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\begin{center}
ON A CYLINDRICAL BENDING OF A PRISMATIC SHELL 
WITH TWO CUSPED EDGES UNDER THE ACTION OF 
AN INCOMPRESSIBLE VISCOUS  FLUID\footnote{This research was supported by
a grant of Georgian President for the university}
 
\vspace*{0.2cm}
{\it N. Chinchaladze}

\vspace*{0.1cm}
{\it I.Vekua Institute of Applied Mathematics}\\
{\it I.Javakhishvili Tbilisi State University}
\end{center}

\vspace*{0.1cm}
\par This paper deals with the proof of a uniqueness theorem of a problem
of the interaction of the incompressible viscous fluid and
of a thin elastic cusped plate of variable thickness (i.e., prismatic shell).
\par Let the projection of a prismatic shell be an infinite strip
$$
\Omega=\{(x_1,x_2):\;-\infty <x_1<\infty,\; 0\le x_2\le \ell\}, 
$$
and let the shell thickness be given by the equation 
$$
2h(x_2)=h_0x_2^{\alpha/3}\left(\ell-x_2\right)^{\beta/3},\;\;h_0,\alpha,\beta={\rm const},\;\;
h_0>0,\;\;\alpha,\beta\ge 0.
$$
The edges $x_2=0$ and $x_2=\ell$ are cusped if $\alpha>0$ and $\beta>0$,
respectively.
\par The geometry of the elastic solid along thickness is taken into account in 
the coefficients of the following bending equation [1]
\begin{equation}
\frac{\partial^2}{\partial x_2^2}\left[x_2^2(\ell-x_2)^\beta\frac{\partial^2 w(x_2,t)}
{\partial x_2^2}\right]=-\frac{2\rho^s h(x_2)}{D_0}\frac{\partial^2 w(x_2,t)}
{\partial t^2}+\frac{q(x_2,t)}{D_0},
\end{equation}
\par{\hspace{-0.6cm}}where $D(x_2)=D_0x_2^\alpha(\ell-x_2)^\beta,$ 
$D_0=const>0,$
is a flexural rigidity of the plate, 
$\rho^s$ is a density of solid, 
$w(x_2,t)\in C^4(\Omega)\cap C^2(t>0)\cap C^1(t\ge 0)$ 
is a deflection of the plate, $q(x_2,t)\in C(\bar\Omega\times\{t\ge 0\})$ is 
an intensivity of a lateral load.
\par On $\partial\Omega$ the admissible boundary conditions of the plate theory
should be given [1,2].
\par
Let a flow of the viscous and incompressible
fluid be independent of $x_1$ and parallel to $0x_2x_3$ 
(i.e., $v_1\equiv 0$) with the following conditions at infinity
\begin{equation}
\begin{array}{c}
v_i(x_2,x_3,t)\vert_{x_3\rightarrow\infty}=v_{i\infty}(x_2,t),
\;\;p(x_2,x_3,t)\vert_{x_3\rightarrow\infty}=p_\infty(x_2,t),\\
v_{i\infty}(x_2,t)\vert_{x_3\rightarrow-\infty}=O(1),\;\;
p_{\infty}(x_2,t)\vert_{x_3\rightarrow-\infty}=O(1),\\
v_{i\infty}(x_2,t)\vert_{x_2^2\rightarrow\infty}=O(1),\;\;
p_{\infty}(x_2,t)\vert_{x_2^2\rightarrow\infty}=O(1),\;\;i=2,3,
\end{array}
\end{equation}
here $v:=(v_2,v_3)\in C^2(\Omega^f)\cap C^1(t>0)\cap C(t\ge 0)$ is a velocity vector of 
the fluid, $p(x_2,x_3,t)\in C^2(\Omega^f)\cap C(t\ge 0)$ is a
pressure, $v_{2\infty}(x_2,t),$ $v_{3\infty}(x_2,t),$ and
$p_\infty(x_2,t)\in C(\{]-\infty,0[\cap
]\ell,+\infty[\}\times\{t\ge 0\})$ are given functions.
