MR. R. J. McLEAN asked if viscosity measurements using the pipeline viscometer were valid for heavy, low purity massecuites.
DR. NESS replied that heavy, low purity massecuites tended to show viscoelastic properties, but these were only of interest when a change of shear rate is forced on the material, e.g. start of pumping. For steady state operation the viscosity values determined the pipeline method are valid for the design of factor pipework. [at p. 101]
Of course, the simplification for steady state operation Ness refers to does not apply to complex geometries.
Viscoelasticity is also referred to in the original paper (J. N. Ness 1984, ASSCT):
under some processing conditions, e.g. presence of dextrans and other high molecular weight polymers, and with very low purity materials, other rheological phenomena such as elastic and time dependent behaviour may become significant. More elaborate viscometers are needed to quantify these phenomena, and Devillers and Phélizot (1971) report some measurements on viscoelasticity in molasses using a Weissenberg rheogoniometer. Apart from that study these aspects of molasses and massecuite rheology have received little attention. [at p. 275]
In the section `The equations of change for incompressible non-Newtonian flow':
the only thing we have done here [in writing the power-law model in tensor form, for use in a general equation of motion] is to write the expressions in such a way that they transform properly in going from one coordinate system to another. Otherwise, we could not be assured that the parameters---[consistency] and [flow behaviour index], for example---would be the same when determined in various geometrical arrangements. ...Furthermore, we have restricted the discussion even more be the assumption that Eq. 3.6.-2 [τ=-ηΔ] can be used to describe non-Newtonian behaviour [from p. 103]
if one desires only a partial solution to a problem and is willing to accept the fact that his solutions may not give good results, the [rheological] equation of state
pij=α(Y)e*ij, (4.78)
Y=½e*nm e*mn (4.79)
[i.e. the generalized Newtonian model] may be used for incompressible fluids. In Chapter 7 we shall show that Eq. (4.78) may be used for the steady flow of viscoelastic fluids. [from p. 86]
Fredrickson here uses steadiness in a different sense than is usual: each particle must experience an unchanging stress state, rather than the velocity at a point remaining the same, cf. the following excerpt from a discussion of boundary layer flow on p. 247:
Since the strain rate seen by a moving particle in the situation considered [viz. boundary layer flow] changes with time, it is clear that any viscoelasticity must be manifested in some fashion
Other quotations from Fredrickson's book:
it was shown that the Reiner—Rivlin equations for a purely viscous fluid predict a type of Weissenberg effect...so that the effect [viz. rod-climbing] is not associated only with viscoelastic fluids. However, all fluids which yield Weissenberg effects are also viscoelastic, so that one is probably not wrong in associating the effect primarily with viscoelasticity [at p. 122]
As a matter of fact, the experimental evidence indicates that there are no Stokesian fluids, save those which are also Newtonian [at p. 70.]
A Stokesian fluid is a purely viscous fluid which is isotropic; i.e. a fluid for which the stress tensor is an (isotropic) function of the rate-of-strain tensor (but not the history of the rate-of-strain tensor). This includes all generalized Newtonian fluids, for which the stress tensor is just half the rate-of-strain tensor times a scalar field called the viscosity (which may depend on the scalar invariants of the rate-of-strain or stress tensors). The power-law fluid is then a very special case of the Stokesian model.
The final geometry in the present study involves channel flow that is disturbed by a cylindrical obstacle (see figure 2 f). The obstacle is in a slightly asymmetric position with respect to the channel. For the Newtonian liquid (figure 17 a, plate 13) the asymmetry in the streamlines is barely discernible to the naked eye (if at all) whereas the asymmetry is exaggerated for the elastic liquid (figures 17 b, c). This is yet another dramatic demonstration of the differences in flow characteristics found in Newtonian and elastic liquids. It supports our conclusion in regard to combined mixing and separating flow that symmetry is elusive for elastic liquids whenever there is a mechanism in the flow for promoting an asymmetry of any degree. [at p. 170]
Because of the failure [of the Stokesian, a.k.a. Reiner—Rivlin, constitutive equation, σij = -pδij + 2ηdij + 4ν2dikdkj, where η and ν2 are functions of the second and third invariants of dij] to model the dominant normal stress difference, it is not possible to use this equation in any flow where this factor is important. It is also not a realistic description of the transient response of polymer fluids since when motion ceases τij immediately vanishes which is contrary to observation. In fact, the only flow which is described accurately by this equation is a steady elongational flow. It is nevertheless often used when the flow is dominated by a variable viscosity. The function ν2 [the second normal stress coefficient] is then usually set equal to zero and the viscosity taken as a function of the second invariant of dij only. One is then back to a simple non-Newtonian fluid, often called a generalized Newtonian fluid, in which the viscosity varies with the shear rate. The only flows of polymer fluids which are strictly suitable for modelling in this way are those which are very close to steady viscometric; for slurries where elastic effects are minimal it is more useful. It would be futile to attempt to catalogue all solutions to flow problems which use this limited and atypical type of flow behaviour. Most often the viscosity function is described by a `power-law' behaviour...However, it must be reiterated that the inelastic fluid model often omits important phenomena, and hence it is not reasonable to devote great efforts to the analytical solution of this system of equations.
W. J. Kelly & A. E. Humphrey (1998) Computational fluid dynamics model for predicting flow of viscous fluids in a large fermentor with hydrofoil flow impellers and internal cooling coils, Biotechnol. Prog. 14:248—258.
X.-L. Luo (1996) Operator splitting algorithm for viscoelastic flow and numerical analysis for the flow around a sphere in a tube. J. Non-Newtonian Fluid Mech. 63:121—140.
A. C. Pipkin & R. I. Tanner (1977) Steady non-viscometric flows of viscoelastic liquids, Ann. Rev. Fluid Mech. 9:13—32.
Last modification Thu Aug 10 11:57:32 EST 2000 Geordie McBain.