A Mathematical Interaction Model between an Ice Cover and a Hydrodynamic Dipole in a Channel
Abstract
In this paper, hydroelastic waves generated by a hydrodynamic dipole in a channel covered with ice are studied. The stationary dipole placed in a flow of a liquid with constant speed and the dipole that moves uniformly along the channel are described. A mathematical model is based on the Kelvin-Voigt differential equation of viscoelastic plate and the Laplace equation for a velocity potential under the ice cover. The velocity potential is equal to the sum of the dipole potential in the channel and a potential of the flow caused by the deflection of the plate. The equation for the dipole potential in the channel is obtained by a method of mirror images for four walls. These equations are supplemented by impermeability boundary conditions on the walls and bottom of the channel, clamped boundary conditions of ice on the walls of the channel, and kinematic and dynamic conditions at the ice-liquid interface. The dynamic condition is described by the linearized Cauchy-Lagrange integral. A formulation of the stationary problem is studied. The solution is sought in the form of a traveling wave, which is independent of time in a coordinate system moving with the dipole. Streamlines of the fluid motion and shapes of the flow are studied for the case of flowing around the dipole in the channel. Obtained results are compared with forms of flow in an unbounded fluid.
DOI 10.14258/izvasu(2017)1-30
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