Source-type solutions to thin-film equations in 

higher dimensions

Ferreira, Raúl; Bernis, Francisco

doi:10.1017/s0956792597003197

Cited by 61 publications

(79 citation statements)

References 22 publications

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“…In some sense, these terms slow down the dynamics and move the symmetry point to infinity in de-singularized coordinates, which makes the construction of solutions by a shooting argument, starting at the contact line, less accessible and a shooting argument as in [22], starting at the point of symmetry, more favorable.…”

Section: 2mentioning

confidence: 99%

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A dynamical systems approach for the contact-line singularity in thin-film flows

2016

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Abstract. We are interested in a complete characterization of the contact-line singularity of thin-film flows for zero and nonzero contact angles. By treating the model problem of source-type self-similar solutions, we demonstrate that this singularity can be understood by the study of invariant manifolds of a suitable dynamical system. In particular, we prove regularity results for singular expansions near the contact line for a wide class of mobility exponents and for zero and nonzero dynamic contact angles. Key points are the reduction to center manifolds and identifying resonance conditions at equilibrium points. The results are extended to radially-symmetric source-type solutions in higher dimensions. Furthermore, we give dynamical systems proofs for the existence and uniqueness of self-similar droplet solutions in the nonzero dynamic contact-angle case.

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Section: 2mentioning

confidence: 99%

“…Existence, uniqueness, and leading-order asymptotics of this problem were studied by Bernis and Ferreira in [22] under the assumption…”

Section: Appendix a Higher Dimensionsmentioning

confidence: 99%

A dynamical systems approach for the contact-line singularity in thin-film flows

2016

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“…This case is nonphysical but bears the same features as the original system and admits a selfsimilar analytical solution, 19 which can be used to assess the validity of the numerical solution. Diez and Kondic 11 showed that for an initial profile which approximates the delta function and Neumann boundary conditions, the evolution of the maximum film thickness h val satisfies Figure 3 shows the analytical and numerical solutions with V val = 3 and a 0.01 thick precursor film.…”

Section: Validation Of the Solution Proceduresmentioning

confidence: 99%

Modeling the coalescence of sessile droplets

Sellier

Trelluyer

2009

Biomicrofluidics

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This paper proposes a simple scenario to describe the coalescence of sessile droplets. This scenario predicts a power-law growth of the bridge between the droplets. The exponent of this power law depends on the driving mechanism for the spreading of each droplet. To validate this simple idea, the coalescence is simulated numerically and a basic experiment is performed. The fluid dynamics problem is formulated in the lubrication approximation framework and the governing equations are solved in the commercial finite element software COMSOL. Although a direct comparison of the numerical results with experiment is difficult because of the sensitivity of the coalescence to the initial and operating conditions, the key features of the event are qualitatively captured by the simulation and the characteristic time scale of the dynamics recovered. The experiment consists of inducing coalescence by pumping a droplet through a substrate which grows and ultimately coalesces with another droplet resting on the substrate. The coalescence was recorded using high-speed imaging and also confirmed the power-law growth of the neck.

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“…Ferreira and Bernis [14] study the equation in higher space dimensions: u t = −∇ · (u n ∇∆u). They consider radially-symmetric solutions and find that if 0 < n < 3 there exist compactly supported source-type solutions with zero contact angles.…”

Section: 2mentioning

confidence: 99%

“…The authors present three methods for solving the ODEs that the profiles U must satisfy. They use fixed point arguments (as in [14]) to prove the existence of the source-type solutions. They use a shooting argument to find the dipole solutions in the n = 1 case.…”

Section: 2mentioning

confidence: 99%

Selfsimilar blowup of unstable thin-film equations

Slepčev¹,

Pugh²

2005

Indiana Univ. Math. J.

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ABSTRACT. We study selfsimilar blowup of long-wave unstable thin-film equations with critical powers of nonlinearities: ut = −(u n uxxx + u n+2 ux)x. We show that the equation cannot have selfsimilar solutions (with zero contact angles) that blow up in finite time if n ≥ 3/2. We show that for 0 < n < 3/2 there are compactly supported, symmetric, selfsimilar solutions (with zero contact angles) that blow up in finite time. Moreover there exist families of these solutions with any number of local maxima. We also study the asymptotic behaviour of these selfsimilar solutions as n approaches 3/2 and obtain a sharp lower bound on the height of solutions one time unit before the blowup. We also prove qualitative properties of solutions; for example, the profile of a selfsimilar solution that blows up in finite time can always be bounded from below by a compactly supported steady state.

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Source-type solutions to thin-film equations in higher dimensions

Cited by 61 publications

References 22 publications

A dynamical systems approach for the contact-line singularity in thin-film flows

A dynamical systems approach for the contact-line singularity in thin-film flows

Modeling the coalescence of sessile droplets

Selfsimilar blowup of unstable thin-film equations

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