A thin liquid layer flowing due to gravity and a surface tension gradient is taken into account. On the liquid/gas interface one of the boundary conditions reduces to the fact that the normal stress equals the atmospheric pressure.
This is the main difference between our study and those in which the same boundary condition expresses the fact that the normal stress is proportional to the curvature. In these, by using the standard lubrication theory, a fourth-order nonlinear parabolic equation for the fluid film height is obtained.
In ours, by using the same theory, a nonlinear conservation law with a non-convex flux function is deduced for the same variable. For this equation a similarity solution is carried out. It shows that the behavior of the liquid layer depends essentially upon the gradient of surface tension and is quite insensitive to the viscosity of the liquid. “Viscous” and weak formulations for the conservation law are also carried out. An entropy condition to pick out physically relevant weak solutions is used.
Marangoni flow; normal stress; atmospheric pressure; nonlinear conservation law; non-convex flux; similarity solution; entropy condition;
See the expanding block below.
C.I. Gheorghiu, On the behavior of a thin liquid layer flowing due to gravity and a surface tension gradient, I. Mathematical aspects, Studia Univ. Babeş-Bolyai Math., XLI (1996) 47-54