The Rheology Handbook. Thomas Mezger

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       Figure 2.4: Double-tube test

      Experiment 2: The double-tube test, or the contest of the two fluids (see Figure 2.4)

      In the beginning, fluid F1 is flowing faster than fluid F2. With decreasing fluid level, F1 shows reduced flow velocity. F2, however, continues to flow with a hardly visible change in velocity. Therefore finally, F2 empties its tube before F1 does. F1, a wallpaper paste, is an aqueous methylcellulose solution, and F2 is a mineral oil. Flow behavior of polymer solutions such as the wall paper paste is explained in Chapter 3.3.2.1: shear-thinning flow behavior.

2.3.1Ideal-viscous flow behavior

      a) Viscosity law

      Formally, ideal-viscous flow behavior is described by the viscosity law:

      Equation 2.11

      τ = η ⋅ γ ̇

      Isaac Newton (1643 to 1727) wrote in 1687 in his textbook Principia [2.18] in a quite inaccurate form about the flow resistance of liquids (“defectus lubricitatus”; see also Chapter 14.1: 1687). Therefore, and especially in English spoken countries, ideal-viscous flow behavior often is also called Newtonian flow behavior. In rheology, both terms have the same meaning. Based on later research on fluid dynamics by D. Bernoulli (in 1738, Hydrodynamica [2.19]), L. Euler (in 1739/1773, Scientia Navalis, and Construction des vaisseaux [2.20]), Joh. Bernoulli (in 1740, Hydraulica [2.21]), and C. L. M. H. Navier (in 1823 [2.22]), finally G. G. Stokes (in 1845 [2.12]) stated the modern form of what was called later Newton’s viscosity law. Therefore sometimes, the viscosity law is also termed the “Newton/Stokes law” [2.23].

      Examples of ideal-viscous materials

      Low-molecular liquids (and this means here: with a molar mass below 10,000 g/mol) such as water, solvents, mineral oils (without polymer additives), silicone oils, viscosity standard fluids (of course!), blood plasma; but also pure and clean bitumen (without associative superstructures, and at a sufficiently high temperature).

      Flow behavior is illustrated graphically by flow curves (previously sometimes also called rheograms) and viscosity curves. Flow curves are showing the interdependence of shear stress τ and shear rate γ ̇ . Usually, γ ̇ is presented on the x-axis (abscissa), and τ on the y-axis (ordinate). However, τ might also be displayed on the x-axis and γ ̇ on the y-axis, but this is meanwhile rarely used in industrial laboratories.

      Viscosity curves are derived from flow curves. Usually, η is presented on the y-axis and γ ̇ on the x-axis. Alternatively, the function η(τ) can be shown with η on the y-axis and τ on the x-axis, however, this is less frequently carried out in industrial labs.

      Generally, the slope value of each point (x; y) of a curve can be calculated as: y/x. This counts for each point of a flow curve with the pair of values ( γ ̇ ; τ). The result of this calculation again corresponds to the viscosity value, this is because: η = τ/ γ ̇ . Therefore, the η( γ ̇ )-curve can be calculated point by point from the τ( γ ̇ )-curve. Correspondingly, a steeper slope of the flow curve results in a higher level of the viscosity curve (see Figures 2.5 and 2.6). Usually today, this calculation is performed by a software program.

       Figure 2.5: Flow curves of two ideal-viscous fluids

       Figure 2.6: Viscosity curves of two ideal-viscous fluids

      The values of the shear viscosity of ideal-viscous fluids or Newtonian fluids are independent of the degree and duration of the shear load applied.

      Viscosity values of ideal-viscous liquids are often measured using flow cups, capillary viscometers, falling-ball viscometers or Stabinger viscometers (see Chapters 11.3 to 11.6). However, when using these simple devices, the results do not accurately mirror the more complex behavior of non-Newtonian liquids (see for example Chapter 11.3.1.2c: change of shear rates in capillaries).

      For “Mr. and Ms. Cleverly”

       Figure 2.7: The dashpot model to illustrate ideal-

      viscous behavior

       Figure 2.8: A shock absorber which can be loaded from both sides [2.24]

      b) The dashpot model

      The dashpot model is used to illustrate the behavior of ideal-viscous fluids or Newtonian liquids, respectively (see Figure 2.7). Mechanically similar examples are gas or liquid shock absorbers (see Figure 2.8).

      Ideal-viscous flow behavior,

      explained by the behavior of a dashpot

      1) When under load

      Under a constant force, СКАЧАТЬ