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.

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.

Derivation of a Thin Film Equation by a Direct Approach

1996 ◽  
Vol 63 (2) ◽  
pp. 467-473
Author(s):  
F. Y. Huang ◽  
C. D. Mote
Keyword(s):  
Thin Film ◽  
Velocity Field ◽  
Viscous Fluid ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Film Structure ◽  
Navier Stokes ◽  
Slip Boundary ◽  
Thin Film Equations ◽  
Thin Film Equation

A new model of the thin viscous fluid film, constrained between two translating, flexible surfaces, is presented in this paper: The unsteady inertia of the film is included in the model. The derivation starts with the reduced three-dimensional Navier-Stokes equations for an incompressible viscous fluid with a small Reynolds number. By introduction of an approximate velocity field, which satisfies the continuity equation and the no-slip boundary conditions exactly, into weighted integrals of the three-dimensional equations over the film thickness, a two-dimensional thin film equation is obtained explicitly in a closed form. The 1th thin film equation is obtained when the velocity field is approximated by 21th order polynominals, and the three-dimensional viscous film is described with increasing accuracy by thin film equations of increasing order. Two cases are used to illustrate the coupling of the film to the vibration of the structure and to show that the second thin film equation can be applied successfully to the prediction of a coupled film-structure response in the range of most applications. A reduced thin film equation is derived through approximation of the second thin film equation that relates the film pressure to transverse accelerations and velocities, and to slopes and slope rates of the two translating surfaces.

CONVERGENCE OF THIN FILM APPROXIMATION FOR A SCALAR CONSERVATION LAW

2005 ◽  
Vol 02 (01) ◽  
pp. 183-199 ◽  
Author(s):  
FELIX OTTO ◽  
MICHAEL WESTDICKENBERG
Keyword(s):  
Thin Film ◽  
Conservation Law ◽  
Approximate Solutions ◽  
Strictly Convex ◽  
Scalar Conservation Law ◽  
Scalar Conservation

In this paper we consider the thin film approximation of a 1D scalar conservation law with strictly convex flux. We prove that the sequence of approximate solutions converges to the unique Kružkov solution.

Transient Thin-Film Flow on a Moving Boundary of Arbitrary Topography

2009 ◽  
Vol 131 (10) ◽  
Author(s):  
Roger E. Khayat ◽  
Tauqeer Muhammad
Keyword(s):  
Thin Film ◽  
Steady State ◽  
Moving Boundary ◽  
Initial Conditions ◽  
Flow Behavior ◽  
Galerkin Projection ◽  
Thin Film Flow ◽  
Flat Substrate ◽  
Thin Film Equations ◽  
Two Dimensional Flow

The transient two-dimensional flow of a thin Newtonian fluid film over a moving substrate of arbitrary shape is examined in this theoretical study. The interplay among inertia, initial conditions, substrate speed, and shape is examined for a fluid emerging from a channel, wherein Couette–Poiseuille conditions are assumed to prevail. The flow is dictated by the thin-film equations of the “boundary layer” type, which are solved by expanding the flow field in terms of orthonormal modes depthwise and using the Galerkin projection method. Both transient and steady-state flows are investigated. Substrate movement is found to have a significant effect on the flow behavior. Initial conditions, decreasing with distance downstream, give rise to the formation of a wave that propagates with time and results in a shocklike structure (formation of a gradient catastrophe) in the flow. In this study, the substrate movement is found to delay shock formation. It is also found that there exists a critical substrate velocity at which the shock is permanently obliterated. Two substrate geometries are considered. For a continuous sinusoidal substrate, the disturbances induced by its movement prohibit the steady-state conditions from being achieved. However, for the case of a flat substrate with a bump, a steady state exists.

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

1997 ◽  
Vol 8 (5) ◽  
pp. 507-524 ◽  
Author(s):  
RAUL FERREIRA ◽  
FRANCISCO BERNIS
Keyword(s):  
Thin Film ◽  
Negative Solution ◽  
Finite Speed Of Propagation ◽  
Finite Speed ◽  
First Order ◽  
Source Type ◽  
Thin Film Equations ◽  
Thin Film Equation ◽  
Self Similar ◽  
Speed Of Propagation

We prove that the thin film equation ht+div (hn grad (Δh))=0 in dimension d[ges ]2 has a unique C1 source-type radial self-similar non-negative solution if 0<n<3 and has no solution of this type if n[ges ]3. When 0<n3 the solution h has finite speed of propagation and we obtain the first order asymptotic behaviour of h at the interface or free boundary separating the regions where h>0 and h=0. (The case d=1 was already known [1]).

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