Viscometric flow may be defined as that which is indistinguishable from steady simple shear flow. Additional information may be obtained from a different type of flow: extensional flow that yields an extensional viscosity. Pure extensional flow does not involve shearing and is sometimes referred to as "shear free" flow. In published literature, elongational viscosity and Trouton viscosity are frequently used onyms for extensional viscosity. Similarly, elongational flow is a syn-onym for extensional flow.
Many food processing operations involve extensional deformation and the molecular orientation caused by extension, versus shear, can produce unique food products and behavior. The reason shear and extensional flow have a different influence on material behavior may be explained by the way in which flow fields orient long molecules of high molecular weight. In shear flow, the preferred orientation corre-sponds to the direction of flow; however, the presence of a differential velocity across the flow field encourages molecules to rotate thereby reducing the degree of stretching induced in molecular chains. The tendency of molecules to rotate, versus elongate, depends on the mag-nitude of the shear field: There is relatively more elongation, less rotation, at high shear rates. In extensional flow, the situation is very different. The preferred molecular orientation is in the direction of the flow field because there are no competing forces to cause rotation. Hence, extensional flow will induce the maximum stretching of the molecules producing a chain tension that may result in a large (compared to shear flow) resistance to deformation.
The nature of the molecule, branched versus linear, may signifi-cantly influence flow behavior in extension. In comparable fluid systems (i.e., high-density polyethylene, a linear molecule, versus low-density polyethylene, a branched molecule) branched molecules will cause a fluid to be less tension-thinning then linear molecules. A similar
argument can be made in comparing the relative stiffness of biopolymer molecules: Stiffer molecules are more quickly oriented in an extensional flow field. This phenomenon may be a factor in the choice of a thickening agent for pancake syrup: Stringiness can be reduced, while maintaining thickness, when stiffer molecules are selected as additives. Reduced stringiness leads to what can be called a clean "cut-off" after pouring syrup from a bottle. An example of a stiff molecule would be the rod-like biopolymer xanthan compared to sodium alginate or carboxymethyl-cellulose which exhibit a randon-coil-type conformation in solution (Padmanabhan, 1995).
Extensional flow is an important aspect of food process engineering and prevalent in many operations such as dough processing. Sheet stretching, as well as extrudate drawing, provides a good example of extensional flow (Fig. 1.21). Converging flow into dies, such as those found in single and twin screw extruders, involves a combination of shear and extensional flow; the extensional component of deformation is illustrated in Fig. 1.21. The analysis of flow in a converging die (see Sec. 4.4) allows one to separate the pressure drop over the die into the shear and extensional components. Converging flow may also be observed when fluid is sucked into a pipe or a straw, or when applying a food spread with a knife.
One of the most common examples of extensional flow is seen when stretching warm mozzarella cheese while pulling a slice of pizza away from the serving pan. Sometimes this behavior is subjectively referred to as stringiness. A similar observation can be made when pulling apart a caramel filled candy bar or a pastry with fruit filling. Extensional deformation is also present in calendering (Fig. 1.22), a standard operation found in dough sheeting. Gravity induced sagging (Fig. 1.22) also embodies extensional deformation. This may be observed in a cut-off apparatus associated with fruit filling systems for pastry prod-ucts. Extensional flow in this situation is undesirable because it may contribute to inconsistent levels of fill or an unsightly product appearance due to smeared filling. Bubble growth from the production of carbon dioxide gas occurring during dough fermentation, extrudate expansion from the vaporization of water, and squeezing to achieve product spreading involve extensional deformation (Fig. 1.23). Exten-sional flow is also a factor in die swell and mixing, particularly dough mixing with ribbon blenders.
Figure 1.21. Extensional flow found in sheet stretching (or extrudate drawing) and convergence into an extruder die.
Figure 1.22. Extensional flow in calendering and gravity induced sagging.
Sheet Stretching Extruder Die
Calendering Sagging
Figure 1.23. Extensional flow found in bubble growth and squeezing flow between lubricated plates.
