 (1966). 

Thus far we have given exclusive attention to the flow of purely viscous fluids. In practice the chemical engineer often encounters non-Newtonian fluids exhibiting elastic as well as viscous behavior. Such viscoelastic fluids can be extremely complex in their rheological response. The le vel of mathematical complexity associated with these types of fluids is much more sophisticated than that presented here. Within the limits of space allocated for this article, it is not feasible to attempt a summary of this very extensive field. The reader must seek information elsewhere. Here we shall content ourselves with fluids that do not exhibit elastic behavior. 

The major characteristic of a polymeric reactor that is different from most other types of reactors discussed earlier is the viscous and often non-Newtonian behavior of the fluid. Shear-dependent rheological properties cause difficulties in the estimation of the design parameters, particularly when the viscosity is also time-dependent. While significant literature on the design parameters for a mechanically agitated vessel containing power-law fluid is available, similar information for viscoelastic fluid is lacking. 

The cone-plate geometry is widely used in rheological measurements of viscoelastic fluids. The fluid is placed between a plate of radius and a cone of the same radius. The angle, a, between the cone and the plate is usually smaller than 3° (see Fig. 13.19). 

In the previous sections, the non-Newtonian viscosity rj) was used to characterize the rheology of the fluid. For a viscoelastic fluid, additional coefficients are required to determine the state of stress in any flow. For steady simple shear flow, the additional coefficients are given by... 

Surface rheology Viscoelasticity of the monolayer differentiation between fluid and solid phases. Surface elasticity and viscosity in the transversal and longitudinal mode wave damping characteristics. Relaxation processes in monolayers. Mechanical stability of the monolayer. Interpretation often complicated because several molecular processes may be involved and because viscous and elastic components may both contribute. 

Soft glasses are known to exhibit remarkable nonlinear shear rheology. They are yield-stress fluids that respond either like an elastic solid when the applied stress is zero or below the yield stress, or a like a viscoelastic fluid when a stress greater than the yield value of the material is applied [185]. Above their yield stresses, soft glasses are shear thinning fluids and very often the shear stress increases with the shear rate raised to the one-half power. This is well documented for the case of concentrated emulsions [102, 182, 186], microgel suspensions [31], and multilamellar... 

An analogous representation for the viscoelastic fluids is possible, see Fig. 1.33. For the experimental determinations of the rheological material properties the... 

The vast majority of concentrated dispersions, such as LADDs, exhibit both viscous and elastic properties. These systems are therefore referred to as viscoelastic. The flow properties discussed in the previous section are not sufficient for complete rheological characterization of viscoelastic fluids. Dynamic mechanical properties, characterized by the storage modulus (G ) and loss modulus (G"), are normally... 

There are a number of other rheological methods used for characterizing viscoelastic fluids. A detailed discussion can be found in Chapter 4 or a rheology text such as Barnes et al. [12]. 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

Viscoelastic fluids. In the monograph [181], the exact solution of Stokes first problem (6.10.1)—(6.10.3) was obtained for Maxwellian fluids with the rheological law (6.1.10), which has the following form for the problem in question ... 

The relaxation time in rheology, and particularly in rotational rheometry, is a measure of the rate at which the viscoelastic fluid changes in response to the change in flow due to the oscillatory movements of the fluid. Typically, an apparent relaxation time is defined as the time for the disturbances to decrease by a factor of 1/e, that is, 0.368. 

When an aqueous solution of a high-molecular-weight polymer is used in a practical engineering system, the solvent is generally predetermined by the system. However, the importance of the solvent on the pressure drop and heat transfer behavior with these viscoelastic fluids has often been overlooked. Since the heat transfer performance in turbulent flow is critically dependent on the viscous and elastic nature of the polymer solution, it is important to understand the solvent effects on the rheological properties of a viscoelastic fluid. 

It should be noted here that in polymer rheology, for viscoelastic fluids the commonly used dimensionless parameter to characterize the ratio of elastic force to viscous force is the Deborah number denoted by the symbol De. This parameter is essentially just the Peclet number. In terms of characteristic times, it is equal to the ratio of the largest time constant of the molecular motions or other appropriate relaxation time of the fluid compared to the characteristic flow time. 

It is possible to model the deformation of film bubbles with a system of dimensionless equations that is derived according to the following assumptions [13] steady-state and axisymmetrical flow (z-axis) of an incompressible fluid thin and flat film external forces on the bubble are neglected Newtonian, pseudoplastic, or viscoelastic fluids and linear temperature profiles between die exit and freezeline position. The system of dimensionless fundamental equations can be represented, irrespective of the rheological constitutive equation used, as shown in the following equations ... 

We begin the discussion by defining the technique itself. Rheology is a branch of science concerned with the characterization of viscoelastic fluids. A more scientific definition of rheology is the study of materials which are not completely described by either classic Hookean mechanics, which describe solids, or Newton s law of viscosity, which describe liquids, but instead fall somewhere in-between. So, when we talk about the rheology of viscoelastic solids/liquids, what we mean is that we are monitoring the viscosity, elasticity, and other flow properties, and the change in those properties over time in the presence of stimuli. 

However, in general the rheological properties of a viscoelastic fluid are shear rate (y) dependent. Thus the various dimensionless parameters are also shear rate dependent. For example, the Elasticity number El is now dependent on the change in shear rate y, which could be written as... 

If the shear rates are constants, the non-Newtonian fluids can also be classified according to their viscosity dependence on time. This classification has been widely applied to describe the rheological characteristics of coatings. For the development of deformation, the time evolution corresponds to the effect of the increase of shear rate. Three typical cases occur with the time evolution the thixotropic fluids exhibit the decrease of viscosity, corresponding to pseudo-plastic fluids the rheopectic fluids exhibit the increase of viscosity, corresponding to dilatant fluids while the viscoelastic fluids exhibit partial recovery of the deformation of pseudo-plastic fluids after the removal of the stress. Since polymers can perform a large scale of elastic deformation, this character appears extremely significant. 

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