Analytical Study Of The Head-on Collision Between Hydroelastic Solitary Waves By Means Of Singular Perturbation Methods

The study of the deformation of a thin elastic plate floating on the surface of the liquid has many applications in polar regions and ocean engineering.In the polar areas,the ice sheet is converted into runways and roads for transportation purposes.Such kinds of floating materials bolong to a category of naval architect,ure known as Very large floating structures(VLFSs),which is applicable in different purposes,i.e.,military bases,exploiting ocean resources,entertainment facilities,recreation parks,mobile offshore structures,industrial space,and even for habitation.The nonlinear head-on collision between two hydroelastic solitary waves in shallow water is studied in this thesis.The motion of the fluid is assumed to be irrotational with incompressible and inviscid properties.An infinite t,hin elastic plate is floating on the surface of the water.The mathematical modeling of thin elastic plate iy based on Euler-Bernoulli beam theory.To determine the head-on collision process,we utilize a singular perturbation method.The Poincare-Lighthill-Kuo(PLK)method is used to obtain the analytical solutions of the highly nonlinear partial differential equations.This method has the capability to determine the head-on collision process between solitary waves.Within the framework of Ursell's theory for shallow water waves with the Boussinesq approximation,we assume the scaling of the horizontal wavelength.The resulting solut,ions are presented separately for the left-and rightt-going waves.The behavior of all the emerging parameters are presented mathematically and discussed graphically for the phase shift,maximum run-up amplitude,distortion profile,wave speed,and solitary wave profile.The thesis is formulated in the following manner.1)In the first part,head-on collision between two hydroelastic solitary waves is studied.An infinite thin e.lastic plate is floating on the surface of the water in the presence of compression.PLK method is applied to analyze the collision process between solitary waves.The asymptotic series solutions are presented up to third-order approximation.A third-order KdV equation is obtained for the hydroelastic solitary wave profile.2)In the second part,we studied the head-on collision process under a thin ice sheet in the presence of surface tension.The ice sheet is represented by the Plotnikov-Toland model with the help of the special Cosserat theory of hyperelastic shells and the Kirchhoff-Love plate theory,which yields the nonlinear and conservative expression for the bending forces.The mathematical formulation of the ice prob-lem is similar to t.hat of CG wave with the curvature term due to surface tension replaced by a term modeling the nonlinear elasticity of thin plates.The shallow wat,er assumption is taken for the fluid motion with the Boussinosq approxima-tion.The resulting governing equations are solved asymptotically with the aid of the Poincare-Lighthill-Kuo method,and the solutions up to the third order are explicitly presented.A graphical comparison is presented with published results,and the graphical comparison between linear and nonlinear elastic plate models is also shown as a special case.3)The third part discusses an analytical simulation of the head-on collision between a pair of hydroelastic solitary waves propagating in opposite directirns in the presence of a uniform current.An infinite thin elastic plate is floating on the surface of the wat,er.The mathematical modeling of the thin elastic plate is based on the Euler-Bernoulli beam model.The resulting kinematic and dynamic boundary conditions are highly nonlinear,which are solved analytically with the help of a singular perturbation method.The Poincare-Lighthill-Kuo method is applied to obtain the solution of the nonlinear partial differential equations.The resulting solutions are presented separately for the left-and right-going waves.The behavior of all the emerging parameters are presented mathematically and discussed graphically for the phase shift,maximum run-up amplitude,distortion profile,wave speed,and solitary wave profile.A graphical comparison with pure-gravity waves is also presented as a particular case.4)In the forth part,the head-on collision between two hydroelastic solitary waves in plate-covered water with Xwogou's Boussinesq model for the nonlinear fluid mot,ion.This model contains a parameter "?"that is associated with horizontal velocities according to the chosen level of horizontal velocity variables.A thin elastic cover is considered as the Euler-Bernoulli beam model.To derive the series solution,we apply the Poincare-Lighthill-Kuo(PLK)method to solve an-alytically the highly nonlinear coupled partial diifferential equations.The impact of all the physical parameters is discussed with the help of asymptotic solutions and graphic representations.In particular,ww address the behavior of plate de-flection,maximum run-up during a collision,phase shift,distortion profile,and wave speed.5)In the fifth part,Head-on collision between hydroelastic solitary waves propagat-ing in a two-layer fluid beneath a thin elastic plate are analytically investigated.The plate structure is modeled using the Euler-Bernoulli beam theory.The ef-fect of compressive stress is taken into account for the elastic plate.We consider that the lower-and upper-layer fluids having different constant densities are in-compressible and the motion is irrotational.The asymptotic series solution of the resulting highly nonlinear coupled differential equations are deduced with the combination of a method of strained coordinates and the Poincare-Lighthill-Kuo(PLK)method.The series solutions obtained are presented up to the third-order approximation.The inclusion of all the emerging parameters is discussed graph-ically and mathematically against interfacial waves,plate deflection,wave speed.phase shift,maximum run-up amplitude,and the velocity functions.According to the analytical results obtained by the singular perturbation,it is found that solitary waves propagating in the opposite direction regain their original shapes and speed after the collision process.The hydroelastic response of a thin elastic plate significantly impacts the wave profile and provides a remarkable in the amplitude of the wave profile.It is also noticed that the hydroelastic responses of nonlinear and linear elastic plates are the same on the wave profile.The presence of uniform current not only affects the wavelength and wave amplitude but also produces a remarkable impact on the wave speed.The hydroelastic response of an elastic plate is also elabo rated using X-wogu's Boussinosq fluid model.It is noticed that the wave profiles behave similarly.However,these results can be further extended for deep water waves.The wave profiles in the upper-and lower-layer fluid become diminutive due to the more significant influence of density ratio and plate deflection.