\par Because of the incompressibility
\begin{equation}
{\rm div}\;v(x_2,x_3,t)=0,\;\;(x_2,x_3)\in\Omega^f,\;\;t\ge 0,
\end{equation}
and (see [5], p.5)
\begin{equation}
\sigma_{ij}(x_2,x_3,t)=-p\delta_{ij}+\mu\left(\frac{\partial v_i}{\partial x_j}+
\frac{\partial v_j}{\partial x_i}\right),
\end{equation}
where $\sigma_{ij}$ is a stress tensor, $\mu$ is a coefficient of velosity,
$\delta_{ij}$ is Kroneker delta.
\par Further, if the plate is thin, we can assume that: 
\par $-$ the fluid occupies the whole 
space $R^3$ but the middle plane $\Omega$ of the plate, i.e., 
$\Omega^f=R^3\backslash\Omega$. 
\par $-$ values of a normal component of stress tensor, 
$\sigma_{n3}^f\left(x_2,
{\mathop h\limits^{(\pm)}}(x_2),t\right)$
in the fluid 
part $\Omega^f$  are transferred at appropriate 
points of $\Omega$ from corresponding sides, i.e.,  
\begin{equation}
q(x_2,0,t)=\sigma^f_{33}(x_2,0_+,t)-
\sigma^f_{33}(x_2,0_-,t),\;\; x_2\in[0,\ell],\;\;t\ge 0;
\end{equation}
\par $-$ transmission conditions for $v_i(x_2,x_3,t)$ $(i=2,3)$ 
have the forms as follows (see [3], [4])
\begin{equation}
v_2(x_2,0,t)=0,\;\;\;
v_3(x_2,0,t)=\frac{\partial w(x_2,t)}{\partial t},
\;\; x_2\in[0,\ell],\;\;t\ge 0.
\end{equation}
\par
From (3) and (4) a normal component of the stress tensor is equal to
\begin{equation}
\sigma^f_{33}(x_2,x_3,t)=
-p(x_2,x_3,t)-2\mu\frac{\partial v_2(x_2,x_3,t)}{\partial x_2}.
\end{equation}
\par By virtue of (6), from (7) we get 
$$
\sigma^f_{33}(x_2,0_\pm,t)=-p(x_2,0_\pm,t).
$$
Hence, (5) has the following form
\begin{equation}
-p^+(x_2,t)+p^-(x_2,t)=q(x_2,t),\;\;x_2\in[0,\ell],\;\;t\ge 0.
\end{equation}
\par Let the motion of the fluid flow be sufficiently slow i.e., $v_i$ and
$v_{i,j}$ $(i,j=2,3)$ be so small that linearization of
Navier-Stokes equations (see [4,5,6,7]) be admissible. Hence,
$$
\frac{\partial v_i}{\partial t}=-\frac{1}{\rho^f}{\rm grad} p
+\nu\Delta v_i+F_i(x_2,x_3,t),\;\;\rho^f,\nu=const,\;i=2,3,
\eqno{\rm(9_i)}
$$
where $\rho^f$ is a density of fluid, $\nu=\mu/\rho$,  
$F:=(F_2,F_3)$ is a volume force.
\setcounter{equation}{9}
\par Let the initial conditions be
\begin{equation}
\begin{array}{l}
v_i(x_2,x_3,0)=v_i^0(x_2,x_3),\;\;
 i=2,3,\;\;(x_2,x_3)\in \Omega^f,\\
\ \\\displaystyle
w(x_2,0)=w^0(x_2),\;\;\frac{\partial w(x_2,t)}{\partial t}=
v_3^0(x_2,0), \;\; x_2\in[0,\ell].
\end{array}
\end{equation}
\par {\bf Theorem 1.} The solution of the initial transmission boundary value 
problem (1), (3), $\rm(9_i)$
$(i=2,3)$,  (10), (6), (8), (2) is unique.
