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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 whole shear gap. Therefore, the shear rate is independent of the position of any flowing layer in the shear gap. Of course, this applies only if the shear conditions are met as mentioned in the beginning of Chapter 2.2. However, this does not apply to the velocity v which decreases from the maximum value vmax on the upper, movable plate to the minimum value vmin = 0 on the lower, immovable plate. Therefore, when testing pure liquids, sometimes as a synonym for shear rate the term velocity gradient is used (e. g. in ASTM D4092).