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\begin{center}
ON A CYLINDRICAL BENDING OF A PRISMATIC SHELL WITH TWO CUSPED EDGES UNDER ACTION OF 
AN IDEAL  FLUID

\vspace*{0.2cm}
{\it N. Chinchaladze, G. Jaiani }

\vspace*{0.2cm}
{\it I. Vekua Institute of Applied Mathematics\\
I. Javakhishvili Tbilisi State University}
\end{center}

\vspace*{0.1cm}
\par
{\it 1. Some General Remarks on Solid-Fluid Transmission Conditions (G.Jaiani).}
\par
In the three-dimensional Solid-Fluid models usually the following transmission conditions
are used [1]:
\begin{equation}
[u]:=u^s-u^f=0,\;\;[\sigma_{ij}n_j]:=\sigma^s_{ij}n_j-\sigma^{f}_{ij}n_j=0,
\end{equation}
where $u^s$, $\sigma^s_{ij}$ and $u^f$, $\sigma^f_{ij}$ are displacement vector
and
stress tensor in the solid and fluid parts correspondingly, $n$ is outward normal
of $\partial(\Omega\backslash I)$, $\Omega=\Omega^s\cup\Omega^f$, $I$ is interface
between solid part $\Omega^s$ and fluid part $\Omega^f$.
\par
The Solid-Fluid model with unilateral transmission conditions as follows:
\begin{equation}
[u_iN_i]\ge 0,\;\; \sigma^f_{ij}n_j=\sigma^f_{NN}N_i,
\end{equation}
\begin{equation}
\sigma^s_{ij}n_j=-\sigma^f_{NN}N_i,
\end{equation}
\begin{equation}
\sigma^f_{NN}\le 0,
\end{equation}
\begin{equation}
{\rm and}\;\; \sigma^f_{NN}=0\;\;{\rm if}\;\; [u_iN_i]>0
\end{equation}
has been investigated in [2]. (2) and (3) mean that stresses on interface $I$ are
normal (tangential components are equal to zero, $\sigma_{NN}$ is a normal
component, $N$ is an outward normal to $\Omega^f$ on $I$) and balancing each 
other, (4) means that we have compression, (5) means
that if crack between solid and fluid parts is open than at that point normal force
equals zero.
\par
Our aim is to construct transmission conditions between three dimensional fluid
part, and solid part of plate or shell type, e.g. based on the classical bending theory
of plates.
Since in this case all mechanical quantities in solid part can be calculated by deflection,
as transmission conditions can be used either (1) or (2)-(5). Another possibility
of transmission conditions. If middle plane of the
plate lies in the plane $0x_1x_2$ and flow of moving fluid is parallel to $0x_3$, 
it could also have the form:
$$
\begin{array}{c}
\sigma_{33}^f\left(x_1,x_2,
{\mathop h\limits^{(+)}}(x_1,x_2),t\right)-
\sigma_{33}^f\left(x_1,x_2,
{\mathop h\limits^{(-)}}(x_1,x_2),t\right)=q(x_1,x_2,t),\\
\displaystyle
v_3\left(x_1-{\mathop h\limits^{(+)}}(x_1,x_2)
w,_1(x_1,x_2,t),x_2-{\mathop h\limits^{(+)}}(x_1,x_2)w,_2(x_1,x_2,t),
{\mathop h\limits^{(+)}}(x_1,x_2)+
\right.
\end{array}
$$
\begin{equation}
\begin{array}{c}
\left.
+w(x_1,x_2,t),t\right)=
\displaystyle
v_3\left(x_1-{\mathop h\limits^{(-)}}(x_1,x_2)
w,_1(x_1,x_2,t),x_2-\right.
\end{array}
\end{equation}
$$
\begin{array}{c}
\left.
\displaystyle
-{\mathop h\limits^{(-)}}(x_1,x_2)w,_2(x_1,x_2,t),
{\mathop h\limits^{(-)}}(x_1,x_2)-w(x_1,x_2,t),t\right)=
\displaystyle\frac{\partial w(x_1,x_2,t)}{\partial t},\\
\end{array}
$$
(the first of the last pair of equalities is valid since deflection of plate $w$ is independent of $x_3$), where $(x_1,x_2)\in \Omega$, 
$\Omega$ is projection of plate before deformation, $q(x_1,x_2,t)$ is intensivity of
lateral load, $v:=(v_1,v_2,v_3)$ is a velocity vector of fluid (compare (6) with the transmission
conditions considered in [3], where motion of fluid in elastic pipe has been 
investigated). On $\partial \Omega$ admissible boundary conditions of plate theory
should be given.
\par
{\it 2. Title Problem (N. Chinchaladze)}.
\par
Let the projection on the plane $0x_1x_2$ of plate under consideration be
$$\Omega=\{(x_1;x_2):-\infty<x_1<+\infty;\;0\le x_2\le b\}.$$
\par
Let us consider the interface problem of the interaction of the plate, whose flexural rigidity
is given by the equation
$$
D=D_{0}x_2^{\alpha}(b-x_2)^{\beta},\;\; \;\;D_0,\; \alpha,\;\beta=const>0,
$$
caused by the thickness
$$
2h(x_2)=h_0x_2^{\alpha/3}(b-x_2)^{\beta/3},\;\;\;h_0=const>0,
$$
and of a flow of the incompressible ideal fluid.
\par Let us consider the case of the flow of the fluid independent
of $x_1$ and parallel to $0x_3$ 
(directed upwards), with the following conditions at infinity
$$
v_3(x_2,x_3,t)\vert_{x_2^2+x_3^2\rightarrow\infty}=v_{3\infty}(t),\;\;
v_2(x_2,x_3,t)\vert_{x_2^2+x_3^2\rightarrow\infty}=0,
$$

