# Motion of Submerged Body in a Frozen Channel with Compressed Porous Ice
## 2 Formulation of the Problem  
The three-dimensional problem of interaction between an ice cover and a moving submerged body in a channel is considered. The channel has a rectangular cross-section with a width of \(2b\) (\(-b<y<b\)), a depth of \(H\left(-H<z<0\right)\), and its length is unlimited. The channel is filled with an ideal, incompressible fluid, and its surface is covered with a thin ice cover. The submerged body is modeled by a three-dimensional dipole moving with a speed \(U(t)\). The scheme of the problem is illustrated in Figure 1.  
The problem is solved within the linear theory of hydroelasticity. In the linear formulation, the deflection of the heterogeneous ice cover \(w(x,y,t)\) is modeled by the vertical displacement of the plate from the state of rest (\(z=0\)) and satisfies the equation of the thin viscoelastic plate  
\[\rho_{i}h_{i}(y)w_{tt}+\left(1+\tau\frac{\partial}{\partial t}\right)\Lambda w +Q_{x}\frac{\partial^{2}w}{\partial x^{2}}+Q_{y}\frac{\partial^{2}w}{\partial y ^{2}}+Q_{xy}\frac{\partial^{2}w}{\partial x\partial y}=p(x,y,0,t), \tag{1}\]  
where \(\Lambda\) is a differential operator in the form  
\[\Lambda=D(y)\Delta^{2}+2D_{y}\bigg{(}\frac{\partial^{3}}{\partial y^{3}}+\frac {\partial^{3}}{\partial x^{2}\partial y}\bigg{)}+D_{yy}\bigg{(}\frac{\partial^ {2}}{\partial y^{2}}+v\frac{\partial^{2}}{\partial x^{2}}\bigg{)},\]  
\(\Delta^{2}=\partial^{4}/\partial x^{4}+2\partial^{4}/\partial x^{2}\partial y ^{2}+\partial^{4}/\partial y^{4}\), \(\rho_{i}\) is the density of ice, \(h_{i}(y)\) is the thickness of the plate, \(\tau=\eta/E\) is the retardation time, \(\eta\) is the viscosity of ice, \(E\) is the Young's modulus, \(D=El_{i}^{3}(y)/[12(1-\
u^{2})]\) is the bending rigidity of the plate, \(\
u\) is the [PERSON]'s ratio, \(p(x,y,0,t)\) is the pressure of the fluid on the ice/fluid interface, \(Q_{x}\) and \(Q_{y}\) are the  
Figure 1: Scheme of the problem. compressing stresses in the plate plane along the \(x\) and \(y\) directions, respectively, and \(Q_{xy}\) is the combined compressing stress. The ice cover is frozen to the channel walls, which is modeled by clamped conditions  
\[w=w_{y}=0\quad(-\infty<x<\infty,y=\pm b). \tag{2}\]  
The pressure of the fluid at the ice/fluid interface is given by the linearized Bernoulli integral  
\[p(x,y,0,t)=-\rho_{l}\Phi_{t}(x,y,0,t)-\rho_{l}gw\quad(-\infty<x<\infty,-b<y<b), \tag{3}\]  
where \(\Phi(x,y,z,t)\) is the velocity potential of the fluid flow in the channel, \(g\) is the acceleration due to gravity, and \(\rho_{l}\) is the density of the fluid. The velocity potential of the fluid flow \(\Phi(x,y,z,t)\) satisfies the Laplace equation in the channel domain, as well as the kinematic condition at the fluid/porous plate interface  
\[\Phi_{z}(x,y,0,t)=w_{t}+\alpha p(x,y,0,t), \tag{4}\]  
where \(\alpha\) is a parameter of porosity and is given by the formula \(\alpha=\frac{K}{\mu h_{i}(y)}\), \(K\) is the permeability coefficient of ice, and \(\mu\) is the dynamic viscosity of ice. At the walls and bottom of the channel, the potential satisfies the no-slip conditions  
\[\Phi_{z}=0\quad(z=-H),\qquad\Phi_{y}=0\quad(y\pm b). \tag{5}\]  
The submerged body is modeled by a three-dimensional dipole placed in the channel beneath the ice cover. To determine the shape of the submerged body, it is necessary to define the dipole potential \(\varphi^{D}(x,y,z,t)\) in the channel. For a dipole of small strength, its motion describes the motion of a sphere of small radius \(R\)[49]. The dipole potential satisfies the Laplace equation in the channel and decays at infinity, \(\varphi^{D}\to 0\) as \(|x|\to+\infty\). The normal derivative of the potential is equal to \(0\) at the walls \(y=\pm b\), at the bottom \(z=-H\), and at the ice/fluid interface \(z=0\). The potential \(\varphi^{D}(x,y,z,t)\) is obtained using the method of mirror images [49]. First, the potential \(\varphi^{D1}\) is derived, taking into account images from the vertical channel walls, which satisfies the boundary conditions \(y=\pm b\).