The Rheology Handbook. Thomas Mezger

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       Figure 2.1: The Two-Plates model for shear tests to illustrate the velocity distribution of a flowing fluid in the shear gap

       Figure 2.2: Laminar flow in the form of planar fluid layers

      The Two-Plates model is used to define fundamental rheological parameters (see Figure 2.1). The upper plate with the (shear) area A is set in motion by the (shear) force F and the resulting velocity v is measured. The lower plate is fixed (v = 0). Between the plates there is the distance h, and the sample is sheared in this shear gap. It is assumed that the following shear conditions are occurring:

      Accurate calculation of the rheological parameters is only possible if both conditions are met.

      Experiment 1: The stack of beer mats

      Each one of the individual beer mats represents an individual flowing layer. The beer mats are showing a laminar shape, and therefore, they are able to move in the form of layers along one another (see Figure 2.2). Of course, this process takes place without vortices, thus without showing any turbulent behavior.

      The real geometric conditions in rheometer measuring systems (or measuring geometries) are not as simple as in the Two-Plates model. However, if a shear gap is narrow enough, the necessary requirements are largely met and the definitions of the following rheological parameters can be used.

2.2.1Shear stress

      Definition of the shear stress:

      Equation 2.1

      τ = F/A

      τ (pronounced: tou); with the shear force F [N] and the shear area (or shearing surface area) A [m2], see Figure 2.1. The following holds: 1 N = 1 kg · m/s2

      The unit of the shear stress is [Pa], (pascal).

      Blaise Pascal (1623 to 1662 [2.1]) was a mathematician, physicist, and philosopher.

      For conversions: 1 Pa = 1 N/m2 = 1 kg/m · s2

      A previously used unit was [dyne/cm2]; with: 1 dyne/cm2 = 0.1 Pa

      Note: [Pa] is also the unit of pressure

      100 Pa = 1 hPa (= 1 mbar); or 100,000 Pa = 105 Pa = 0.1 MPa (= 1 bar)

      Example: In a weather forecast, the air pressure is given as 1070 hPa (hecto-pascal; = 107 kPa).

      Some authors take the symbol σ for the shear stress (pronounced: sigma) [2.2] [2.3]. However, this symbol is usually used for the tensile stress (see Chapters 4.2.2, 10.8.4.1 and 11.2.14). To avoid confusion and in agreement with the majority of current specialized literature and standards, here, the symbol τ will be used to represent the shear stress (see e. g. ISO 3219-1, ASTM D4092 and DIN 1342-1).

2.2.2Shear rate

      Definition of the shear rate:

      Equation 2.2

       γ ̇ = v/h

       γ ̇ (pronounced: gamma-dot); with the velocity v [m/s] and the distance h [m] between the plates, see Figure 2.1.

      The unit of the shear rate is [1/s] or [s -1 ], called “reciprocal seconds”.

      Sometimes, the following terms are used as synonyms: strain rate , rate of deformation, shear gradient , velocity gradient .

      Previously, the symbol D was often taken instead of γ ̇ . Nowadays, almost all current standards are recommending the use of γ ̇ (see e. g. ISO 3219-1, ASTM D4092). Table 2.1 presents typical shear rate values occurring in industrial practice.

      For “Mr. and Ms. Cleverly”

      a) Definition of the shear rate using differential variables

      Equation 2.3

       γ ̇ = dv/dh

      flowing layers, СКАЧАТЬ