\label{plBC}$$ and supply – with initial data for the velocity field $v=(v^1; v^2; v^3)$ and the shell displacement vector $u=(u^1; u^2; w)$ of the form[^3] $$v\big|_{t=0}=v_0,~~
u\big|_{t=0}=u_0, ~~
w_t\big|_{t=0}=w_1, ~~
{{\varrho}}\left[{\bar{u}}_t\big|_{t=0}-{\bar{u}}_1\right]=0, \label{IC}$$ where ${\bar{u}}=(u^1; u^2)$. Here $v_0=(v^1_0; v^2_0; v^3_0)$, $u_0=(u^1_0; u^2_0; w_0)$, $w_1$, and ${\bar{u}}_1=(u^1_1; u^2_1)$ are given vector functions which we specify later.

We note that and imply the following compatibility condition $$\label{Com-con}
\int_{{\Omega}}w_t(x',t) dx'=0 \quad \mbox{for all}~~ t\ge 0.$$ This condition fulfills when $$\int_{{\Omega}}w(x',t) dx'=const \quad \mbox{for all}~~ t\ge 0$$ and can be interpreted as preservation of the volume of the fluid.

We emphasize that even in the linear case we cannot split system – into two sets of equations describing longitudinal and transversal plate movements separately, i.e., we cannot reduce the model considered to the cases studied in [@Chu_2010; @ChuRyz2011]. The point is that the surface force $T_f(v)$ is not the sum of the corresponding loads in the models [@Chu_2010] and [@ChuRyz2011]. The models in [@Chu_2010; @ChuRyz2011] are much simpler in several respects. For instance, in the case of longitudinal plate deformations only (see [@Chu_2010]) the equations which correspond to do not contain the terms $v^3_{x_i}$ and the model does not require any compatibility conditions like (\[Com-con\]) 