Single Drop Impacts of Complex Fluids: A Review - Drops and Bubbles in Contact with Solid Surfaces - page 290

Figure 19. Qualitative flow curves for Newtonian fluids (long dashed), shear-thinning (short dashed)
and shear-thickening (dotted) fluids, Bingham fluids (continuous), and Herschel-Bulkley fluid
(dash-dot).

where μ 0 and μ ∞ are the viscosities at zero shear and infinite shear, respectively,
and C is the Cross time constant. This parameter is the reciprocal of the strain rate
at which the zero-strain rate component and the power-law component of the flow
curve intersect.
Finally, the Ellis model is obtained by setting μ ∞ = 0 in the Cross model:

τ
τ 1 / 2

n − 1

μ 0

μ =

1 + , (A4)

where τ is the applied shear stress, and τ 1 / 2 is the shear stress at which μ is exactly
half of the zero-shear viscosity value.
Qualitative flow curves displaying the shear stress with respect to the shear rate
for shear-thinning and shear-thickening fluids, as compared with Newtonian fluids,
are shown in Fig. 19.

A1.2. Yield-Stress Fluids

An important type of non-Newtonian fluid is the viscoplastic or yield-stress fluid,
which responds like elastic solids for applied stresses lower a certain threshold
value, called the yield stress, and flows only when the yield stress is overcome.
Practically, such flow behavior occurs in many situations, including slurries and
suspensions, certain polymer solutions, crystallizing lavas, muds and clays, heavy
oils, avalanches, cosmetic creams, hair gel, liquid chocolate, pasty materials, foams
and emulsions.
The simplest constitutive model describing viscoplastic fluids was introduced
by Bingham to characterize the behavior of paints [67], and represents the shear
stress component as a linear function of the velocity gradient, with the intercept τ c
corresponding to the threshold yield point:
˙

γ = 0 , τ τ c ,

(A5)

τ = τ c + μ γ, ˙ τ>τ c .