Single Drop Impacts of Complex Fluids: A Review - Drops and Bubbles in Contact with Solid Surfaces - page 291

A more refined model is the Herschel-Bulkley equation [68], given by:
τ

= Gγ, τ τ c ,

(A6)

γ n , τ = τ c + K ˙ τ >τ c ,

where G is the shear modulus, γ is the shear deformation and τ c is the yield-stress
magnitude. Whilst this model is well established and amongst the most widely used
when analyzing yield-stress behavior, another popular model was proposed by Cas-
son [69]:

μ

√ τ

= √ τ c + γ, τ>τ c . ˙ (A7)

Figure 1 shows the qualitative flow curves for Bingham fluids and Herschel-
Bulkley fluids, as compared with power-law fluids.
According to Eqs (A5)-(A7), the transition from the elastic regime to the fluid
regime is abrupt, which means that the shear stress derivative with respect to the
shear rate exhibits a first-order discontinuity. This represents a major technical issue
when the yield stress fluid constitutive equation is implemented to find analytical
or (especially) numerical solutions of fluid flow problems. To remove this disconti-
nuity, Papanastasiou proposed a constitutive equation featuring a smooth transition
between the two regimes [70], which provides a better description of real materials:

γ n , τ = τ c [ 1 − exp ( − m γ) ˙ ]+ C ˙ (A8)

where m is a material-dependent constant with values of the order of 10 2 .
Research into viscoplastic fluids, their measurement and characterization is ex-
tensive and has been summarized in numerous reviews [71-74]. One matter still
subject of debate is the definition of yield-stress fluid itself, that is, whether flu-
ids can actually exhibit such a physical property as the yield-stress. The review by
Barnes [71] examines the evidence for and against its existence, and argues that
whereas the concept of a definable yield-stress has proven and continues to prove
useful in a whole range of applications, if viscosity is plotted as a function of the
shear stress, one can clearly identify a Newtonian plateau when the velocity gradient
tends to zero (typically less than 10 − 5 s − 1 ), which implies that the material contin-
ues to creep although this can be observed only on very long timescales. However,
in many practical situations time frame of observation is much shorter than the time
necessary for viscoplastic fluids to exhibit measurable flow characteristics.
Whilst several fluids exhibit an apparent yield stress, Carbopol dispersions are
probably the most thoroughly studied model viscoplastic fluid system. Carbopol
consists of highly cross-linked polymer particles, with dangling free ends of poly-
mer gel strands that strongly interact with adjacent microgel particles, resulting into
to a very high viscosity at low shear stress [28, 73]. Carbopol dispersions and gels
are found in dozens of everyday products, ranging from toothpastes, through hair
and shower gels, to artificial tears.