11,693 research outputs found
A waiting time phenomenon for thin film equations
We prove the occurrence of a waiting time phenomenon for solutions to fourth order degenerate parabolic differential equations which model the evolution of thin films of viscous fluids. In space dimension less or equal to three, we identify a general criterion on the growth of initial data near the free boundary which guarantees that for sufficiently small times the support of strong solutions locally does not increase. It turns out that this condition only depends on the smoothness of the diffusion coefficient in its point of degeneracy. Our approach combines a new version of Stampacchia's iteration lemma with weighted energy or entropy estimates. On account of numerical experiments, we conjecture that the
aforementioned growth criterion is optimal
Existence of solutions for a higher order non-local equation appearing in crack dynamics
In this paper, we prove the existence of non-negative solutions for a
non-local higher order degenerate parabolic equation arising in the modeling of
hydraulic fractures. The equation is similar to the well-known thin film
equation, but the Laplace operator is replaced by a Dirichlet-to-Neumann
operator, corresponding to the square root of the Laplace operator on a bounded
domain with Neumann boundary conditions (which can also be defined using the
periodic Hilbert transform). In our study, we have to deal with the usual
difficulty associated to higher order equations (e.g. lack of maximum
principle). However, there are important differences with, for instance, the
thin film equation: First, our equation is nonlocal; Also the natural energy
estimate is not as good as in the case of the thin film equation, and does not
yields, for instance, boundedness and continuity of the solutions (our case is
critical in dimension in that respect)
Growth saturation of unstable thin films on transverse-striped hydrophilic-hydrophobic micropatterns
Using three-dimensional numerical simulations, we demonstrate the growth
saturation of an unstable thin liquid film on micropatterned
hydrophilic-hydrophobic substrates. We consider different transverse-striped
micropatterns, characterized by the total fraction of hydrophilic coverage and
the width of the hydrophilic stripes. We compare the growth of the film on the
micropatterns to the steady states observed on homogeneous substrates, which
correspond to a saturated sawtooth and growing finger configurations for
hydrophilic and hydrophobic substrates, respectively. The proposed
micropatterns trigger an alternating fingering-spreading dynamics of the film,
which leads to a complete suppression of the contact line growth above a
critical fraction of hydrophilic stripes. Furthermore, we find that increasing
the width of the hydrophilic stripes slows down the advancing front, giving
smaller critical fractions the wider the hydrophilic stripes are. Using
analytical approximations, we quantitatively predict the growth rate of the
contact line as a function of the covering fraction, and predict the threshold
fraction for saturation as a function of the stripe width.Comment: 11 pages, 5 figure
Liquid transport generated by a flashing field-induced wettability ratchet
We develop and analyze a model for ratchet-driven macroscopic transport of a
continuous phase. The transport relies on a field-induced dewetting-spreading
cycle of a liquid film with a free surface based on a switchable, spatially
asymmetric, periodic interaction of the liquid-gas interface and the substrate.
The concept is exemplified using an evolution equation for a dielectric liquid
film under an inhomogeneous voltage. We analyse the influence of the various
phases of the ratchet cycle on the transport properties. Conditions for maximal
transport and the efficiency of transport under load are discussed.Comment: 10 pages, 5 figure
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