The spreading dynamics of Newtonian liquids have been extensively studied in hydrophilic and hydrophobic surfaces and its behavior has been extensively explored over the last years. However, drop impact of Non-Newtonian liquids still needs further study. Luu and Forterre (J. Fluid Mech., 632, 2009) successfully found scaling laws for yield-stress fluids on hydrophilic surfaces. They also uncovered interesting and yet unexplained regimes when the impact was performed on a superhydrophobic surface. In this work, we perform drop impact experiments on micro-patterned surfaces with two types of non-Newtonian liquids: one showing shear-thickening behavior and another one showing shear-thinning. Our results show that a typical shear-thickening liquid as cornstarch -at least a the relatively low concentration of 30%w/w-spreads according to the scaling laws of Newtonian liquids, whereas visco-elastic liquids as Carbopol behave as predicted by Luu and Forterre for impact on hydrophilic surfaces, but show different scaling laws when they impact on superhydrophobic surfaces.
This work considers a nearly-spherical bubble and a nearly-flat free surface interacting under buoyancy at vanishing Bond number Bo. For each perturbed surface, the deviation from the unperturbed shape is asymptotically obtained at leading order on Bo. The task appeals to the normal traction exerted on the unperturbed surface by the Stokes flow due to a spherical bubble translating toward a flat free surface. The free surface problem is then found to be well-posed and to admit a solution in closed form when gravity is still present in the linear differential equation governing the perturbed profile through a term proportional to Bo. In contrast, the bubble problem amazingly turns out to be over-determined. It however becomes well-posed if the requirement of horizontal tangent planes at the perturbed bubble north and south poles is discarded or if the term proportional to Bo is omitted. Both previous approaches turn out to predict for small Bond number quite close solutions except in the very vicinity of the bubble poles. The numerical solution of the proposed asymptotic analysis shows, in the overlapping range Bo = O(0.1) and for both the bubble and the free surface perturbed shapes, a good agreement with a quite different boundary element approach developed in [Phys. Fluids 23, 092102 (2011)]. It also provides approximated bubble and free surface shapes whose sensitivity to the bubble location is examined.
The axisymmetric gravity-driven dynamics of a bubble rising toward a free surface is addressed for gas-liquid interfaces having unequal surface tensions. The liquid flow is governed by the Stokes equations which are here solved using a boundary element method in axisymmetric configuration. Within this framework, two dimensionless numbers arise: the Bond number Bo 1 based on the surface tension of the bubble interface and the surface tension ratioγ comparing the free surface and bubble surface tensions. Under a careful and discussed selection of the code key settings (number of boundary elements, initial bubble location, and distance beyond which the free surface is truncated), it has been possible to numerically and accurately track in time the bubble and free surface shapes for several values of (Bo 1 ,γ). The long-time shapes are found to deeply depend upon both Bo 1 andγ and also to compare well with the shapes predicted in Princen and Mason ["Shape of a fluid drop at a fluid-liquid interface. II. Theory for three-phase systems," J. Colloid. Sci. 20, 246-266 (1965)] using a hydrostatic model in which both surfaces are touching. Similarly, the drainage dynamics of the liquid film thickness between the bubble and the free surface depends on (Bo 1 ,γ). The long-time film thickness exponentially decays in time and a so-called thinning rate α for which the numerical behaviors and a simple model reveal two basic behaviors: (i) at small Bond number, α behaves as 1/Bo 1 and (ii) at large Bond number, α is nearly constant. In addition, it is found that in the entire range of the quantity χ = (1 +γ)Bo 1 /(2γ), the thinning rate α is well approximated by the function 1/(18 χ) + α ∞ with α ∞ ≈ 0.158. Such a result also permits one to estimate the typical drainage time versus the initial bubble radius a, the liquid density ρ and viscosity µ, the gravity and the free surface, and bubble surface tensions.
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