Although extensional viscosity is clearly a factor in food processing, our use of this rheological property in engineering design of processes and equipment is still at an early stage of development. Extensional flow is also an important factor in the human perception of texture with regard to the mouthfeel and swallowing of fluid foods and fluid drugs.
In addition, many plastic manufacturing operations involve extensional flow: compression moulding, thermoforming, blow moulding, fiber spinning, film blowing, injection moulding, and extrusion.
Extensional viscosity has been measured for various food products.
Leighton et al. (1934) used the sagging beam method developed by Trouton (1906) to measure the extensional viscosity of ice cream.
Results were presented in terms of apparent viscosity by using the well known Trouton ratio showing that extensional viscosity is equal to three times the shear viscosity (see Eq. [1.78]). This appears to be the first reported measurement of the extensional flow of a food product. Moz-zarella cheese has been tested in uniaxial tension by Ak and Gunase-karan (1995). Biaxial extensional flow, produced by squeezing material between parallel plates, has been used in evaluating cheese (Campanella et al., 1987; Casiraghi et al., 1985), wheat flour doughs (Huang and Kokini, 1993; Wikström et al., 1994), gels (Bagley et al.
1985; Christianson et al. 1985), and butter (Rohn, 1993; Shuka et al., 1995). Data from the Chopin Alveograph, a common dough testing device where a spherical bubble of material is formed by inflating a sheet, can be interpreted in terms of biaxial extensional viscosity (Faridi
Bubble Growth Squeezing
and Rasper, 1987; Launay and Buré, 1977). This technique requires an accurate determination of the sample geometry before and during inflation. Doughs have also been evaluated by subjecting them to uniaxial extension (de Bruijne et al., 1990).
The spinning test (also called extrudate drawing) was applied to measure the stretchability of melted Mozzarella cheese (Cavella et al., 1992). Entrance pressure drop from converging flow into a die has been used to evaluate an extensional viscosity for corn meal dough (Bhat-tacharya et al., 1994; Padmanabhan and Bhat(Bhat-tacharya, 1993; See-thamraju and Bhattacharya, 1994) and bread dough (Bhattacharya, 1993). Additional methods have been proposed for evaluating the extensional behavior of polymeric materials (Ferguson and Kemb-lowski, 1991; James and Walters, 1993; Jones et al., 1987; Macosko, 1994; Petrie, 1979; Tirtaamadja and Sridhar, 1993; Walters, 1975):
bubble collapse, stagnation flow in lubricated and unlubricated dies, open siphon (Fano flow), filament stretching, spinning drop tensiometer, and converging jets. Extensional viscosities for some Newtonian and non-Newtonian fluids are presented in Appendices [6.15] and [6.16], respectively. Measurement methods, and example problems, are dis-cussed in Chapter 4.
Types of Extensional Flow. There are three basic types of extensional flow (Fig. 1.24): uniaxial, planar, and biaxial. During uniaxial extension material is stretched in one direction with a corresponding size reduction in the other two directions. In planar extension, a flat sheet of material is stretched in the direction with a corresponding decrease in thickness ( decreases) while the width ( direction) remains unchanged. Biaxial extension looks like uniaxial compression, but it is usually thought of as flow which produces a radial tensile stress.
Uniaxial Extension. With a constant density material in uniaxial extension (Fig. 1.24), the velocity distribution in Cartesian coordinates, described with the Hencky strain rate, is
x1
x2 x3
Figure 1.24. Uniaxial, planar, and biaxial extension.
