A fluid is a substance that deforms constantly under shear stress, or, to put it in another way, one that does not have a specific shape. Fluid mechanics deals with fluids in motion and at rest, as well as the effect of the fluid on the boundaries, which may be either solid surfaces or fluid-fluid interfaces. When it comes to the various uses of fluids in the world of science and technology ranging from regulating the passage of air through the lungs to pumping water through a pipeline, there is no limit to the variety of fluid applications. Our environment consists of mostly fluids. Most fundamental concepts of conservation of mass, energy, and momentum of applied mechanics are utilised to study fluid behaviour. Fluid properties such as density, volume, temperature, and pressure are all influenced by the one key assumption termed as Continuum Hypothesis. It is considered that fluids are made up of continuous matter, which allows fluid features to be defined in infinitesimally small areas and to change continuously from one point to the next. The fluid velocity matches the solid boundary velocity when it reaches a solid barrier. The outermost molecules of the fluid may be seen as adhering to the surfaces it passes across. Mathematically, when a boundary value problem is prescribed with a known solution at certain points, then that condition is called Dirichlet boundary condition. Matching the boundary velocity across a rigid barrier in fluid flow problem leads to Neumann boundary condition. The term no-slip condition is used to describe such no flow boundary condition in fluid mechanics.
As far as fluid mechanics is concerned, there are two methods of investigation that may be used. An initial approach known as the Lagrangian type involves the tracking of the movement of a single fluid particle from an initial location. It is also possible to specify
the mechanics in anEulerian way, which uses a fixed spatial coordinate system and looks at fluid properties as time-dependent functions of flow velocity. Although the Lagrangian specification is useful in certain cases, but it may take some time to analyse and does not instantly reveal the spatial gradients of velocity within the fluid. Therefore, Eulerian specifications are often preferred. Streamlines,pathlines, andstreaklinesare three common curves used in the Eulerian description of fluid motion. Using these curves as a starting point for monitoring seed particles or dye filaments in experiments is a common technique in fluid dynamics research. At all times, astreamline’s curve is tangent to the fluid velocity of the flow field. It is known as a streakline when the path of all fluid particles passing through a given area is traced out. A pathline is the path that a particle takes while it moves. One may find path lines and streaklines at the same spot.
When there is a steady stream, viscosity is the fluid attribute that makes a fluid more resistant to shear forces. Theoretical investigations of fluid mechanics often begin with the concept of an ideal fluid, where two contacting layers experience no tangential forces (i.e., shearing stresses) but only normal forces (i.e., pressure). It implies that there is no internal resistance in an ideal fluid. Real fluids, on the other hand, are susceptible to both tangential and normal stresses in their inner layers. Tangential forces explain the existence of viscosity in a fluid. Various characteristics of a fluid flow allow for a plethora of classifications. When pressure and temperature fluctuate, the density of a fluid changes substantially, but when pressure and temperature do not change, the density of a fluid remains the same. As an example, the Mach number (the ratio of an object’s speed to the local speed of sound) is employed in aerodynamics as a measure of compressible and incompressible flows. Negligible viscosity flow is referred to as a non- viscous or inviscid flow of the fluid, else the fluid is termed as viscous which describes a fluid having resistance to deform. While unsteady flow is characterised by a change in the properties of the fluid over time, asteady flow is the one in which the fluid properties remain constant over time. Recirculation, whirling, and unpredictability in the fluid are all factors that contribute to the formation ofturbulence. Laminar fluid flow occurs when there is little or no turbulence in the fluid stream. Fluid is categorised into two groups based on the mathematical connection between stress and strain: Newtonian fluids (water, air, etc.) and non-Newtonian fluids (blood, polymer, etc.). There is a very close linear connection between stress (internal force) and strain (normalised measure of deformation) for a Newtonian fluid. For a non-Newtonian fluid, stress and strain are not related to each other. Subsonic, transonic, supersonic, and hypersonic are all types of fluid flow based on the flow velocity (Mach number). If curl ~v = 0 in a flow field with a velocity vector
~v, the flow is called irrotational; otherwise, it is calledrotational. A scalar function must be present in order that a flow is irrotational. It is called the velocity potential usually with the notation Φ. Laplacian of the velocity potential must be zero if the flow is both irrotational and incompressible, resulting in an equation of continuity for a potential flow.
1.1. Preamble 3 The water wave theory of Airy was developed under the assumption of incompressible fluid with inviscid and irrotational flow. A similar assumption is being followed in our research which leads to the ideal fluid by definition. However, water in real life is not an ideal fluid. Moreover, practically water is an incompressible fluid with negligible viscosity.
In this context, water can be assumed as an ideal fluid.
