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On the basis of theoretical considerations (mechanistic model), an equation was determined that allowed to calculate the free energy (Helmholtz) of a spherical droplet deposited on a flat surface in a system without external forces. Assuming isochoric and isothermal transformation of the system and a very fast conversion of mechanical energy into heat, the obtained equation allows to determine the trajectory of thermodynamic transformation consisting of the spreading of the droplet on the surface of the substrate. The similarities and differences in the behaviour of spherical droplets described by the mechanistic model and Young's model, together with its improvements, were discussed. The trajectories of free energy changes during the spreading of droplets in a system in which the adhesive force acting perpendicular to the wetted surface was considered as well. [ABSTRACT FROM AUTHOR]

In this paper, a new object in the form of a theoretical network is presented, which is useful as a benchmark for particle filtering algorithms designed for multivariable nonlinear systems (potentially linear, nonlinear, and even semi-Markovian jump system). The main goal of the paper is to propose an object that potentially can have similar to the power system grid properties, but with the number of state variables reduced twice (only one state variable for each node, while there are two in the case of power systems). Transition and measurement functions are proposed in the paper, and two types of transition functions are considered: dependent on one or many state variables. In addition, 10 types of measurements are proposed both for branch and nodal cases. The experiments are performed for 14 different, four-dimensional systems. Plants are both linear and highly nonlinear. The results include information about the state estimation quality (based on the mean squared error indicator) and the values of the effective sample size. It is observed how the higher effective sample size resulted in the better estimation quality in subsequent cases. It is also concluded that the very low number of significant particles is the main problem in particle filtering of multivariable systems, and this should be countered. A few potential solutions for the latter are also presented. [ABSTRACT FROM AUTHOR]