Simple waves and shocks in a thin film of a perfectly soluble anti-surfactant solution

Abstract

We consider the dynamics of a thin film of a perfectly soluble anti-surfactant solution in the limit of large capillary and Péclet numbers in which the governing system of nonlinear equations is purely hyperbolic. We construct exact solutions to a family of Riemann problems for this system, and discuss the properties of these solutions, including the formation of both simple-wave and uniform regions within the flow, and the propagation of shocks in both the thickness of the film and the gradient of the concentration of solute.

Delta shock wave and wave interactions in a thin film of a perfectly soluble anti-surfactant solution

“…common table salt, when added to water [19,18], water when added to short-chain alcohols [11], and certain resins that are included in solvent-based paints [29,12]. Here, we consider the Riemann problem for a thin film of a perfectly soluble anti-surfactant solution [6] in the limit of large capillary and Péclet numbers and the solution of which belongs to some measure space…”

Existence results of solitary wave solutions for a delayed Camassa-Holm-KP equation

“…Conn et al [3] proposed a mathematical model for an anti-surfactant solution. Subsequently Conn et al [4] obtained exact solutions to a family of Riemann problems for a reduced version of their model which describes a thin film of a perfectly wetting anti-surfactant solution (i.e., the case in which the surface concentration of anti-surfactant is identically zero). Specifically, Conn et al [4]…”

“…In a fundamental work, Conn et al [3] considered linear stability of an infinitely deep layer of initially quiescent fluid and discussed the occurrence of an instability driven by surface-tension gradients, which occurs for anti-surfactant solutions. Further, Conn et al [4] derived a set of Riemann solutions to the system (5) for different Riemann initial data and discussed their properties.…”

“…Elementary waves. It may be noted that Conn et al [4] considered the system of PDEs (5) subject to the Riemann data (9) and derived the corresponding Riemann solution. However, we emphasize here that they have followed the method of characteristics to obtain the solution with a restriction on the initial data, i.e., b l < 0 and b r < 0.…”

“…represents the conservation of mass of the fluid whereas (4) represents the conservation of mass of the solute. Let us denote the gradient of concentration of the solute by b(x, t), i.e., ∂c/∂x = b(x, t) and differentiate (4) with respect to x to obtain a system of PDEs in a conservative form[4] …”