P. Pallav, PhD

The upper body is moving with a velocity (V), while the lower body stands still. The gap (h) is filled with a fluid. The arrows indicate the velocity of each layer v(z).

When the fluid is a viscosity fluid, the shear stress, is calculated as

Velocity Gradient

In the figure the velocity in the fluid v(z) increases from zero at the lower surface to V at the upper surface. In fluid mechanics this increase is described as the velocity gradient v', which is identical to the shear strain rate, or simply shear rate known in solid mechanics. Mathematically it is the derivative of the velocity v(z),

In the simple case in the figure, v' = V/h, so Equation 1a can be rewritten as Newton's law of viscosity

t = µ x v', Equation 1,

The velocity gradient is an important parameter to erosive (CFA) wear, because it indicates how much the velocity increases from zero at the  surface to greater values slightly away from it. In this way it is a measure of the velocity of the erosive particles in the fluid near the surface. (see Erosive Actvity)

Power Law Fluids

When the gap is filled with a viscosity fluid, the shear stress (t) can be calculated with Newton's law of viscosity (Equation 1)

There are many fluid to which Newton's law does not (fully) apply. In many cases it is possible to use the power law formula, which was derived from Newton's law:

t = µ x (v')^b, Equation 3,

Where b is an exponent, which characterizes the type of fluid.

Obviously, if b=1, this equation becomes Newton's law of viscosity, the fluid is classified as a Newtonian or a viscosity fluid and µ represents the (dynamic or absolute) viscosity.

The standard example of a dilatant fluid were b>1 is sand. When Newton's law is (incorrectly) applied, the viscosity appears to increase with the shear rate: walking slowly requires hardly less energy than walking fast.

If b=0, the fluid is a plastic (solid) and µ represents the shear strength.

The range of so-called pseudo plastic materials, where 0<b<1, covers the many substances, which have a flow behavior in between that of solid and a viscosity fluid. In this category fall a wide range of food-like substances, paper slurries, paint, etc., and of course the slurry used with the Acta Wear Machine.

Of the straight lines the black and the violet one are viscosity fluids (b=1 in Equation 3, just above). The black might be water (µ low) and the violet heavy oil (µ high). The yellow line is a plastic (b≈0), its shear strength is the height of the horizontal line.

The blue (µ low) and the green line (µ high) are pseudo plastics (b=½). The brown upwards bending line represents a dilatant fluid (b=2).

The red line is a Bingham plastic (below) with its minimum flow stress at the point where it hits the t-axis.

With the exception of the Bingham Plastic, all curves are drawn with the Power Law equation, equation 3, above.

Bingham fluids

Pastes are often Bingham fluids. If you put a bit on a mixing pad, you will find that it doesn't run off, not even when you hold the pad upside down. This is because with Bingham fluids the stress needs to exceed a (small) minimum value, before flow will occur. It is as if these materials have a little bit of strength.

The flow with Bingham fluids, when occurring, may be of any type (dilatant, pseudo plastic, etc.). Both the flow behavior and the strength may depend on the 'stirring' history, which results in a combination with thixotropic or rheopectic flow (below).

Many food like substances have Bingham properties, but because the 'strength' is usually rather small with respect to stresses caused by shearing, it is often not necessary to take it into account.

Other Fluids

Although this falls outside of the scope of this page, some other fluid behavior will be mentioned, because many people seem to confuse pseudo plasticity with thixotropy and dilatancy with rheopectic behavior.

In contrast to pseudo plastic and dilatant flow, thixotropic and rheopectic flow are time-dependent. 

When thixotropic or rheopectic fluids are stirred for some time, the viscosity will change. This change often remains some time after the stirring has stopped. 

Whipping cream is an example of rheopectical behavior: stirring (whipping) will make the cream stiffer, but it will get more fluid again after some time (quicker if you keep it out of the fridge).

Tomato ketchup is both pseudo plastic and thixotropic: if you shake it really hard, the bottle will need less beating to get the ketchup out.