PENDIENTES DEL TERRENO

For a Newtonian fluid the uniaxial extensional viscosity is always three times that of the shear viscosity (Barnes et al. _{(1989)). This is not true, however, for non-}

Newtonian fluids. It is not possible to estimate the extensional viscosity from the shear viscosity or vice versa for a non-Newtonian fluid because of the huge variation in properties. Throughout this investigation extensional rheology is used to determine a relaxation time, from which both Deborah and Weissenberg numbers may be estimated and hence Elasticity numbers may also be obtained. It is now well known that the extensional properties of fluids have a strong influence on flow through contractions (see, for example, Debbaut and Crochet (1988), Purnode and Crochet (1998)).

The Thermo Haake Capillary Break-up Extensional Rheometer (CaBER) exerts a

uniaxial step strain, e.g. by creating a fluid filament, on a sample of the working fluid and measures the reduction in filament diameter due to surface tension over time (Rodd et al. _{(2005)). In the configuration used here a small column of fluid}

(less than 0.2ml) is placed between two cylindrical platens with diameters of 4mm, the upper platen is moved away from the lower platen almost instantaneously (approximately 50-100ms) (see Figure 2.5 for schematic). An extensional strain is exerted on the fluid sample and an unstable cylindrical fluid filament is formed. Once the stretching has stopped the fluid is subject to an extensional strain rate, which is determined by the extensional properties of the fluid (i.e. not controlled by the instrument). The midpoint of the filament diameter decreases over time due to surface tension and the extensional stresses within the fluid element resist this

thinning. A laser micrometer, resolution 10µm, measures the reduction in the

Fluid Characterisation

the fluid. Analysis of the filament diameter decay over time gives an estimated relaxation time, which can be used to estimate a Deborah number, the relaxation

time can be found using (from Oliveira et al. _(2006))

[ ]

(

)

where t _{is time,}

_λ

‘characteristic time scale for viscoelastic stress growth in a uniaxial elongational flow’ due to the fact that the stress does not relax as such, it grows as the filament diameter decays (Rodd et al. _(2005)). D_1, t_1, k_1, V_{2 and} t_{2 are fitting parameters}

(determined using the least-squares-fitting method described in 2.1.1.). Equation 2.14 describes the three regimes usually observed during capillary break-up of flexible polymer solutions, the initial necking, the exponential thinning and the final drainage regimes.

The CaBER experiment and the results are affected by several factors, such as the initial and final aspect ratios and the strike time (Rodd et al. _{(2005)). The initial}

aspect ratio (Λ_i) is the ratio of the initial sample height to the sample (or platen) diameter. If the initial aspect ratio of the column of fluid is too large sagging will occur because of gravitational effects, i.e. the column will not be cylindrical as there will be more fluid towards the base of the sample than the top. Numerical

simulations suggest that the optimal range for the initial aspect ratio is 0.5≤Λ_i ≤1

(Yao & McKinley (1998) and Rodd et al. _{(2005)). The final aspect ratio (}

f

Λ ) is the

ratio of the final sample height to the platen diameter. If the final aspect ratio is too small the filament that is formed will not be cylindrical causing the flow at the mid point not to be purely extensional, but if it is too large the sample will break during the platen separation and the required cylindrical filament will not form. The strike time is the time taken for the top platen to move from its initial position to its final position. It can be varied according to requirements; normally the strike time is of the order 50-100ms. When the strike time is low, e.g. 50ms, and a very elastic fluid is being tested, e.g. 0.05% polyacrylamide, oscillations can be seen within the sample once the filament has formed, this means that the rheometer will not always be measuring the correct part of the fluid filament. If the strike time is increased slightly the oscillations become smaller and the results more reliable.

Fluid Characterisation

The Thermo Haake CaBER should be used in conjunction with a high-speed camera in order that the required final aspect ratio and strike time can be correctly selected and amended as necessary. It is useful to be able to view a magnified image of the sample before performing the test to assess whether the correct amount of fluid has been used, and to confirm that the sample is cylindrical and neither slightly concave nor convex. The camera also shows any small air bubbles that may not be visible to the naked eye and the sample can be reloaded before performing the test. Use of a high-speed camera can also provide another method of estimating the filament break up time (by estimating the equivalent length of each pixel) and this result can be compared with results from the laser micrometer. Unfortunately, however, the resolution of the camera used during these investigations was not good enough to precisely determine the variation in the filament diameter over time.