A Review of the Research into the Mathematical Problems of Mechanics and Control Theory at the Institute of Mathematics of the National Academy of Sciences of Ukraine

Ukrainian Mathematical Journal - We present a review of the research into the mathematical problems of mechanics and control theory carried out at Institute of Mathematics of the National Academy...     

Linear and nonlinear sloshing in a circular conical tank

Linear and nonlinear fluid sloshing problems in a circular conical tank are studied in a curvilinear coordinate system. The linear sloshing modes are approximated by a series of the solid spheric harmonics. These modes are used to derive a new nonlinear modal theory based on the Moiseyev asymptotics. The theory makes it possible to both classify steady-state waves occurring due to horizontal resonant excitation and visualise nonlinear wave patterns. Secondary (internal) resonances and shallow fluid sloshing (predicted for the semi-apex angles >60) are extensively discussed.     

A Note on the Variational Formalism for Sloshing with Rotational Flows in a Rigid Tank with Unprescribed Motion

Ukrainian Mathematical Journal - The Bateman–Luke-type variational formulation of the free-boundary “sloshing” problem is generalized to the case of irrotational flows and...     

Two-dimensional variational vibroequilibria and Faraday’s drops

When contacting with acoustically-vibrated structures a fluid volume can take a [time-averaged] geometric shape differing from capillary equilibrium. In accordance with theorems by Beyer et al. (2001) this shape (vibroequilibrium) furnishes a local minimum of a [quasi-potential energy] functional. The variational problem contains five dimensionless parameters evaluating the fluid volume, the wave number of acoustic field in the fluid domain, the contact angle and two newly-introduced numbers (1, 2) giving relationships between (surface tension, gravitation) and Kapitsas vibrational forces/energy. The paper focuses on negligible small wave numbers (incompressible fluid) and two-dimensional flows. Although the variational problem may in some isolated cases have analytical solutions, it requires in general numerical approaches. Numerical examples simulate experiments by Wolf (1969) and Ganiyev et al. (1977) on vibroequilibria in horizontally vibrating tanks. These show that there appear at least two types of stable vibroequilibria associated with symmetric (possible non-connected) and asymmetric surface shapes. The paper represents also numerical results on flattening and vibrostabilisation of a drop hanging beneath a vibrating plate (experiments by Faraday (1831)).Received: October 17, 2002; revised: June 30, 2003     

Two-dimensional variational vibroequilibria and Faraday’s drops

When contacting with acoustically-vibrated structures a fluid volume can take a [time-averaged] geometric shape differing from capillary equilibrium. In accordance with theorems by Beyer et al. (2001) this shape (vibroequilibrium) furnishes a local minimum of a [quasi-potential energy] functional. The variational problem contains five dimensionless parameters evaluating the fluid volume, the wave number of acoustic field in the fluid domain, the contact angle and two newly-introduced numbers (1, 2) giving relationships between (surface tension, gravitation) and Kapitsas vibrational forces/energy. The paper focuses on negligible small wave numbers (incompressible fluid) and two-dimensional flows. Although the variational problem may in some isolated cases have analytical solutions, it requires in general numerical approaches. Numerical examples simulate experiments by Wolf (1969) and Ganiyev et al. (1977) on vibroequilibria in horizontally vibrating tanks. These show that there appear at least two types of stable vibroequilibria associated with symmetric (possible non-connected) and asymmetric surface shapes. The paper represents also numerical results on flattening and vibrostabilisation of a drop hanging beneath a vibrating plate (experiments by Faraday (1831)).