Coalescing drops spontaneously jump out of plane on a variety of biological and synthetic superhydrophobic surfaces, with potential applications ranging from self-cleaning materials to self-sustained condensers. To investigate the mechanism of self-propelled jumping, we report three-dimensional phase-field simulations of two identical spherical drops coalescing on a flat surface with a contact angle of 180 • . The numerical simulations capture the spontaneous jumping process, which follows the capillary-inertial scaling. The out-of-plane directionality is shown to result from the counter-action of the substrate to the impingement of the liquid bridge between the coalescing drops. A viscous cutoff to the capillary-inertial velocity scaling is identified when the Ohnesorge number of the initial drops is around 0.1, but the corresponding viscous cutoff radius is too small to be tested experimentally. Compared to experiments on both superhydrophobic and Leidenfrost surfaces, our simulations accurately predict the nearly constant jumping velocity of around 0.2 when scaled by the capillary-inertial velocity. By comparing the simulated drop coalescence processes with and without the substrate, we attribute this low non-dimensional velocity to the substrate intercepting only a small fraction of the expanding liquid bridge.
Self-propelled jumping upon drop coalescence has been observed on a variety of textured superhydrophobic surfaces, where the jumping motion follows the capillary–inertial velocity scaling as long as the drop radius is above a threshold. In this paper, we report an experimental study of the self-propelled jumping on a Leidenfrost surface, where the heated substrate gives rise to a vapour layer on which liquid drops float. For the coalescence of identical water drops, we have tested initial drop radii ranging from 20 to $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}500\ \mu \mathrm{m}$, where the lower bound is related to the spontaneous takeoff of individual drops and the upper bound to gravitational effects. Regardless of the approaching velocity prior to coalescence, the measured jumping velocity is around 0.2 when scaled by the capillary–inertial velocity. This constant non-dimensional velocity holds for the experimentally accessible range of drop radii, and we have found no cutoff radius for the scaling, in contrast to prior experiments on textured superhydrophobic surfaces. The Leidenfrost experiments quantitatively agree with our numerical simulations of drop coalescence on a flat surface with a contact angle of 180°, suggesting that the cutoff is likely to be due to drop–surface interactions unique to the textured superhydrophobic surfaces.
The rheology of a dilute two-dimensional suspension of vesicles (closed bags of a lipid bilayer membrane) is studied by numerical simulations. The numerical methods used are based on the boundary integral formulation (Green's function technique) and the phase field approach, which has become a quite popular and powerful tool for the numerical study of free-boundary problems. The imposed flow is an unbounded linear shear. The goal of the present study is to elucidate the link between the rheology of vesicle suspensions and the microscopic dynamics of the constituent particles (tanktreading and tumbling motions). A comparison with emulsion rheology reveals the central role played by the membrane. In particular, at low viscosity ratio λ (defined as the viscosity of the internal fluid over that of the ambient one), the effective viscosity decreases with λ, while the opposite trend is exhibited by emulsions, according to the classical Taylor result. This fact is explained by considering the velocity field of the ambient fluid. The area-incompressibility of the vesicle membrane modifies the surrounding velocity field in a quite different manner than what a drop does. The overall numerical results in two dimensions are in reasonable agreement with the three-dimensional analytical theory derived recently in the small deformation limit (quasi-spherical shapes). The finding that the simulations in two dimensions capture the essential features of the three-dimensional rheology opens the way for extensive and large-scale simulations for semi-dilute and concentrated vesicle suspensions. We discuss some peculiar effects exhibited by the instantaneous viscosity in the tumbling regime of vesicles. Finally, the rheology is found to be relatively insensitive to shear rate.
International audienceWe perform particle scale simulations of suspensions submitted to shear reversal. The simulations are based on the Force Coupling Method, adapted to account for short range lubrication interactions together with direct contact forces between particles, including surface roughness, contact elasticity and solid friction. After shear reversal, three consecutive steps are identified in the viscosity transient: an instantaneous variation, followed by a rapid contact force relaxation, and finally a long time evolution. The separated contributions of hydrodynamics and contact forces to the viscosity are investigated during the transient, allowing a qualitative understanding of each step. In addition, the influence of the contact law parameters (surface roughness height and friction coefficient) on the transient are evaluated. Concerning the long time transient, the difference between the steady viscosity and minimum viscosity is shown to be proportional to the contact contribution to the steady viscosity, allowing in principle easy determination of the latter in experiments. The short time evolution is studied as well. After the shear reversal, the contact forces vanish over a strain that is very short compared to the typical strain of the long time transient, allowing to define an apparent step between the viscosity before shear reversal and after contact force relaxation. This step is shown to be an increasing function of the friction coefficient between particles. Two regimes are identified as a function of the volume fraction. At low volume fraction, the step is small compared to the steady contact viscosity, in agreement with a particle pair model. As the volume fraction increases, the value of the viscosity step increases faster than the steady contact viscosity, and, depending on the friction coefficient, may approach it
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