Rod climbing

In some ways a simpler approach to obtaining normal stresses from the concentric cylinder system is to use the rod climbing phenomenon quantitatively. Because the flow is complex, analysis requires the assumption of some constitutive relation. Joseph and Fosdick (1973) have done this using the second-order fluid, which should be an exact representation of elastic liquids in the limit of slow flows (see Section 4.3). They derive a power series for the 

Experimental values of h(R, Q) for several aluminum Folds in polyisobutylene in oil (STP oil additive). Replotted from Beavers and Joseph (1975). 

Note that according to eq. 5.3.36 there is an optimum radius to give the maximum climbing effect. 

One of the disadvantages of the rod climbing experiment is that it requires another physical property measurement surface 

However, the sensitivity and simplicity of the experiment are great advantages. Furthermore, the limiting normal stress values from are useful data for molecular theories of rheological phenomena. 

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Many materials of practical interest (such as polymer solutions and melts, foodstuffs, and biological fluids) exhibit viscoelastic characteristics they have some ability to store and recover shear energy and therefore show some of the properties of both a solid and a liquid. Thus a solid may be subject to creep and a fluid may exhibit elastic properties. Several phenomena ascribed to fluid elasticity including die swell, rod climbing (Weissenberg effect), the tubeless siphon, bouncing of a sphere, and the development of secondary flow patterns at low Reynolds numbers, have recently been illustrated in an excellent photographic study(18). Two common and easily observable examples of viscoelastic behaviour in a liquid are ... 

A graphic example of the consequences of the existence of in stress in simple steady shear flows is demonstrated by the well-known Weissenberg rod-climbing effect (5). As shown in Fig. 3.3, it involves another simple shear flow, the Couette (6) torsional concentric cylinder flow,3 where x = 6, x2 = r, x3 = z. The flow creates a shear rate y12 y, which in Newtonian fluids generates only one stress component 112-Polyisobutelene molecules in solution used in Fig. 3.3(b) become oriented in the 1 direction, giving rise to the shear stress component in addition to the normal stress component in. 

Fig. 3.3 A 9, 52-mm D aluminum rod rotating at 10 rps in a wide-diameter cylinder containing (a) Newtonian oil, and (b) polyisobutylene (PIB) solution, which exhibits the rod-climbing Weissenberg effect [from G. S. Beavers and D. D. Joseph, J. Fluid Mech., 69, 475 (1975)]. (c) Schematic representation of the flow direction flow-induced causing rod climbing. For Newtonian fluids, Tn = 0, since the small and simple Newtonian fluid molecules are incapable of being oriented by the flow.

The elastic-melt extruder makes use of the Weissenberg or rod climbing effect—which is observed when an elastic fluid is sheared or rotated inside a container by a rod. Because of viscoelasticity, the fluid climbs the rod. 

While chemical engineers are well-grounded in the mechanics of Newtonian fluids, it is the non-Newtonian character of polymers that controls their processing. Three striking examples [6] of the differences between Newtonian and typical polymeric liquids (either melts or concentrated solutions) are shown schematically in Fig. 1. The upper portion of the figure refers to the Weissenberg effect [7], or rod-climbing, exhibited by polymers excellent photos may be found in Bird et al. [4] as well. When a rod is rotated in a Newtonian fluid, a vortex develops near the rod due to centripetal acceleration of the fluid. When the same experiment is repeated with a polymeric fluid, however, the fluid climbs the rod. In the center... 

Rigid rodcrystallisation, 706 Rod climbing effect, 526 Rod-like molecules, 252 Rod-like polymer molecules, 274 Rod-shaped particle, 276 Rubber elasticity, 401 Rubbery plateau, 400 Rudin equations, 272 Rudin-Strathdee equation, 602 Rules of thumb for substituting an H-atom by a group X, 182... 

The equations of motion and the constitutive relation are then linearized with respect to . To avoid some possible unphysical instabilities, the range of parameter a, which o priori belongs to (—1,1), is restricted to the set a > 1/2,/ < 1. The first restriction ensures that the model is consistent with rod climbing, the later that one stays on the increasing part of the curve of the shear stress as a function of the shear rate, which heis a maximum at A = 1. 

Figure 13.20 (a) Rod-climbing effect of a viscoelastic fluid caused by rotation of the rod in the fluid, (b) The same situation for a Newtonian fluid. 

Figure 13.21 Schematic representation of a rod-climbing experiment, for a polymer solution viewed from the top. The dashed line represents a force component resulting from the retraction force of the chain.