\par {\bf Proof.} Let the difference of two admissible solutions of the problem 
under consideration be
$v_i(x_2,x_3,t)\;(i=2,3),$ $p(x_2,x_3,t),$ $q(x_2,t),$ $w(x_2,t).$
\par For these functions 
%$q(x_2,t)$, $w(x_2,t),$ $v_2(x_2,x_3,t)$, $v_3(x_2,x_3,t)$, 
%$p(x_2,x_3,t)$  
we get equations $(1)$, $(3)$, and 
$$
\frac{\partial v_i}{\partial t}=-\frac{1}{\rho^f}{\rm grad} p
+\nu\Delta v_i,\;\;i=2,3,
\eqno{\rm(11_i)},
$$
under the following conditions at infinity
\setcounter{equation}{12}
$$
p(x_2,x_3,t)=0\;{\rm when}\; x_3\rightarrow\infty;\;\;
p(x_2,x_3,t)=O(1) \;{\rm when}\;
x_2^2+x_3^2\rightarrow\infty,\eqno{\rm(12_1)}
$$
$$
v_i(x_2,x_3,t)=0\;{\rm when}\;\; x_3\rightarrow\infty;
$$
$$
v_i(x_2,x_3,t)=O(1)\;{\rm when}\;
x_2^2+x_3^2\rightarrow\infty,\;i=2,3,\eqno{\rm(12_i)}
$$
initial conditions
\begin{equation}
\begin{array}{l}
v_i(x_2,x_3,0)=0\;(i=2,3),\;\;
p(x_2,x_3,0)=0,\;\;(x_2,x_3)\in\Omega^f,\\
\displaystyle
q(x_2,0)=0,\; w(x_2,0)=0\;\;w,_t(x_2,0)=0,\;\;x_2\in[0,\ell],
\end{array}
\end{equation}
and transmission conditions (6), (8).
\par
After differentiation of $\rm(11_i)$ with respect to $x_i$ and termwise 
summation, in virtue of (3), we obtain that $p(x_2,x_3,t)$ is a harmonic 
function, i.e.,
\begin{equation}
\Delta p(x_2,x_3,t)=0,
\end{equation}
where $\Delta=\frac{\partial^2}{\partial x_2^2}+
\frac{\partial^2}{\partial x_3^2}$ is a Laplace operator.
\par The solution of the problem (14), (8), $\rm(12_1),$
has the following form (see [8])
$$
p(x_2,x_3,t)=-\frac{1}{2\omega\pi}{\mathop\int\limits_0^\ell}
\frac{q(\xi,t)x_3}{(\xi-x_2)^2+x_3^2}d\xi;
$$
\par $v_i(x_2,x_3,t)$ $(i=2,3)$ can be written as follows
$$
v_i(x_2,x_3,t)=-\frac{1}{\rho^f}{\mathop\int\limits_0^t}p,_{i}(x_2,x_3,\tau)
d\tau+\phi_i(x_2,x_3,t),\eqno{(15_i)}
$$
where the first term is a particular solution of $\rm (11_i)$, and 
$\phi_i(x_2,x_3,t)$ $(i=2,3)$, taking into 
account $\rm (11_i)$, (13), (3), (12), (6),  
are solutions of the problems
$$
\phi_{i,t}-\nu\Delta\phi_i=0, 
$$
\setcounter{equation}{16} 
$$
\phi_i(x_2,x_3,0)=0; \;\;\phi_i\vert_{x_3\rightarrow\infty}=0,
\;\;\phi_i\vert_{x_2^2+x_3^2\rightarrow\infty}=O(1),\eqno{\rm(16_i)}
$$
$$
\phi_2(x_2,0,t)=0,\;
\phi_3(x_2,0,t)=w,_t(x_2,t)+
\frac{1}{\rho^f}{\mathop\int\limits_0^t}p,_{3}(x_2,0,\tau)
d\tau,\;x_2\in[0,\ell],
$$
\begin{equation}
\phi_{2,2}(x_2,x_3,t)+\phi_{3,3}(x_2,x_3,t)=0.
