The Rheology Handbook. Thomas Mezger

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:

      1 The sample shows adhesion to both plates without any wall-slip effects.

      2 There are laminar flow conditions, i. e. flow can be imagined in the form of layers. Therefore, there is no turbulent flow, i. e. no vortices are appearing.

      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, and the “infinitely” (differentially) small thickness dh of a single flowing layer (see Figure 2.2).

Table 2.1: Typical shear rates of technical processes
Process	Shear rates γ ̇ (s-1)	Practical examples
physical aging, long-term creep within days and up to several years	10-8 ... 10-5	solid polymers, asphalt
cold flow	10-8 ... 0.01	rubber mixtures, elastomers
sedimentation of particles	≤ 0.001 ... 0.01	emulsion paints, ceramic suspensions, fruit juices
surface leveling of coatings	0.01 ... 0.1	coatings, paints, printing inks
sagging of coatings, dripping, flow under gravity	0.01 ... 1	emulsion paints, plasters, chocolate melt (couverture)
self-leveling at low-shear conditions in the range of the zero-shear viscosity	≤ 0.1	silicones (PDMS)
mouth sensation	1 ... 10	food
dip coating	1 ... 100	dip coatings, candy masses
applicator roller, at the coating head	1 ... 100	paper coatings
thermoforming	1 ... 100	polymers
mixing, kneading	1 ... 100	rubbers, elastomers
chewing, swallowing	10 ... 100	jelly babies, yogurt, cheese
spreading	10 ... 1000	butter, spreadcheese
extrusion	10 ... 1000	polymer melts, dough,ceramic pastes, tooth paste
pipe flow, capillary flow	10 ... 104	crude oils, paints, juices, blood
mixing, stirring	10 ... 104	emulsions, plastisols,polymer blends
injection molding	100 ... 104	polymer melts, ceramic suspensions
coating, painting, brushing, rolling, blade coating (manually)	100 ... 104	brush coatings, emulsion paints, wall paper paste, plasters
spraying	1000 ... 104	spray coatings, fuels, nose spray aerosols, adhesives
impact-like loading	1000 ... 105	solid polymers
milling pigments in fluid bases	1000 ... 105	pigment pastes for paints and printing inks
rubbing	1000 ... 105	skin creams, lotions, ointments
spinning process	1000 ... 105	polymer melts, polymer fibers
blade coating (by machine), high-speed coating	1000 ... 107	paper coatings, adhesive dispersions
lubrication of engine parts	1000 ... 107	mineral oils, lubricating greases

      There is a linear velocity distribution between the plates, since the velocity v decreases linearly in the shear gap. Thus, for laminar and ideal-viscous flow, the velocity difference between all neighboring layers are showing the same value: dv = const. All the layers are assumed to have the same thickness: dh = const. Therefore, the shear rate is showing a constant value everywhere between the plates of the Two-Plates model since

       γ ̇ = dv/dh = const/const = const (see Figure 2.3).

       Figure 2.3: Velocity distribution and shear rate in the shear gap of the Two-Plates model

      Both γ ̇ and v provide information about the velocity of a flowing fluid. The advantage of selecting the shear rate is that it shows a constant value throughout the