Vol 72, No 6 (2017)

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 (Integral Boundary Layer) model. The case n ≥ 1 corresponds to its modifications called the WIBL (Weighted Integral Boundary Layer) 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 accurate 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.

The boundary conditions are studied for nematic liquid crystals in the case of weak anchoring. The cases of the general expression and one-constant approximation are considered for the Frank energy of elastic distortions of the director field. It is shown that the one-constant approximation is correct for one-dimensional problems only and, for two- and three-dimensional problems, this model significantly simplifies the boundary conditions and changes their type.