\end{equation}
\par After differentiating $\rm (15_i)$ with respect to 
$x_j$ $(i,j=2,3,$ $i\ne j)$
and using (14) it is easy to show that
there exists the function $\psi(x_2,x_3,t)$ given by the expression 
$$
\psi,_i(x_2,x_3,t)=v_i(x_2,x_3,t)-\phi_i(x_2,x_3,t),\;\;i=2,3.\eqno{\rm(18_i)}
$$
By virtue of $\rm(11_i)$ and (17), we can find $\psi(x_2,x_3,t)$ from the
the following problem
\setcounter{equation}{18}
\begin{equation}
\Delta\psi=0,\;\;\psi,_2(x_2,0,t)=0,\;\;\psi,_2\vert_{x_2^2+x_3^2\rightarrow
\infty}=O(1).
\end{equation}
(19) means (see [8]) that $\psi,_2(x_2,x_3,t)=0$. Therefore
$\psi,_{33}(x_2,x_3,t)\equiv 0,$ and
$$
\psi(x_2,x_3,t)=c_1(t)x_3+c_2(t).
$$
Taking into account $\rm(18_3),$ $\rm(12_3),$ and $\rm(16_3),$ we get
$$
\psi,_3=c_1(t)=v_3-\psi_3=0,\;\;{\rm when}\;\;x_3\rightarrow\infty.
$$
Finally, we obtain $\psi,_3\equiv 0$,
\begin{equation}
v_i(x_2,x_3,t)-\phi_i(x_2,x_3,t)=0,\;\;i=2,3.
\end{equation}
\par By virtue of $\rm (15_i)$, (20), and 
$p(x_2,x_3,t)$ is a function only of $t$. On the other hand (see $\rm(12_i)$)
 this function is equal to zero 
at infinity, i.e.,
\begin{equation}
p(x_2,x_3,t)=A(t)=0.
\end{equation}
\par Further, the transmission condition (8) and equation (1) mean that
\begin{equation}
q(x_2,t)=0,\;\;
w(x_2,t)=0.
\end{equation}
\par In view of (21), (22), from  $\rm (11_i)$ and (6) we obtain
$$
v_{i,t}(x_2,x_3,t)-\nu\Delta v_i(x_2,x_3,t)=0,\;\;i=2,3.\eqno{\rm(23_i)}
$$
\setcounter{equation}{23}
\begin{equation}
v_i(x_2,0,t)=0,\;\;x_2\in[0,\ell],\;\;i=2,3.
\end{equation}
It is easy to show that the problem
(3), $\rm (23_i)$, (24), (13), (12)
has only trivial solution, i.e., $v_i\equiv 0$, $i=2,3.$

\newpage
\footnotesize
\begin{center}
{\bf R e f e r e n c e s}

\vspace*{0.2cm}
\end{center}
\par
[1] Chinchaladze N., Cylindrical Bending of the Prismatic Shell with Two Sharp Edges
in Case of a Strip. Reports of Enlarged Session of the Seminar of 
I.Vekua Institute of Applied Mathematics,  10, $N^{_{\underline 0}}$ 1, 1995, 21-23.
\par
[2] Jaiani, G.V., Elastic Bodies with Non-smooth Boundaries-Cusped
  Plates and Shells. ZAMM, 76, Suppl. 2, 1996, 117-120.
\par
[3] Chinchaladze N., Jaiani G., On a Cylindrical Bending of a Prismatic Shell
with Two cusped Edges Under Action of an Ideal Fluid. Bulletin of TICMI, 2, 1998, 
30-34.\\ (Web-site:http//www.viam.hepi.edu.ge/Others/TICMI)
\par
[4] Wollmir A., Shells on the Flow of Fluid and Gas. Problems of
Hydro-elasticity. Moscow, 1981, (Russian).
\par
[5] Duutray R., Lions J.-L.: Mathematical Analysis and Numerical Methods for 
Science and Technology. Vol.1, Springer-Verlag, Berlin, Heidelber, New-York, London, 
Paris, Tokyo, Hong Kong, 1990.
\par
[6] Loitsianskii L., Mechanics of Fluid and Gas. Moscow, 1960, (Russian).
\par
[7] Solonikov V.A., On Quasistationary Approximation in the Problem of Motion
of a Capillary Drop. Preprint 7/98, Pr\'e-publica\c{c}\v{o}es de Matem\'atica, Universidade de Lisboa, 1998.
\par
[8] Muskhelishvili N., Singular Integral Equations. Noordhoff, 1953.


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