Visco-elastic fluid behaviour

In the classical theory of elasticity, the stress in a sheared body is directly proportional to the strain. For tension, Hooke s law applies and the coefficient of proportionality is known as Young s modulus, G,  

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 

Qualitative differences between a viscous fluid and an elastic 

A typical dependence of the first normal stress difference on shear rate is shown in Figme 1.15 for a series of polystyrene-in-toluene solutions. Usually, the rate of decrease of [ri with shear rate is greater than that of the apparent 

The two normal stress difTerences defined in this way are characteristic of a material, and as such are used to categorise a fluid either as purely viseous N 0) or as visco-elastic, and the magnitude of N in comparison with Xy, is often used as a measure of visco-elasticity. 

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It has been a common practice to describe visco-elastic fluid behaviour in steady shear in terms of a shear stress Ty and the first normal stress difference (N ) both of which are functions of shear rate. Generally, a fluid relaxation or characteristic time, Xf, (or a spectrum) is defined to quantify the viscoelastic behaviour. There are several ways of defiiung a characteristic time by combining shear stress and the first normal stress difference, e.g. the so-called Maxwellian relaxation time is given by ... 

As discussed before ( 5.2), a molten polymer shows also elastic behaviour, particularly on a short time-scale the fluid is visco-elastic. This can, in a simple experiment, be demonstrated in two ways. When we let a bar rotate around its axis in a viscoelastic fluid, then, after removal of the driving torque, it will rotate back over a certain angle. Moreover the fluid will, during rotation, creep upward along the bar, which indicates the existence of normal stresses next to shear stresses. 

Unfortunately, this group depends on the assignment of a single characteristic time to the fluid (a relaxation time ). While this is better than no description at all, it appears to be inadecpiate for many visco-elastic materials which show different relaxation behaviour under differing conditions. 

Little is known about the effect of visco-elasticity on the motion of bubbles and drops in non-Newtonian fluids, though a preliminary study suggests that spherical bubbles are subject to a larger drag in a visco-elastic than in an inelastic liquid. Recent surveys clearly reveal the paucity of reliable experimental data on the behaviour of fluid particles in non-Newtonian liquids [Chhabra, 1993a DeKee et al., 1996]. 

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