$$
p(x_2,x_3,t)\vert_{x_2^2+x_3^2\rightarrow\infty}=p_\infty (t).
$$
\par The transmission conditions in a view of smallness of deflection and 
of smallness of plate's thickness, taking into account (6), are
\begin{equation}
v_3(x_2,0,t)=\frac{\partial w(x_2,t)}{\partial t},
\end{equation}

\begin{equation}
-p(x_2,{\mathop h\limits^{(+)}}(x_2),t)+
p(x_2,{\mathop h\limits^{(-)}}(x_2),t)=q(x_2,t).
\end{equation}
\par In case of the potential motion of the flow there exists a complex
function $\Phi=\psi+i\varphi$, where 
\begin{equation}
\frac{\partial \varphi}{\partial x_2}=\frac{\partial \psi}{\partial x_3}=v_2,\;\;\;
\frac{\partial \varphi}{\partial x_3}=-\frac{\partial \psi}{\partial x_2}=v_3.
\end{equation}
\par Pressure is given by the equation
\begin{equation}
\displaystyle
p(x_2,x_3,t)=\rho^f\left[\frac{v^2_\infty}{2}+\frac{p_\infty}{\rho}+\frac{\partial\varphi_\infty}{\partial t}-
\frac{\partial \varphi}{\partial t}-\frac{1}{2}(v_2^2+
v_3^2)\right].
\end{equation}
\par In case under consideration for $w(x_2,t)$, in virtue of (8), we get the following differential
equation [4]
\begin{equation}
\begin{array}{c}
\displaystyle
\frac{d^2}{dx_2^2}\left[x_2^\alpha(b-x_2)^\beta w''_{x_2}(x_2,t)\right]=
\frac{2\rho^s h}{D_0}\frac{\partial^2 w(x_2,t)}{\partial t^2}+\\
\\
\displaystyle
+\frac{-p\left(x_2,\frac{1}{2}h_0x_2^{\alpha/3}(b-x_2)^{\beta/3},t\right)+
p\left(x_2,-\frac{1}{2}h_0x_2^{\alpha/3}(b-x_2)^{\beta/3},t\right)}{D_0}.
\end{array}
\end{equation}
\par
For $\Phi'_{x_2}(x_2,x_3,t)=-v_3+iv_2$, in view of (7), we get the following expression (see[5])
$$
\begin{array}{c}
\displaystyle
\Phi'_{x_2}=-\frac{1}{\pi i \sqrt{(x_2+ix_3)(x_2+ix_3-b)}}\int\limits_{x_0}^{x_2}
\frac{\sqrt{(\tau_2+ix_3)(\tau_2+ix_3-b)}}{(\tau_2-x_2)-ix_3}
\frac{\partial w(\tau_2,t)}{\partial t}d\tau_2+
\end{array}
$$
\begin{equation}
\begin{array}{c}
\displaystyle
+v_{3\infty}\frac{x_2+ix_3-b/2}{\sqrt{(x_2+ix_3)(x_2+ix_3-b)}}.
\end{array}
\end{equation}
\par Let 
$$
w(x_2,t)=e^{i\omega t}w_0(x_2),\;\;\;
p(x_2,x_3,t)=e^{i\omega t}p_0(x_2,x_3),
\;\;\;\psi(x_2,x_3,t)= e^{i\omega t}\psi_0(x_2,x_3),
$$
\begin{equation}
v_2(x_2,x_3,t)= e^{i\omega t}v_2^0(x_2,x_3),\;\;\;
