Integral boundary layer relations in the theory of wave flows for capillary liquid films

A generalized method of deriving the model equations is considered for wave flow regimes in falling liquid films. The viscous liquid equations are used on the basis of integral boundary layer relations with weight functions. A family of systems of evolution differential equations is proposed. The integer parameter $n$ of these systems specifies the number of a weight function. The case $n=0$ corresponds to the classical IBL model. The case $n\geq 1$ corresponds to its modifications called the WIBL models. The numerical results obtained in the linear and nonlinear approximations for $n=0,1,2$ are discussed. The numerical solutions to the original hydrodynamic differential equations are compared with experimental data. This comparison leads us to the following conclusions: as a rule, the most exact solutions are obtained for $n=0$ in the case of film flows on vertical and inclined solid surfaces and the accuracy of solutions decreases with increasing $n$. Hence, the classical IBL model has an advantage over the WIBL models.