Boundary value problems arising in multi-layered fluid system

Abstract

Boundary value problems (BVPs) occur in a natural way while studying the scattering of water waves as well as the fluid flow problems. The objective of the thesis is to solve such BVPs by using the most appropriate mathematical techniques and to analyze: (i) scattering of small amplitude water waves by different types of undulations present at the porous bottom in a two-layer fluid of finite depth, where the upper layer is either free to the atmosphere or covered by a thin ice-sheet or affected by the surface tension; (ii) the behavior of the free-surface as well as the interfaces at the outset in the case of the flow of fluid in a multi-layered channel having an arbitrary bottom topography. It is assumed that the fluid is inviscid, incompressible and immiscible with constant but different densities; and the motion of the fluid is irrotational and simple harmonic in time. In Part-I of the thesis, various problems involving scattering of water waves in a two- layer fluid are considered for their solutions where the upper layer is either free to the atmosphere or covered by a thin ice-sheet or affected by the surface tension. Depending upon the angle of incidence, the problem is one of these kinds: one describing the scatter- ing of waves when the progressive waves incident normally to the undulating bottom, and the other describing the scattering of waves when the progressive waves incident obliquely. In each situation, the physical problem formulated as mixed and coupled boundary value problem to determine the velocity potentials corresponding to each layer of the fluid. The governing partial differential equation happens to be Laplace’s equation for the case of normal incidence, whereas it is Helmholtz equation for the case of oblique incidence with mixed boundary conditions at the top surface and at the porous bottom and two interface conditions out of which one is mixed boundary condition. Due to the mixed boundary condition at the interface, the BVPs become coupled. As the fluid region extends to infin- ity, a far-filed condition in each layer arises which affirms the uniqueness of the problem. The coupled and mixed BVP is reduced to a simpler coupled BVP of the first-order by employing a perturbation analysis. The simpler coupled BVP is uncoupled which gives rise to two independent BVPs. These independent BVPs are solved by using the Fourier transform technique and residue theorem to derive the first-order velocity potentials. These potentials are utilized to determine the first-order reflection and the transmission coefficients. These coefficients are obtained in terms of the integrals involving the shape of the undulating porous bottom. Different types of shape functions are considered to evaluate these integrals explicitly. The numerical values of the first-order reflection and transmission coefficients for various system parameters are derived and shown graphically for a special kind of bottom topography, namely a patch of sinusoidal ripples due to its applications in the coastal and marine engineering. The graphical forms illustrate the transformation of the incident wave energy from one layer to another layer. In addition, the energy identity, an important feature of the study involving scattering of water waves in a two-layer fluid, is investigated in detail with the help of the Green’s integral theorem. This identity provides the correctness of the analytical and numerical results derived in this thesis. The Fourier transform method is found to be a comparatively easier method for solving the scattering problems. It may be emphasized that, when the density of the upper layer fluid is equal to the density of the lower layer fluid, the two-layer fluid problem reduces to a single-layer fluid problem which is also investigated here. The results devel- oped here are expected to be helpful for a large class of scattering problems in a two-layer ocean with an uneven porous bottom in the areas of coastal and marine engineering. In Part-II of the thesis, various problems involving the flow of fluid in a channel are considered for their solutions. Here the behavior of the free-surface as well as the interface profiles (for multi-layer fluid) which are unknown at the outset, are analyzed. First, the problem involving the flow of an inviscid and incompressible fluid in a single-layer channel having small obstruction is studied, and then the problem involving two-layer flow is studied to determine the analytical expressions for the profiles of interface and the free- surface. Further, the two-layer flow problem is generalized as the three-layer flow problem as the single or the two-layer approximation becomes insufficient due to the continuous stratification of the fluid. The BVPs arising in a natural way are solved using perturbation analysis in conjunction with Fourier transform technique. In the case of the flow in a three- layer fluid, the following flow problems are studied: (A) when the uppermost fluid layer is bounded by a rigid lid and (B) when it is free to the atmosphere. In this case, in order to study the influence of natural and individual variation of the bottom profile on the essence of the interface profiles, two different kinds of bottom profiles, such as concave and convex are considered in both the problems separately. In both problems (A) and (B), the nature of the roots of the dispersion relation, which is an important relation of the study involving fluid flow, is investigated in detail using the Rouche’s theorem. The numerical results are tabulated and demonstrated in graphical forms to understand the effect of some important physical quantities on the behavior of the interfaces. It is observed that the interfacial profiles are oscillatory in nature, representing waves of variable amplitude with different wave numbers which lead to several interesting features like wavy solutions, beating-like behavior, downstream resonances etc. These phenomena are noticed in this study, which are not reported in earlier studies. In addition, it is observed here that the interface profiles exhibit a wave-free region at the upstream of the obstacle followed by a wavy region at the downstream of the obstacle. These studies are based on the linear theory. An effort has also been made in this thesis to solve a nonlinear flow problem involving free-surface flow over arbitrary bottom in a single-layer fluid. A simple numerical approach for solving Fredholm integral equations of the second kind is developed here. This new method helps in finding the solution of this nonlinear flow problem. This nonlinear flow problem is formulated mathematically in terms of a nonlinear BVP which is reduced to a Dirichlet problem after using certain transformations. After utilizing a suitable application of Plemelj-Sokhotski formulae, the Dirichlet problem is solved when a pair of second kind Fredholm integral equations are solved. The new approximate method developed here is utilized to solve the coupled integral equations and hence the free-surface profile is determined. A number of observations are made to illustrate the effect of nonlinearity. The effects of the Froude number and the arbitrary bottom topography on the free-surface profile are discussed here. In addition, the nonlinear results are compared with the results obtained by using the linear theory. The study in the thesis may be helpful for handling different situations arising in the broad area of ocean engineering, atmospheric sciences and related branches of mathematical physics.