v_3(x_2,x_3,t)= e^{i\omega t}v_3^0(x_2,x_3).
\end{equation}
\par
If velocity of the flow is too 
small, then, in virtue of (10), $p$ can be represented in the form 
(compare [3], p. 14)
\begin{equation}
p(x_2,x_3,t)=\rho^f\left[\frac{p_\infty}{\rho}+\frac{\partial\varphi_\infty}{\partial t}-
\frac{\partial \varphi}{\partial t}\right].
\end{equation}
From (12), we have expression for $v_3$. By means of the latter, in view of (9), we
can calculate $\varphi$ which we have to substitute in (14).\par
Finally, taking into account (14), and (13) from (11) after four times integration,
 for $w_0(x_2)$ we get the following integral equation
\begin{equation}
\displaystyle
w_0(x_2)+\frac{\omega^2}{D_0\pi}\int\limits_{x_2^0}^{x_2}K(\tau_2,x_2)w_0(\tau_2)d\tau_2=
f(x_2),
\end{equation}
where 
$$
\begin{array}{c}
\displaystyle
f(x_2)=c_3(x_2-x_2^0)+c_4+{\mathop \int\limits^{x_2}_{x_2^0}}
\frac{(c_1+c_2)y(x_2-y)}{y^\alpha(b-y)^\beta}dy-\\
\displaystyle
-\frac{\omega^2\rho^f}{\pi D_0}{\mathop \int\limits^{x_2}_{x_2^0}}
\frac{(x_2-y)dy}{y^\alpha(b-y)^\beta}
{\mathop \int\limits^{y}_{x_2^0}}d\tau
{\mathop \int\limits^{{\mathop h\limits^{(+)}}(\tau)}_{-{\mathop h\limits^{(+)}}(\tau)}}
\left\{(\tau-b/2)cos\frac{\phi(\tau,\tau_3)}{2}
+\tau_3sin\frac{\phi(\tau,\tau_3)}{2}\right\}
\frac{v_{3\infty}^0 d\tau_3}{\sqrt{r(\tau,\tau_3)}},
\end{array}
$$
$$
\begin{array}{l}
\displaystyle
K(\tau_2,x_2)=
{\mathop\int\limits_{\tau_2}^{x_2}}
\frac{x_2-y}{y^\alpha(b-y)^\beta}\left\{
-2(y-\tau_2)\pi\rho^s h(\tau_2)+
\rho^f{\mathop\int\limits_{x_2^0}^{\tau_2} }(y-\tau)d\tau\right.
{\mathop \int\limits^
{\mathop h\limits^{(+)}(\tau)}
_{-{\mathop h\limits^{(+)}}(\tau)}}
\frac{\sqrt{r(\tau_2,\tau_3)}}{\sqrt{r(\tau,\tau_3)}}\times\\
\\
\left.\displaystyle
\times\frac{(\tau-\tau_2) cos\{(\phi(\tau_2,\tau_3)-\phi(\tau,\tau_3))/2\}+\tau_3
sin\{(\phi(\tau_2,\tau_3)-\phi(\tau,\tau_3))/2\}}{(\tau_2-\tau)^2+\tau_3^2}d\tau_3
\right\}dy,
\end{array}
$$
$$cos\phi(x_2,x_3)=(x_2^2-x_3^2-bx_2)/r(x_2,x_3),$$ $$r(x_2,x_3)=\sqrt{(x_2^2-x_3^2-bx_2)^2+(2x_2-b)x_3^2}.$$
\par
The integral equation (15) can be solved by method of successive approximations, and
 then constants $c_i$ $(i=1,...,4)$ will be defined from the admissible boundary value conditions [4]:
