219 research outputs found
Coarsening rates for the dynamics of slipping droplets
We derive reduced finite dimensional ODE models starting from one dimensional
lubrication equations describing coarsening dynamics of droplets in nanometric
polymer film interacting on a hydrophobically coated solid substrate in the
presence of large slippage at the liquid/solid interface. In the limiting case
of infinite slip length corresponding in applications to free films a
collision/absorption model then arises and is solved explicitly. The exact
coarsening law is derived for it analytically and confirmed numerically.
Existence of a threshold for the decay of initial distributions of droplet
distances at infinity at which the coarsening rates switch from algebraic to
exponential ones is shown
Self-similar finite-time singularity formation in degenerate parabolic equations arising in thin-film flows
A thin liquid film coating a planar horizontal substrate may be unstable to perturbations in the film thickness due to unfavourable intermolecular interactions between the liquid and the substrate, which may lead to finitetime rupture. The self-similar nature of the rupture has been studied before
by utilising the standard lubrication approximation along with the Derjaguin (or disjoining) pressure formalism used to account for the intermolecular interactions, and a particular form of the disjoining pressure with exponent n = 3 has been used, namely, Π(h) ∝ −1/h3, where h is the film thickness. In
the present study, we use a numerical continuation method to compute discrete solutions to self-similar rupture for a general disjoining pressure exponent n (not necessarily equal to 3), which has not been previously performed. We focus on axisymmetric point-rupture solutions and show for the first time that pairs of solution branches merge as n decreases, starting at nc ≈ 1.485. We verify that this observation also holds true for plane-symmetric line-rupture solutions for which the critical value turns out to be slightly larger than for the axisymmetric case, nplane c ≈ 1.499. Computation of the full time-dependent problem also demonstrates the loss of stable similarity solutions and the subsequent onset of cascading, increasingly small structures
Thermal rupture of a free liquid sheet
We consider a free liquid sheet, taking into account the dependence of
surface tension on temperature, or concentration of some pollutant. The sheet
dynamics are described within a long-wavelength description. In the presence of
viscosity, local thinning of the sheet is driven by a strong temperature
gradient across the pinch region, resembling a shock. As a result, for long
times the sheet thins exponentially, leading to breakup. We describe the quasi
one-dimensional thickness, velocity, and temperature profiles in the pinch
region in terms of similarity solutions, which posses a universal structure.
Our analytical description agrees quantitatively with numerical simulations
Stability and Dynamics of Self-Similarity in Evolution Equations
A methodology for studying the linear stability of self-similar solutions is discussed. These fundamental ideas are illustrated on three prototype problems: a simple ODE with finite-time blow-up, a second-order semi-linear heat equation with infinite-time spreading solutions, and the fourth-order Sivashinsky equation with finite-time self-similar blow-up. These examples are used to show that self-similar dynamics can be studied using many of the ideas arising in the study of dynamical systems. In particular, the use of dimensional analysis to derive scaling invariant similarity variables is discussed, as well as the role of symmetries in the context of stability of self-similar dynamics. The spectrum of the linear stability problem determines the rate at which the solution will approach a self-similar profile. For blow-up solutions it is demonstrated that the symmetries give rise to positive eigenvalues associated with the symmetries, and it is shown how this stability analysis can identify a unique stable (and observable) attracting solution from a countable infinity of similarity solutions
Universal Self-Similar Attractor in the Bending-Driven Leveling of Thin Viscous Films
We study theoretically and numerically the bending-driven leveling of thin
viscous films within the lubrication approximation. We derive the Green's
function of the linearized thin-film equation and further show that it
represents a universal self-similar attractor at long times. As such, the
rescaled perturbation of the film profile converges in time towards the
rescaled Green's function, for any summable initial perturbation profile. In
addition, for stepped axisymmetric initial conditions, we demonstrate the
existence of another, short-term and one-dimensional-like self-similar regime.
