We investigate the free boundary propagation of a non-linear degenerate parabolic problem with both fourth and second order non-linearities, which may be used to model viscous thin-film flow in the regime of weak slippage with an additional contributions from a singular disjoining pressure brought forth by inter-molecular forces. By carefully treating the individual scaling behaviors of the non-linearities, we obtain asymptotically optimal upper and lower bounds on the rate of forward support propagation. These rates show that the fourth-order operator dominates the propagation rate on short time scales, but the long-term rate of propagation is controlled by the second-order operator. We estimate further the time at which this trade-off in propagation rate occurs. Lastly, we examine sufficient conditions to ensure that the free boundary does not immediately advance in a given direction, leading to a detailed picture of how the competition between the second/fourth-order operators influence the free boundary dynamics of solutions.
We present a formulation for the linear stability of the miscible Rayleigh-Taylor instability, retaining viscosity, Fickian mass diffusion, and non-Boussinesq effects. Rather than adopting the diffusive corrections of Duff, Harlow, and Hirt (1962) or relying on the viscous eigenvalue problem of Chandrasekhar (1961), th...
Marildo Kola, Daniel Israel, Aaron Towne· 0 citations
In this work, we develop a nonlinear continuum model for wave propagation in two-dimensional granular materials, building on key ideas from Nesterenko's work. Starting from the Hertzian contact law and neglecting contact friction, we derive an effective continuum description through homogenization, assuming spherical,...
S. Cominelli, R. Cottereau, S. Gavrilyuk et al.· 0 citations
We study the nonlinear initial boundary value problem (IBVP) and that governs thin film flows. The generally accepted version is shown not to lead to an energy bound for all parameter values. Firstly, we introduce modifications to the continuous IBVP. The first modification retains the original thin film flow equation but rel...
J. Nordström, C. Harley, E. Momoniat· Journal of Computational Phy...· 0 citations
In the Navier--Stokes equations, incompressibility allows rewriting the viscous term in various forms leading to distinct numerical properties and flow descriptions. Furthermore, models accounting for non-Newtonian, thermal or turbulent effects often break the constant-viscosity assumption, thereby producing additional...
Benjamín Cisternas, Douglas R. Q. Pacheco, F. Galarce et al.· 0 citations
Starting from localized energy estimates, we prove finite speed of propagation for kinetic solutions to stochastic porous media equations with nonlinear conservative noise, the existence and uniqueness of which has recently been established. In particular, we propose a novel iteration technique which allows us to obtai...
G. Grün, Max Sauerbrey, Joshua Utley· 2 citations
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