[1.55]
[1.56]
[1.57]
where . Since this flow is axisymmetric, it may also be described in cylindrical coordinates (it may be helpful to visualize this situation with the positive axis aligned with the axis, Fig. 1.24):
[1.58]
[1.59]
[1.60]
Pure extensional flow does not involve shear deformation; therefore, all the shear stress terms are zero:
[1.61]
AFTER DEFORMATION BEFORE DEFORMATION
uniaxial extension
planar extension
biaxial extension z
r
1 2
3
x x
x
u1=ε˙hx1 u2=−ε˙hx2
2 u3=−ε˙hx3
2 ε˙h>0
z x1
uz=˙εhz ur=−ε˙hr
2 uθ=0
σ12= σ13= σ23= σrθ= σrz= σzθ=0
Stress is also axisymmetric:
[1.62]
resulting in one normal stress difference that can be used to define the tensile extensional viscosity:
[1.63]
Materials are considered tension-thinning (or extensional-thinning) if decreases with increasing values of . They are tension-thickening (extensional-thickening) if increases with increasing values of . These terms are analogous to shear-thinning and shear-thickening used previously (Sec. 1.5.1) to describe changes in apparent viscosity with shear rate.
Biaxial Extension. The velocity distribution produced by uniaxial compression causing a biaxial extensional flow (Fig. 1.24) can be expressed in Cartesian coordinates as
[1.64]
[1.65]
[1.66]
where . Since , biaxial extension can actually be viewed as a form of tensile deformation. Uniaxial compression, however, should not be viewed as being simply the opposite of uniaxial tension because the tendency of molecules to orient themselves is stronger in tension than compression. Axial symmetry allows the above equations to be rewritten in cylindrical coordinates (Fig. 1.24) as
[1.67]
[1.68]
[1.69]
Biaxial extensional viscosity is defined in terms of the normal stress difference and the strain rate:
[1.70]
Planar Extension. In planar extension (Fig. 1.24), the velocity distribution is
[1.71]
[1.72]
[1.73]
This type of flow produces two distinct stress differences: and . Planar extensional viscosity is defined in terms of the most easily measured stress difference, :
[1.74]
It is difficult to generate planar extensional flow and experimental tests of this type are less common than those involving tensile or biaxial flow.
Relation Between Extensional and Shear Viscosities. The fol-lowing limiting relationships between extensional and shear viscosities can be expected for non-Newtonian fluids at small strains (Dealy, 1994;
Walters, 1975; Petrie, 1979):
[1.75]
[1.76]
[1.77]
Reliable relationships for non-Newtonian fluids at large strains have not been developed. The above equations may be precisely defined for the special case of Newtonian fluids:
[1.78]
[1.79]
[1.80]
Eq. [1.78], [1.79], and [1.80] can be used to verify the operation of extensional viscometers. Clearly, however, a Newtonian fluid must be extremely viscous to maintain its shape and give the solid-like appearance required in many extensional flow tests. Extensional behavior of low viscosity fluids can be evaluated with the method of opposing jets (Sec. 4.5), by spinning (Sec. 4.6), or by investigating tubeless siphon behavior (Sec. 4.7).
u1=ε˙hx1
Trouton established a mathematical relationship between tensile extensional viscosity (he called it the coefficient of viscous traction) and shear viscosity (Trouton, 1906). Presently, data for extensional and shear viscosities are often compared using a dimensionless ratio known as the Trouton number ( ):
[1.81]
Since extensional and shear viscosities are functions of different strain rates, a conventional method of comparison is needed to remove ambiguity. Based on a consideration of viscoelastic and inelastic fluid behavior, Jones et al. (1987) advocated the following conventions in computing the Trouton numbers for uniaxial and planar extensional flow:
[1.82]
[1.83]
meaning that shear viscosities are calculated at shear rates equal to or for uniaxial or planar extension, respectively. Using the similar considerations, Huang and Kokini (1993) showed that the Trouton number for case of biaxial extension should be calculated as
[1.84]
The Trouton ratio for a Newtonian fluid may be determined from Eq.
[1.78], [1.79], and [1.80]: in tensile extension it is equal to 3; it is 6 and 4, respectively, in biaxial and planar flow. Departure from these numbers are due to viscoelastic material behavior. Experimental results may produce considerably higher values.