While the stress tensor component tfor purely viscous fluids can be determined from the instantaneous values of the rate of deformation tensor 4, the past history of deformation together with the current value of 4, may become an important factor in determining t, for viscoelastic fluids. Constitutive equations to describe stress relaxation and normal stress phenomena are also needed. Unusual effects exhibited by viscoelastic fluids include rod climbing (Weis-senberg effect), die swell, recoil, tubeless siphon, drag, and heat transfer reduction in turbulent flow. 

The viscosity and rheological behavior of the continuous phase also modifies the behavior of emulsions and suspensions. In fact. Eqs. [50 -[53] establish that the larger the viscosity of the continuous phase (T) ), the larger the viscosity of the su.spension diJ. Furthermore, if the continuous phase is strongly viscoela,stic. die swelling and rod climbing may appear (26). Shear thickening has also been observed in concentrated emulsions of very viscous internal phase (27). 

Normal stresses originally recognized by Weissenberg [W3] through observation of rod climbing effects in soap-hydrocarbon liquid suspensions and polymer solution systems began to be measured on thermoplastics in the 1960s [C9, K9]. White and Tokita [W24] noted their occurrence in gum... 

The inequality of normal stresses is responsible for a number of visually noticeable Non-Newtonian effects. These include the Weissenberg rod-climbing effect where the fluid climbs up a rotating rod rather than dipping near the rod, and the extrudate swell, see Tanner (2000) and Boger and Walters (1993) for more details. 

The extensional thickening of polymer solutions is one form of viscoelastic behavior. This ability to support a tensile stress can also be demonstrated in a tubeless syphon with dilute aqueous solutions of polsrmers such as polyacrylamide or polyethylene oxide. If you suck up solution with a medicine dropper attached to a water aspirator and then lift the dropper out of the solution, the solution will still be sucked up. In shear, viscoelastic fluids develop normal stresses, which causes rod climbing on a rotating shaft, as opposed to the vortex and depressed surfaces that form with Newtonian liquids. Polsrmer solutions and semiliquid poljnners exhibit other viscoelastic behaviors, where, on short time scales, they behave as elastic solids. Silly putty, a childrens toy, can be formed into a ball and will slowly turn into a puddle if left on a flat surface. But if dropped to the floor it boimces. 

Figure 1 The Weissenberg effect, seen as the rod-climbing in an elastic liquid, as opposed to the inertial dipping in a Newtonian liquid.

Viscoelastic behaviour is then covered chapters 13 and 14), first linear viscoelasticity with its manifestations in the time and frequency domain. Then nonlinear displays of viscoelasticity are introduced, these effects being usually encountered in steady-state flow, where overt normal-force effects such as the Weissenberg rod climbing and the extrudate-swell phenomena are seen. 

N is positive, corresponding to a tendency of the polymeric fluid to push apart the two plates between which the fluid is sheared (Figure 13.24). Such a tendency of polymeric fluids is manifested in interesting phenomena such as die-swell and rod-climbing (Figure 13.25). 

Thus, in simple shear the neo-Hookean model predicts a first normal stress difference that increases quadratically with strain. This also agrees with experimental results for rubber (note Figure 1.1.3). We will see in Chapter 4 that the same kinds of normal stress appear in shear of elastic liquids (recall the rod climbing in Figure 1.3). Note that there is only one normal stress difference, Ni, for the neo-Hookean solid in shear. 

When a viscoelastic material is sheared between two parallel surfaces at an appreciable rate of shear, in addition to the viscous shear stress T 2, there are normal stress differences Wi s Tn - 722 and N2 s 722 - T23. Here 1 is the flow direction, 2 is perpendicular to the surfaces between which the fluid is sheared, as defined by eq 1.4.8, and 3 is the neutral direction. The largest of the two normal stress differences is N, and it is responsible for the rod climbing phenomenon mentioned at the beginning of this book. For isotropic materials, Ni has always been found to be positive in sign (unless it is zero). In a cone and plate rheometer this means that the cone and plate surfaces tend to be pushed q>art. N2 is usually found to be negative and smaller in magnitude than Ni typically the ratio —N2/N1 lies between 0.05 and 0.3 (Keentok et al., 1980 Ramachandran et al., 1985). Figure 4.2.1 shows the... 

This example makes evident the usefulness of the equation of the second-order fluid. Once the three coefficients of the equation have been specified—and measurements in simple shear alone are enough to specify them—predictions can be made for the first deviations from Newtonian behavior for any other flow. This property has proved useful in analyzing slow but complex nonuniform flows, such as those observed in rod climbing (see Chapter 5) and flow over a pressure hole (see Chapter 6). 

Normd stresses hard to measure because of curvature and need to transmit signal through a rotating shaft Rod climbing is another option, eq. 5.3.36... 

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