Besides, we characterize the convergence time towards the long-term universal
attractor in terms of the relevant physical and geometrical parameters, and
provide the local hydrodynamic fields and global elastic energy in the
universal regime as functions of time. Finally, we extend our analysis to the
non-linear thin-film equation through numerical simulations
Axisymmetric self-similar rupture of thin films with general disjoining pressure
A thin film coating a dewetting substrate may be unstable to perturbations in the thickness, which leads to finite time rupture. The self-similar nature of the rupture has been studied by numerous authors for a particular form of the disjoining pressure, with exponent
n = 3. In the present study we use a numerical continuation method to compute discrete solutions to self-similar rupture for a general disjoining pressure exponent n. Pairs of solution branches merge when n is close to unity, indicating that a more detailed examination of the dynamics of a thin film in this regime is warranted. We also numerically evaluate the power law behaviour of characteristic quantities of
solutions in the limit of large branch number
The structure of a dewetting rim with strong slip: the long-time evolution
When a thin viscous film dewets from a solid substrate, the liquid forms a characteristic rim near the contact line as the contact line retracts. The shape of the rim and also the retraction rate vary strongly with the amount of slip that occurs at the liquid-solid substrate. If the slip length is very large compared to the thickness of the film, extensional stresses dominate the shear stresses, and the film evolution can be modelled by a thin-film model similar to the ones that occur in freely suspended films, with a correction from the viscous friction due to the large but finite slip. Asymptotic investigation of this model reveals that the rim has an amazingly rich asymptotic structure that moreover changes as the solution passes through four distinct time regimes. This paper continues previous work that focused on the first of these regimes [Evans, King, Münch, AMRX 2006:25262, 2006]. The structure of the solution is analyzed in detail via matched asymptotics and then the predictions for the contact line and profile evolution are compared with numerical results
The role of exponential asymptotics and complex singularities in self-similarity, transitions, and branch merging of nonlinear dynamics
We study a prototypical example in nonlinear dynamics where transition to self-similarity in a singular limit is fundamentally changed as a parameter is varied. Here, we focus on the complicated dynamics that occur in a generalised unstable thin-film equation that yields finite-time rupture. A parameter, n, is introduced to model more general disjoining pressures. For the standard case of van der Waals intermolecular forces, n = 3, it was previously established that a countably infinite number of self-similar solutions exist leading to rupture. Each solution can be indexed by a parameter, ϵ = ϵ1 > ϵ2 > · · · > 0, and the prediction of the discrete set of solutions requires examination of terms beyond-all-orders in ϵ. However, recent numerical results have demonstrated the surprising complexity that exists for general values of n. In particular, the bifurcation structure of self-similar solutions now exhibits branch merging as n is varied. In this work, we shall present key ideas of how branch merging can be interpreted via exponential asymptotics
Solving Nonlinear Parabolic Equations by a Strongly Implicit Finite-Difference Scheme
We discuss the numerical solution of nonlinear parabolic partial differential
equations, exhibiting finite speed of propagation, via a strongly implicit
finite-difference scheme with formal truncation error . Our application of interest is the spreading of
viscous gravity currents in the study of which these type of differential
equations arise. Viscous gravity currents are low Reynolds number (viscous
forces dominate inertial forces) flow phenomena in which a dense, viscous fluid
displaces a lighter (usually immiscible) fluid. The fluids may be confined by
the sidewalls of a channel or propagate in an unconfined two-dimensional (or
axisymmetric three-dimensional) geometry. Under the lubrication approximation,
the mathematical description of the spreading of these fluids reduces to
solving the so-called thin-film equation for the current's shape . To
solve such nonlinear parabolic equations we propose a finite-difference scheme
based on the Crank--Nicolson idea. We implement the scheme for problems
involving a single spatial coordinate (i.e., two-dimensional, axisymmetric or
spherically-symmetric three-dimensional currents) on an equispaced but
staggered grid. We benchmark the scheme against analytical solutions and
highlight its strong numerical stability by specifically considering the
spreading of non-Newtonian power-law fluids in a variable-width confined
channel-like geometry (a "Hele-Shaw cell") subject to a given mass
conservation/balance constraint. We show that this constraint can be
implemented by re-expressing it as nonlinear flux boundary conditions on the
domain's endpoints. Then, we show numerically that the scheme achieves its full
second-order accuracy in space and time. We also highlight through numerical
simulations how the proposed scheme accurately respects the mass
conservation/balance constraint.Comment: 36 pages, 9 figures, Springer book class; v2 includes improvements
and corrections; to appear as a contribution in "Applied Wave Mathematics II
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