Figure 101. Successive views of a m

Figure 101. Successive views of a mayonnaise sample at different times, pushed out of a nozzle of radius h0 = 3 mm. The initial time (0 s) corresponds to the separation of the previous extrudate. Reprinted with permission from [523]. Copyright 2005 by the American Physical Society. found, which connect between the symmetric Stokes solution branch and the asymmetric Navier–Stokes solution branch far away from n = 1. In the Newtonian case n = 1 itself, no symmetric solutions are known to exist if inertia is present. Just as in the purely Newtonian case, the axial exponent β is no longer determined by dimensional analysis if inertia is absent, but has to be obtained from solving the similarity problem explicitly, as done in detail in [569]. As expected, β increases with decreasing n, making the profile less slender. 6.3. Other non-Newtonian behaviour Another interesting type of rheological behaviour is displayed by materials like toothpaste, bread dough or mayonnaise (cf figures 101 and 4), which are so-called ‘yield-stress’ fluids [529]. This means they do not flow if the viscous shear stress they are subjected to is below a threshold value σ0: σijσij < 2σ2 0 . In this regime the viscosity is effectively infinite. Above the yield stress σ0, the viscosity is given by η = η0 + σ0 √2 ˙γ (322) in the so-called Bingham model [529]. Mayonnaise is more realistically described by the Herschel–Buckley model, for which the divergence of the viscosity for ˙γ → 0 is weaker than by the power of −1 in (322) [523]. Using the Lagrangian methods developed in [296, 571] analyses dripping using a generalized form of (322) (the two- fluid model), neglecting both inertia and surface tension. In the lower part of the drop the material does not move, but once gravitational stresses have become large enough the material starts to yield and pinch-off eventually occurs. The form of pinch-off is very similar to that of a purely Newtonian fluid with the same approximations, i.e. the liquid bridge has to extend to infinity before pinch-off occurs (cf (226)). Experiments, on the other hand, exhibit breakup profiles close to conical (see figure 101) [523], indicating that a description which predicts very long threads is not realistic. Clearly, it will be crucial to include surface tension into the description. Jets of superfluid liquid helium are also a potentially interesting subject, because of the unique hydrodynamic properties of the superfluid phase. Rayleigh breakup has been demonstrated as a useful source of superfluid helium droplets [525], and pinch-off of both normal and superfluid helium has been studied in [572]. Unfortunately, no qualitative difference between the two cases has been detected. An even more exotic ‘fluid’ that has recently attracted interest is formed by granular materials or sand [573]. Curiously, a jet of sand has been one of the earliest subjects investigated by photographic means [34], and one observes a decay into drops or clusters of sand, which is superficially similar to Rayleigh breakup [34, 526, 574]. A recent study suggests this to be a purely kinematic effect, based on the free fall of particles [526], pouring out of a container. Khamontoff [34] already investigated the effect of gravity, projecting the jet horizontally using compressed air. According to his preliminary investigation, the tendency to form clusters remained the same, which appears to contradict the simple kinematic theory [526]. Hence the true mechanism for cluster formation in a granular jet remains an open problem. 6.4. Surfactants The value of the surface tension, in particular for aqueous solutions, depends strongly on the concentration of surfaceactive molecules or ‘surfactants’ [575]. Either surfactant molecules are inherently present, as in inks and DNA arrays [73], or they are added to control the drop sizes in sprays [576]. In general the surfactant distribution is non-uniform, hence non-trivial effects are to be expected, apart from renormalizing the surface tension. Most existing studies consider the insoluble case ( [527] is an exception), for which the surfactant remains on the surface. Then, the surfactant concentration (s, t)is described by [577] ∂ ∂t + ∇s( v) = Ds∇2 s , (323) where ∇s = (I – nn) • ∇ is the surface gradient operator. A non-dimensional measure of the diffusion constant Ds is the surface Peclet number P e = √Rγ0/ρ/Ds, where γ0 is the surface tension in the absence of surfactant. The Peclet number is usually large (≈106 for mm-sized jets [578]), so diffusion can be neglected [579]. The equation most commonly used to describe the dependence of surface tension on concentration is the non-linear Frumkin equation [578] γ = γ0 + mRT ln(1 − / m). (324) At the maximum packing density m the surface tension reaches a plateau, an effect which is not captured by (324). Apart from changing the value of surface tension, the most important effect of (324) is that it introduces surface tension gradients: this results in Marangoni forces [42], which act to restore a uniform distribution of surfactant. The most important parameter is thus the surface activity number β = mRT /γ0, which measures the ‘effectiveness’ of the surfactant. For β = 0 the surfactant has no effect; in the extreme case of β → ∞ the surface becomes rigid in the sense that any tangential flow must be balanced by radial contraction [579]. Typically, β is of order one (β ≈ 0.3 in the experiments [578]). 71