Viscoelasticity of Fluids

Viscoelasticity deals with the dynamic or time-dependent mechanical properties of materials such as polymer solutions. The viscoelasticity of a material in general is described by stresses corresponding to all possible time-dependent strains. Stress and strain are tensorial quantities the problem is of a three dimensional nature (8), but we shall be concerned only with deformations in simple shear. Then the relation between the shear strain y and the stress a is simple for isotropic materials if y is very small so that a may be expressed as a linear function of y, 

Since the linear viscoelasticity of a material is described with a material function G(t), any experiment which gives full information on G(t) is sufficient it is not necessary to give the stresses corresponding to various strain histories. We will restrict the discussion to incompressible isotropic materials. In this case, different types of deformation such as elongation and shear give equivalent information in the range of linear viscoelasticity. Several types of experiments measure relaxation modulus, creep compliance, complex modulus etc which are equivalent to the relaxation modulus (1). 

For fluids of low viscosity such as dilute polymer solutions, it is customary to describe the viscoelasticity with the complex modulus and the experiments have been performed in simple shear flow. The sinusoidally oscillating simple shear flow employed in this paper is illustrated in Fig. 1.1. The velocity of the fluid in the cartesian coordinate 

Simple shear flow. Rate of shear K(t) is given as a w coscot for sinusoidally oscillating simple shear flow 

Thus either G (to) or G (co) as a function of to gives the information equivalent to that included in G(t) as a function of t. The complex modulus is experimentally a more convenient quantity to describe the linear viscoelasticity of low-viscosity fluids than the relaxation modulus (1). The complex modulus is related to the complex viscosity / (co) and the complex compliance J (to)  

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The proportionality constant fe depends upon the stirrer type. For propeller stirrers fe = 10 ]68], for turbine stirrers fe = 11.5 and pitched-blade stirrers fe = 13 [104]. For blade stirrers k = 2.5, for cross-beam stirrer fe = 4.1 and for helical ribbon stirrer fe = 6.0 [411]. Calderbank [66] found that when turbine stirrers were used with Bingham and pseudoplastic fluids fe = 10 and when used with dilatant liquids fe = 12.8 (d/D) 5. Lower fe values were found as the viscoelasticity of fluids increased [104]. In the case of close-clearance anchor stirrer fe depended upon the wall clearance [24]. 

Indeed, to exploit effectively the viscoelasticity of fluids for chaotic flow instability, and thus mixing, sharper and smaller geometries should be employed. Stress singularities developed at such comers have been the source of elastic instabilities in many macroscale experiments [3], while rounded comers tend to suppress elastic behavior. From a practical standpoint, it is necessary to understand the rheological nature of such flow in order to optimize the use of viscoelastic effects... 

The mathematical theory is rather complex because it involves subjecting the basic equations of motion to the special boundary conditions of a surface that may possess viscoelasticity. An element of fluid can generally be held to satisfy two kinds of conservation equations. First, by conservation of mass. 

Mitsoulis, E., 1990. Numerical Simulation of Viscoelastic Fluids. In Encyclopaedia of Fluid Mechanics, Vol. 9, Chapter 21, Gulf Publishers, Houston. 

One of the mechanisms of drag reduction is that transmission of eddies can be damped by the viscoelastic properties of fluids. The transfer process of an isolated eddy in Maxwell fluids with viscoelastic properties was studied, and the expressions describing such phenomena were obtained [1103]. The results of the study showed that eddy transmission was damped significantly with an increase of the viscoelastic properties of the fluids. 

Z. Li. Effect of fluid viscoelasticity on isolated eddy transmission. J Univ Petrol, China, 15(5) 33-38, October 1991. 

The constant Tr is called the Trouton ratio10 and has a value of 3 in this experiment with an incompressible fluid in the linear viscoelastic limit. The elongational behaviour of fluids is probably the most significant of the non-shear parameters, because many complex fluids in practical applications are forced to extend and deform. Studying this parameter is an area of great interest for theoreticians and experimentalists. 

The effect of particle shape on the forces acting when the particle is moving in a shear-thinning fluid has been investigated by Tripathi et alP7>, and by Venumadhav and Ciiiiaisra 41 1. In addition, some information is available on the effects of viscoelasticity of the fl uid135. ... 

Moreover, real polymers are thought to have five regions that relate the stress relaxation modulus of fluid and solid models to temperature as shown in Fig. 3.13. In a stress relaxation test the polymer is strained instantaneously to a strain e, and the resulting stress is measured as it relaxes with time. Below the a solid model should be used. Above the Tg but near the 7/, a rubbery viscoelastic model should be used, and at high temperatures well above the rubbery plateau a fluid model may be used. These regions of stress relaxation modulus relate to the specific volume as a function of temperature and can be related to the Williams-Landel-Ferry (WLF) equation [10]. 

A further development is possible by noting that the high frequency shear modulus Goo is related to the mean square particle displacement (m ) of caged fluid particles (monomers) that are transiently localized on time scales ranging between an average molecular collision time and the structural relaxation time r. Specifically, if the viscoelasticity of a supercooled liquid is approximated below Ti by a simple Maxwell model in conjunction with a Langevin model for Brownian motion, then (m ) is given by [188]... 

It should be noted that the effect of fluid viscoelasticity on transport of mass, momentum, and heat in porous media has not been discussed in this summary. Although some preliminary studies have been performed in this area [21], no definitive governing equations exist. 

With the above information, it becomes possible to combine viscous characteristics with elastic characteristics to describe the viscoelasticity of polymeric materials.86-90 The two simplest ways of combining these features are shown in Figure 2.49, where a spring having a modulus G models the elastic response. The viscous response is modelled by what is called a dashpot. It consists of a piston moving in a cylinder containing a viscous fluid of viscosity r. If a downward force is applied to the cylinder, more fluid flows into it, whereas an upward force causes some of the fluid to flow out. The flow is retarded because of the high viscosity and this element thus models the retarded movement and flow of polymer chains. 

Figure 12.29 shows that the die does not end at the plane z = 0. Because polymer melts are viscoelastic fluids, it extends to downstream to the end of the die lip region so that a uniform recent flow history can be applied on all fluid elements. In deriving the die design equation, we disregarded the viscoelasticity of the melts, taking into account only their shear thinning character. 

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. 

Let us find the resistance force acting on a spherical particle of radius a which moves slowly with velocity u in an incompressible viscoelastic fluid. It means that the Reynolds number of the problem is small, the convective terms are negligibly small, and the equations of fluid motion are... 

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