Undercompressive shocks in thin film flows

Bertozzi, Andrea L.; Münch, Andreas; Shearer, Michael

doi:10.1016/s0167-2789(99)00134-7

Cited by 150 publications

(235 citation statements)

References 37 publications

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“…Here b P is a point in between b l and b r , typically b P = 1 2 (b l + b r ). As in Bertozzi et al 14 , we refer to P as a Poincare section.…”

Section: B Shooting Methods For Traveling Wave Solutionsmentioning

confidence: 99%

“…However, if we introduce another parameter, such as by varying b l or b r , then this extra dimension means it may become possible to find a solution. This will violate one of the inequalities in the Lax entropy condition, hence such solutions are typically called undercompressive 14,15 . Because we must vary a parameter to find a solution, the set of values which admit undercompressive solutions are a curve in (b l , b r )-space.…”

Section: Governing Equations and Traveling Wave Solutionsmentioning

confidence: 99%

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Creating sharp features by colliding shocks on uniformly irradiated surfaces

Holmes-Cerfon¹,

Aziz²,

Brenner³

2012

Phys. Rev. B

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Using a theoretical analysis of the ion beam sputtering dynamics, we demonstrate how ion bombardment on an initially sloped surface can create knife-edge-like ridges on the surface. These ridges arise as nonclassical shock-like solutions that are undercompressive on both sides, and appear to control the dynamics over a large range of initial conditions. The slope of the ridges is selected uniquely by the dynamics, and can be up to 30 or more depending on the orientation dependence of the sputtering yield. For 1keV Ar + on Si(001), the scale of the ridge is ≈ 2nm. This is much smaller than the most unstable lengthscale and suggests a method for creating very steep, very sharp features on a surface spontaneously, by pre-patterning the surface to contain relatively modest slopes on the macroscale.

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“…Here b P is a point in between b l and b r , typically b P = 1 2 (b l + b r ). As in Bertozzi et al 14 , we refer to P as a Poincare section.…”

Section: B Shooting Methods For Traveling Wave Solutionsmentioning

confidence: 99%

Section: Governing Equations and Traveling Wave Solutionsmentioning

confidence: 99%

Creating sharp features by colliding shocks on uniformly irradiated surfaces

Holmes-Cerfon¹,

Aziz²,

Brenner³

2012

Phys. Rev. B

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“…Studies by Bertozzi et al [3] proposed that there are several di↵erent types of shock waves that connect films of di↵erent heights on an inclined plane when the flow is driven by surface shear and gravity. Solutions with more than one type of shock wave were also reported, dependent on the initial configuration of the flow.…”

Section: Introductionmentioning

confidence: 99%

“…The geometry is illustrated in figure 1, where uniform flow solutions can exist corresponding to three possible conjugate height and wave solutions. Validation of the modelling and numerical calculations will be sought from comparisons with previously published literature, including Shuaib [23] and Bertozzi [3], in the limit of negligible inertial terms. To investigate leading order inertial e↵ects on the film profile, especially in regions where the film profile changes rapidly, modelling will be restricted to at most moderate Reynolds numbers (see Kay et al [9]).…”

Section: Introductionmentioning

confidence: 99%

“…To investigate leading order inertial e↵ects on the film profile, especially in regions where the film profile changes rapidly, modelling will be restricted to at most moderate Reynolds numbers (see Kay et al [9]). Depth-averaging as used by Benilov, Bertozzi, and Kay [1,3,9] and many other authors, is used to reduce the governing equations to fewer dependent variables and leads to Shkadov-type integral boundary layer model upon adoption of an assumed velocity profile. A pair of coupled partial di↵erential equations for film thickness and local film flux result.…”

Section: Introductionmentioning

confidence: 99%

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Inertial effects on thin-film wave structures with imposed surface shear on an inclined plane

Sivapuratharasu

Hibberd

Hubbard

et al. 2016

Physica D: Nonlinear Phenomena

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This study provides an extended approach to the mathematical simulation of thin-film flow on a flat inclined plane relevant to flows subject to high surface shear. Motivated by modelling thin-film structures within an industrial context, wave structures are investigated for flows with moderate inertial e↵ects and small film depth aspect ratio ". Approximations are made assuming a Reynolds number, Re ⇠ O " 1 and depth-averaging used to simplify the governing Navier-Stokes equations. A parallel Stokes flow is expected in the absence of any wave disturbance and a generalisation for the flow is based on a local quadratic profile. This approch provides a more general system which includes inertial e↵ects and is solved numerically. Flow structures are compared with studies for Stokes flow in the limit of negligible inertial e↵ects. Both two-tier and three-tier wave disturbances are used to study film profile evolution. A parametric study is provided for wave disturbances with increasing film Reynolds number. An evaluation of standing wave and transient film profiles is undertaken and identifies new profiles not previously predicted when inertial e↵ects are neglected.

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Numerical studies of the fingering phenomena for the thin film equation

Lee

Yoon

et al. 2010

Numerical Methods in Fluids

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SUMMARYWe present a new interpretation of the fingering phenomena of the thin liquid film layer through numerical investigations. The governing partial differential equation is h t +(h 2 −h 3 ) x =−∇ ·(h 3 ∇ h), which arises in the context of thin liquid films driven by a thermal gradient with a counteracting gravitational force, where h = h(x, y, t) is the liquid film height. A robust and accurate finite difference method is developed for the thin liquid film equation. For the advection part (h 2 −h 3 ) x , we use an implicit essentially nonoscillatory (ENO)-type scheme and get a good stability property. For the diffusion part −∇ ·(h 3 ∇ h), we use an implicit Euler's method. The resulting nonlinear discrete system is solved by an efficient nonlinear multigrid method. Numerical experiments indicate that higher the film thickness, the faster the film front evolves. The concave front has higher film thickness than the convex front. Therefore, the concave front has higher speed than the convex front and this leads to the fingering phenomena.

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Undercompressive shocks in thin film flows

Cited by 150 publications

References 37 publications

Creating sharp features by colliding shocks on uniformly irradiated surfaces

Creating sharp features by colliding shocks on uniformly irradiated surfaces

Inertial effects on thin-film wave structures with imposed surface shear on an inclined plane

Numerical studies of the fingering phenomena for the thin film equation

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