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We discuss a box model of migration in channels of networks with possible application for modelling motion of migrants in migration networks. The channel consists of nodes of the network (nodes may be considered as boxes representing countries) and edges that connect these nodes and represent possible ways for motion of migrants. The nodes of the migration channel have different "leakage", i.e. the probability of change of the status of a migrant (from migrant to nonmigrant) may be different in the different countries along the channel. In addition the nodes far from the entry node of the channel may be more attractive for migrants in comparison to the nodes around the entry node of the channel. We discuss below channels containing infinite number of nodes. Two regimes of functioning of these channels are studied: stationary regime and non-stationary regime. In the stationary regime of the functioning of the channel the distribution of migrants in the countries of the channel is described by a distribution that contains as particular case the Waring distribution. In the non-stationary regime of functioning of the channel one observes exponential increase or exponential decrease of the number of migrants in the countries of the channel. It depends on the situation in the entry country of the channel for which scenario will be realized. Despite the non-stationary regime of the functioning of the channel the asymptotic distribution of the migrants in the nodes of the channel is stationary. From the point of view of the characteristic features of the migrants we discuss the cases of (i) migrants having the same characteristics and (ii) two classes of migrants that have differences in some characteristic (e.g., different religions).

Accurate and computationally accessible models of liquid film flows allow for optimizing coating processes such as hot-dip galvanization and vertical slot-die coating. This paper extends the classic three-dimensional integral boundary layer (IBL) model for falling liquid films (FF) to account for a moving substrate (MS). We analyze the stability of the liquid films on vertically moving substrates in a linear and a nonlinear setting. In the linear analysis, we derive the dispersion relation and the temporal growth rates of an infinitesimal disturbance using normal modes and linearized governing equations. In the nonlinear analysis, we consider disturbances of finite size and numerically compute their evolution using the set of nonlinear equations in which surface tension has been removed. We present the region of (linear) stability of both conditions, i.e. FF and MS, and we place the operating conditions of an industrial galvanizing line in these maps. A wide range of flow conditions was analyzed and shown to be stable according to linear and nonlinear stability analyses. Moreover, the nonlinear analysis, carried out in the absence of surface tension, reveals a nonlinear stabilizing mechanism for the interface dynamics of a liquid film dragged by an upward-moving substrate.

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