Abstract
Numerical solutions of a coupled system of nonlinear partial differential equations modelling the effects of surfactant on the spreading of a thin film on a horizontal substrate are investigated. A CFL condition is obtained from a von Neumann stability analysis of a linearised system of equations. Numerical solutions obtained from a Roe upwind scheme with a third-order TVD Runge-Kutta approximation to the time derivative are compared to solutions obtained with a Roe-Sweby scheme coupled to a minmod limiter and a TVD approximation to the time derivative. Results from both of these schemes are compared to a Roe upwind scheme and a BDF approximation to the time derivative. In all three cases high-order approximations to the spatial derivatives are employed on the interior points of the spatial domain. The Roe-BDF scheme is shown to be an efficient numerical scheme for capturing sharp changes in gradient in the free surface profile and surfactant concentration. Numerical simulations of an initial exponential free surface profile coupled with initial surfactant concentrations for both exogenous and endogenous surfactants are considered.
Highlights
In this paper we investigate numerical solutions of a coupled system of hyperbolic/degenerate-parabolic equations [1] modelling the spreading of an insoluble surfactant on the free surface of a thin liquid film
Surfactants are known to decrease the effects of surface tension by creating a spatial variation in the surface tension due to a tangential surface stress
In this paper we have considered numerical solutions of a coupled system of nonlinear equations modelling the effects of surfactant on the free surface of a thin film on a horizontal substrate
Summary
In this paper we investigate numerical solutions of a coupled system of hyperbolic/degenerate-parabolic equations [1] modelling the spreading of an insoluble surfactant on the free surface of a thin liquid film. Levy and Shearer [2] have considered numerical solutions of the coupled system (3) by implementing an implicit scheme and solving the resulting equations using a Newton’s method. Peterson and Shearer [1] consider numerical solutions of a one-dimensional case of (3) for β = κ = 0 and a linear equation of state for the surface tension. This holds true for the case β = 1 and κ = 0 for a linear equation of state σ(Γ) = 1 − Γ considered in this paper It is this compact support that makes it possible to implement an upwind scheme to solve the hyperbolic/degenerate-parabolic coupled system.
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