\par
$1.1.\;\; w_{0}(0)=\displaystyle\frac{dw_{0}(0)}{dy}=w_{0}(b)=\displaystyle\frac{dw_{0}(b)}{dy}=0,$ when 
$0<\alpha,\;\beta<1$;
\par
$1.2.\;\;w_{0}(0)=\displaystyle\frac{dw_{0}(0)}{dy}=
\displaystyle\frac{dw_{0}(b)}{dy}=Q_2(b)=0$, when 
$0<\alpha,\;\beta<1$;
\par
$1.3.\;\; w_{0}(0)=\displaystyle\frac{dw_{0}(0)}{dy}=w_{0}(b)=M_2(b)=0,$ when 
$0<\alpha<1,\;\; 0<\beta<2$;
\par
$1.4.\;\;w_{0}(0)=\displaystyle\frac{dw_{0}(0)}{dy}=
Q_2(b)=M_2(b)=0,$ when
$0<\alpha<1,\;\; \beta>0$;
\par
$2.1.\;\;\displaystyle\frac{dw_{0}(0)}{dy}=Q_2(0)=
w_{0}(b)=\displaystyle\frac{dw_{0}(b)}{dy}=0,$ when 
$0<\alpha,\;\beta<1$;
\par
$2.2.\;\;\displaystyle\frac{dw_{0}(0)}{dy}=Q_2(0)=
w_{0}(b)=M_2(b)=0,$ when 
$0<\alpha<1,\;\; 0<\beta<2$;
\par
$3.1.\;\;w_{0}(0)=M_2(0)=w_{0}(b)=\displaystyle\frac{dw_{0}(b)}{dy}=0,$ when
$0<\alpha<2, \;\; 0<\beta<1$;
\par
$3.2.\;\;w_{0}(0)=M_2(0)=
\displaystyle\frac{dw_{0}(0)}{dy}=Q_2(b)=0,$ when
$0<\alpha<2,\;\;0<\beta<1$;
\par
$3.3.\;\;w_{0}(0)=M_2(0)=
w_{0}(b)=M_2(b)=0$ when
$0<\alpha,\;\;\beta<2$;
\par
$4.1.\;\; M_2(0)=Q_2(0)=
w_{0}(b)=\displaystyle\frac{dw_{0}(b)}{dy}=0,$ when
$\alpha>0,\;\; 0<\beta<1$,\\
where $M_2$ is the bending moment, $Q_2$ is the intersecting force.
\par
In the stationary case (i.e. $\frac{\partial w}{\partial t}=0)$ symmetrical
prismatic shell
$\left({\mathop h\limits^{(+)}}(x_2)=\right.$ $\left.=-{\mathop h\limits^{(-)}}(x_2)\right)$ acts as absolutely rigid $(q(x_2)\equiv 0)$.

\vspace*{0.2cm}
\footnotesize
\begin{center}
{\bf R e f e r e n c e s}

\vspace*{0.1cm}
\end{center}
\par
[1] Sanches-Palencia E., Non-homogenous Media and Vibration Theory.
 Springer-Verlag, 1980.
\par
[2] Chichua G., On a Boundary-Contact Problem for a 
   Solid-Fluid Model. Reports of Enlarged Session of the Seminar of 
   I. Vekua Institute of Applied Mathematics, 1995, v.10, $N^{_{\underline 0}}$ 1, pp.18-20.
\par
[3] Wollmir A., Shells on the Flow of Fluid and Gas. Problems of
Hydro-elasticity. Moscow, 1981 (in Russian).
\par
[4] 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, 1995, v. 10, $N^{_{\underline 0}}$ 1, pp.21-23.
\par
[5] Muskhelishvili N., Singular Integral Equations. Noordhoff, 1953.
\end{document}