in\quad} {{\cal O}}\times(0,+\infty),$$ $$\label{fl.2}
   {{\rm div\, }}v=0 \quad {\rm in}\quad {{\cal O}}\times(0,+\infty),$$ where $\nu>0$ is the dynamical viscosity and $G_f(t)$ is a volume force (which may depend on $t$). We supplement (\[fl.1\]) and (\[fl.2\]) with the (non-slip) boundary conditions imposed on the velocity field $v=v(x,t)$: $$\label{fl.4}
v=0 ~~ {\rm on}~S;
\quad
v\equiv(v^1;v^2;v^3)=(u^1_t;u^2_t;w_t) ~~{\rm on} ~ {{\Omega}},$$ where $u=u(x,t)\equiv (u^1; u^2; w)(x,t)$ is the displacement of the shell occupying ${{\Omega}}$. Here $w$ stands for transversal displacement, ${\bar{u}}=(u^1;u^2)$ — for lateral (in-plane) displacements.

To describe the shell motion we use the full von Karman model which takes into account the rotational inertia of the filaments and possible presence of in-plane acceleration terms (see the literature cited above).

We denote by $T_f(v)$ the surface force exerted by the fluid on the shell, which is equal to $Tn\vert_{{\Omega}}$, where $n$ is a outer unit normal to ${\partial}{{\cal O}}$ at ${{\Omega}}$ and $T=\{T_{ij}\}_{i,j=1}^3$ is the stress tensor of the fluid, $$T_{ij}\equiv T_{ij}(v)=\nu\left(v^i_{x_j}+v^j_{x_i}\right)-p\delta_{ij},
\quad i,j=1,2,3.$$ Since $n=(0;0;1)$ on ${{\Omega}}$, we have that $$    T_f(v)=(\nu(v^1_{x_3}+v^3_{x_1}); \nu(v^2_{x_3}+v^3_{x_2}); 2\nu {\partial}_{x_3}v^3 -p).$$ Below for some simplification we assume that for the case considered Young’s modulus $E$ and Poisson’s ratio $\mu\in(0,1/2)$ are such that $Eh=2(1+\mu)$, where $h$ is the thickness of 