Explicit weighted essentially non-oscillatory scheme for the simulation of surfactant driven thin film equations

2021 ◽  
Author(s):  
Minu Jose ◽  
Satyananda Panda
Keyword(s):  
Thin Film ◽  
Thin Film Equations

scholarly journals Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise

2021 ◽  
Author(s):  
Konstantinos Dareiotis ◽  
Benjamin Gess ◽  
Manuel V. Gnann ◽  
Günther Grün
Keyword(s):  
Thin Film ◽  
Energy Estimates ◽  
Approximation Procedure ◽  
Thin Film Flow ◽  
Thin Film Equations ◽  
Scalar Conservation Law ◽  
Thin Film Equation ◽  
Degenerate Parabolic ◽  
Scalar Conservation ◽  
Martingale Solutions

AbstractWe prove the existence of non-negative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the liquid-solid interface. Since their introduction more than 15 years ago, by Davidovitch, Moro, and Stone and by Grün, Mecke, and Rauscher, the existence of solutions to stochastic thin-film equations for cubic mobilities has been an open problem, even in the case of sufficiently regular noise. Our proof of global-in-time solutions relies on a careful combination of entropy and energy estimates in conjunction with a tailor-made approximation procedure to control the formation of shocks caused by the nonlinear stochastic scalar conservation law structure of the noise.

scholarly journals Patterns formed in a thin film with spatially homogeneous and non-homogeneous Derjaguin disjoining pressure

2021 ◽  
pp. 1-25
Author(s):  
ABDULWAHED S. ALSHAIKHI ◽  
MICHAEL GRINFELD ◽  
STEPHEN K. WILSON
Keyword(s):  
Thin Film ◽  
Steady State ◽  
Bifurcation Theory ◽  
Disjoining Pressure ◽  
Thin Film Equations ◽  
Planar Substrate ◽  
Spatially Homogeneous ◽  
The One ◽  
Steady State Solutions ◽  
Average Film Thickness

We consider patterns formed in a two-dimensional thin film on a planar substrate with a Derjaguin disjoining pressure and periodic wettability stripes. We rigorously clarify some of the results obtained numerically by Honisch et al. [Langmuir 31: 10618–10631, 2015] and embed them in the general theory of thin-film equations. For the case of constant wettability, we elucidate the change in the global structure of branches of steady-state solutions as the average film thickness and the surface tension are varied. Specifically we find, by using methods of local bifurcation theory and the continuation software package AUTO, both nucleation and metastable regimes. We discuss admissible forms of spatially non-homogeneous disjoining pressure, arguing for a form that differs from the one used by Honisch et al., and study the dependence of the steady-state solutions on the wettability contrast in that case.

scholarly journals A class of thin-film equations with logarithmic nonlinearity

2018 ◽  
Vol 49 (12) ◽  
pp. 1765
Author(s):  
Han Yuzhu ◽  
Shi Xiaowei ◽  
Gao Wenjie
Keyword(s):  
Thin Film ◽  
Thin Film Equations

Influence of Upstream Conditions and Gravity on Highly Inertial Thin-Film Flow

2008 ◽  
Vol 130 (6) ◽  
Author(s):  
Roger E. Khayat
Keyword(s):  
Thin Film ◽  
Free Surface ◽  
Flow Field ◽  
Film Flow ◽  
Flow Direction ◽  
Vertical Wall ◽  
Galerkin Projection ◽  
Thin Film Flow ◽  
Thin Film Equations ◽  
Horizontal Wall

Steady two-dimensional thin-film flow of a Newtonian fluid is examined in this theoretical study. The influence of exit conditions and gravity is examined in detail. The considered flow is of moderately high inertia. The flow is dictated by the thin-film equations of boundary layer type, which are solved by expanding the flow field in orthonormal modes in the transverse direction and using Galerkin projection method, combined with integration along the flow direction. Three types of exit conditions are investigated, namely, parabolic, semiparabolic, and uniform flow. It is found that the type of exit conditions has a significant effect on the development of the free surface and flow field near the exit. While for the parabolic velocity profile at the exit, the free surface exhibits a local depression, for semiparabolic and uniform velocity profiles, the height of the film increases monotonically with streamwise position. In order to examine the influence of gravity, the flow is studied down a vertical wall as well as over a horizontal wall. The role of gravity is different for the two types of wall orientation. It is found that for the horizontal wall, a hydraulic-jump-like structure is formed and the flow further downstream exhibits a shock. The influence of exit conditions on shock formation is examined in detail.

Sign in / Sign up

Export Citation Format

Share Document