because the volume of the fluid obviously preserves in the case of longitudinal deformations. In the case of purely transversal displacements [@ChuRyz2011] the force exerted on the plate by the fluid contains the pressure only.

In the following remark we describe some structural properties of the shell model chosen.

\[re:eq-u1u2\]

[**(A)**]{} One can see that the equations for the in-plane displacement vector ${\bar{u}}=(u^1;u^2)$ can be written in the vector form as follows: $$\label{u12-eq}
{{\varrho}}{\bar{u}}_{tt}+A{\bar{u}}= B(w)+(G_1(t)-\nu(v^1_{x_3}+v^3_{x_1}); G_2(t)-\nu(v^2_{x_3}+v^3_{x_2})),$$ where the operator $A$ in is defined by $$A=
-\left(
    \begin{aligned}
        & (1+\lambda){\partial}^2_{x_1} +{\partial}^2_{x_2} & \lambda{\partial}_{x_1 x_2} \\
        & \lambda{\partial}_{x_1 x_2} &  {\partial}^2_{x_1} +(1+\lambda){\partial}^2_{x_2} \\
    \end{aligned}
\right)$$ with $\lambda=\frac{1+\mu}{1-\mu}$ and $D(A)=[H^2({{\Omega}})\bigcap H^1_0({{\Omega}})]^2$. The nonlinear term $B(w)$ in has the form $$B(w)=\left(
    \begin{aligned}
        &\frac1{1-\mu} {\partial}_{x_1}\left[2\varkappa_1 w+ (w_{x_1})^2 +\mu(w_{x_2})^2\right] +{\partial}_{x_2}\left[w_{x_1}w_{x_2}\right]\\
        &{\partial}_{x_1}\left[w_{x_1}w_{x_2}\right] +\frac1{1-\mu} {\partial}_{x_2}\left[2\varkappa_2 w+(w_{x_2})^2 +\mu(w_{x_1})^2\right]
    \end{aligned}
\right),$$ where $\varkappa_1=k_1+\mu k_2$ and $\varkappa_2=k_2+\mu k_1$. Thus for fixed $w$ and $v$ the equations for the in-plain displacement ${\bar{u}}$ are similar to the standard equations of $2D$ (linear) elasticity theory.

[**(B)**]{} The equations in and can be also written in the form $$\begin{gathered}
(1-\alpha{\Delta}) (w_{tt} +{{\gamma}}w_t)  + {\Delta}^2 w + {\rm trace}\left\{ K {\mathscr{N}}(u)\right\}
 \\
={\rm div}\left\{ {\mathscr{N}}(u)\nabla w\right\}
+ G_3(t)- 2\nu {\partial}_{x_3}v^3 +p\end{gathered}$$ and $${{\varrho}}{\bar{u}}_{tt}  = {\rm div}\left\{ 