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.
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