In physics, mathematics and related fields, a wave is always in disruption (change from equilibrium position) of one or more fields in which values regularly bounce around a stable equilibrium (resting) value. Waves are of two types such as mechanical and electromagnetic. In a mechanical wave, the stress and strain fields vary around a point of mechanical equilibrium (water wave, sound wave, etc.). In an electromagnetic wave, the electric and magnetic fields are alternating. Electromagnetic waves (light) are made up of fluctuating electric and magnetic fields that travel across space. Electromagnetic waveforms include radio waves, infra-red and visible light, ultraviolet and gamma rays, as well as X-rays and other diagnostic imaging modalities. When we throw a stone into a pond, we can see ripples which is a primary example of water waves in action. Examples of a few more such incidence include ripples on a lake’s surface caused by the motion of wind, ripples due to movement of ships in the ocean etc. These waves oscillate at very small frequencies. There are, however, certain exceptions to the rule. Amplitude of waves is not always small, but it may still be perceptible in certain situations. In addition to tsunamis and tidal waves, other high-energy waves include rogue waves.
Researchers have lately proposed a variety of porous breakwaters for the protection of coastal regions and developmental sites. Reduced wave stresses and run-up may be achieved by porous structures. Additionally, these structures are often used to create floating airports, bridges, piers, docks, and wave power conversion systems, among other things. The use of porous structures in the construction of coastal and offshore structures has been found to be more suited. Structures in a sea must resist a variety of atmospheric conditions. As a consequence, hydrodynamic research has focused on building optimal systems in order to find a way to minimise significant hydrodynamic ramifications. When a problem with water wave mechanics arises, it is common to look at water in a region that is bounded by a free surface (in touch with the atmosphere), a rigid surface, a porous surface, or any other media. In addition, for the sake of convenience, we may consider virtual boundaries. The wave interaction with a large floating ice sheet, which may be considered as a massive elastic plate following the Euler-Bernoulli beam equation, is of equal importance in the polar marginal ice zone. A hydroelastic model is necessary in many circumstances when structures are believed to be flexible. Many hydrodynamic fac- tors demand investigation. Several of these structures have advantages over conventional rigid structures because they are recyclable, ecologically friendly, and cost-effective. There has been a reasonable advancement in the knowledge of the dynamics of ocean surface waves and their influence on ocean structures during the past several decades. Different
physical conditions lead to different solutions. Laplace’s equation, Helmholtz or modified Helmholtz equations are mainly used to describe wave motion in water with Dirichlet type (no derivative) or Neumann type (only derivative) or mixture of derivative and non- derivative boundary conditions. Analysis of Fourier transform, complex function theory, Green’s function technique, boundary integral equation method, the least square approxi- mation method, the wide spacing approximation method, and integral transform method can all be utilized as the solution procedures. The same theory holds for the stationary phase as well. In addition, it may be used for porous regions.
A long history of water waves can be found in Craik [23], who wrote extensively on water wave theory. A hypothesis of water waves was initially proposed by Sir Isaac Newton in the 17th century. He suggested a dubious analogy with oscillations in a U- tube and properly deduced that the frequency of deep sea waves was related to the square root of the "breadth of the wave" in Book II, Prop. XLV of Principia (1687). Wilhelm- Jacobs Gravesande (1721) and Charles Bossut (1786) also endorsed Newton’s hypothesis later. Then, Joseph Louis Lagrange (1781, 1786) came up with the linearized governing equations for small-amplitude waves and found the solution to long plane waves in shallow water. In ‘Mechanique Analitique’ (1788), precisely the same theory was established.
Lagrange (1786) said that, for shallow water waves, “the speed of propagation of waves will be that which a heavy body would acquire in falling from ... half the height of the water in canal", i.e., (gh)1/2, whereg represents gravitational acceleration and h denotes liquid depth. Only a handful of publications prior to 1800 dealt with wave motion.
M ˙Flaugergues (1793) and Francois de la Coudraye (1796) created exceptional works, which were subsequently summarised by Weber and Weber (1825). The first accurate nonlinear solution for waves with limited amplitude in deep sea was described by Franz Joseph von Gerstner (1802). Because the motion was not irrotational, the Gerstner wave solution has long been disregarded; even now, it is considered a curiosity rather than a conclusion of practical significance because of this. W. J. Macquorn Rankine (1863) was the one who discovered it on his own. In December, 1813, the French Academie des Sciences launched a mathematical prize competition on the propagation of surface waves in fluids of infinite depths. It was in July 1815 when Augustin-Louis Cauchy filed his submission and in August, Simeon D. Poisson, one of the judges, lodged a memoir of his own to document his independent work. The prize was awarded to Cauchy in 1816. Poisson’s narrative was published in 1818, while Cauchy’s work was finally published in 1827, with 188 more pages of annotations. Horace Lamb presented the Cauchy-Poisson analysis, but confined it to two-dimensional disturbances in "Hydrodynamics" (1895). This book was originally written in 1908 as a short, unfinished, and tiny print book, but he revised it in 1916 for a more comprehensive treatment. The Cauchy-Poisson analysis is currently recognised as an important milestone in the theory of initial value problems.
Ernst Heinrich Weber and Wilhelm Eduard Weber wrote a radically distinct study
1.2. Relevant equations and conditions 5