The 12th International Conference on Hydrodynamics
18 – 23 september 2016, Egmond aan Zee, The Netherlands
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EFFECT OF COMPRESSION ON THE HEAD-ON COLLISION BETWEEN TWO HYDROELASTIC SOLITARY WAVES


Go-down ichd2016 Tracking Number 151

Presentation:
Session: Fluid structural inter-actions IV
Room: Room 4
Session start: 10:30 Tue 20 Sep 2016

M. M. Bhatti   muhammad09@shu.edu.cn
Affifliation: Shanghai University

D.Q. Lu   dqlu@shu.edu.cn
Affifliation: Shanghai University


Topics: - Linear and non-linear waves and current, - Fluid-structural interactions and hydroelasticity

Abstract:

Interaction and collision of different fluids and elastic boundaries originates in various mechanical environments. These types of problems are very much difficult mathematically, due to its coupling between deformable bodies and moving fluids. Another interesting thing we observed that when the solitary waves collide each other, they transfer their position and energies with each other and then regain their original shape and position after separating off. It is noteworthy here that, during the whole process of collision, solitary waves are very much stable in preserving their identities. To obtain the solution of head-on collision between solitary waves, one must employ the asymptotic technique to solve the highly nonlinear partial differential equations. Inverse scattering transform method (IST) is also use to get the solution of KdV equation, but this method is only applicable for overtaking interaction between solitary waves. For this purpose we will employ Poincar$\acute{\textrm{e}}$--Lighthill--Kuo (PLK) method. It is originally the method of strained coordinates which was introduced by Poincare $1982$ for ordinary differential equations, Later Lighthill $(1949)$ and Lin $(1954)$ applied this scheme to hyperbolic differential equations. We investigate the head-on collision between two hydroelastic solitary waves traveling in a fluid covered by an elastic plate under the influence of compression. The fluid having constant density is incompressible and inviscid and the motion is irrotational. The horizontal plane bottom is situated at $z=0$ where the normal velocity of the fluid is considered to be zero. The deflection of the plate is presented at $z=H(x,t)$. With the help of potential flow theory, the governing equation is the Laplace equation. The Euler beam theory is taken for the elastic part in the dynamical boundary conditions. Both the solitary waves are small in amplitude ($a$) and are long in wavelengths $\lambda$, namely $ a/H \ll 1 $ and $\lambda / H \gg 1$. The physical parameters are related to Ursell's ordinary theory of shallow water, i.e $H^3 \approx a \lambda^2 $. The solutions of the nonlinear equation has been obtained with the help of the method of strained coordinates (the PLK method) upto fourth-order approximation. The behavior of all the physical parameters of interest are discussed and demonstrated graphically.