Thin film dynamics: theory and applications

Bertozzi, Andrea L.; Bowen, Mark

doi:10.1007/978-94-010-0510-4_2

Cited by 7 publications

(4 citation statements)

References 93 publications

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“…The dynamics of a liquid film on a flat surface is modelled by [46] the thin film equation, and its study is driven by many applications in physics and chemistry [47,48] as well as in mathematics [49]. Here we look at an anisotropic version of the thin film equation…”

Section: Example 4 Anisotropic Thin Film Equationmentioning

confidence: 99%

Symmetry multi-reduction method for partial differential equations with conservation laws

Anco,

Gandarias

2019

Preprint

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For partial differential equations (PDEs) that have n ≥ 2 independent variables and a symmetry algebra of dimension at least n − 1, an explicit algorithmic method is presented for finding all symmetry-invariant conservation laws that will reduce to first integrals for the ordinary differential equation (ODE) describing symmetry-invariant solutions of the PDE. This significantly generalizes the double reduction method known in the literature. Moreover, the condition of symmetry-invariance of a conservation law is formulated in an improved way by using multipliers, thereby allowing symmetry-invariant conservation laws to be obtained directly, without the need to first find conservation laws and then check their invariance. This cuts down considerably the number and complexity of computational steps involved in the reduction method. If the space of symmetry-invariant conservation laws has dimension m ≥ 1, then the method yields m first integrals along with a check of which ones are non-trivial via their multipliers. Several examples of interesting symmetry reductions are considered: travelling waves and similarity solutions in 1 + 1 dimensions; line travelling waves, line similarity solutions, and similarity travelling waves in 2 + 1 dimensions; rotationally symmetric similarity solutions in n + 1 dimensions. In addition, examples of nonlinear PDEs for which the method yields the explicit general solution for symmetry-invariant solutions are shown.

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Section: Example 4 Anisotropic Thin Film Equationmentioning

confidence: 99%

Symmetry multi-reduction method for partial differential equations with conservation laws

Anco,

Gandarias

2019

Preprint

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“…We note that in the absence of surface tension effects, the system of equations ( 31) and (32) may be solved exactly. In the presence of surface tension (β = 0) with positive initial conditions, (31) is expected to have a smooth solution (see [3]) and (32) becomes a scalar conservation law that can be solved exactly.…”

Section: Dilute Approximationmentioning

confidence: 99%

Surface tension effects for particle settling and resuspension in viscous thin films

et al. 2018

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We consider flow of a thin film on an incline with negatively buoyant particles. We derive a onedimensional lubrication model, including the effect of surface tension, which is a nontrivial extension of a previous model (Murisic et. al [J. Fluid Mech. 2013]). We show that the surface tension, in the form of high order derivatives, not only regularizes the previous model as a high order diffusion, but also modifies the fluxes. As a result, it leads to a different stratification in the particle concentration along the direction perpendicular to the motion of the fluid mixture. The resulting equations are of mixed hyperbolic-parabolic type and different from the well-known lubrication theory for a clear fluid or fluid with surfactant. To study the system numerically, we formulate a semi-implicit scheme that is able to preserve the particle maximum packing fraction. We show extensive numerical results for this model including a qualitative comparison with two-dimensional laboratory experiments.

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“…However, despite the plethora of proposed schemes for the classical advection-diffusion equation in the context of Lagrange finite elements, there is a lack of corresponding treatments for the fourth order diffusion term of the present work, especially in combination with C 1 continuous elements. For linear elements, a relevant regularization for preserving positivity based on an entropy dissipating scheme has been proposed in (Zhornitskaya and Bertozzi, 2000;Bertozzi and Bowen, 2002).…”

Section: Treatment Of Contact Lines and Stabilizationmentioning

confidence: 99%

Finite element method for starved hydrodynamic lubrication with film separation and free surface effects

Poulios

Vølund²,

Klit

2018

Computer Methods in Applied Mechanics and Engineering

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Thin film dynamics: theory and applications

Cited by 7 publications

References 93 publications

Symmetry multi-reduction method for partial differential equations with conservation laws

Symmetry multi-reduction method for partial differential equations with conservation laws

Surface tension effects for particle settling and resuspension in viscous thin films

Finite element method for starved hydrodynamic lubrication with film separation and free surface effects

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