Subash Chandra Martha

A modified approach to numerical solution of Fredholm integral equations of the second kind

A modified approach to obtain approximate numerical solutions of Fredholm integral equations of the second kind is presented. The error bound is explained by the aid of several illustrative examples. In each example, the approximate solution is compared with the exact solution, wherever possible, and an excellent agreement is observed. In addition, the error bound in each example is compared with the one obtained by the Nystrom method. It is found that the error bound of the present method is smaller than the ones obtained by the Nystrom method. Further, the present method is successfully applied to derive the solution of an integral equation arising in a special Dirichlet problem.

Inverse analysis of conductive-convective wet triangular fin for predicting thermal properties and fin dimensions

The present work deals with the application of Homotopy Analysis Method (HAM) in conjunction with Nelder–Mead simplex search method (SSM) to study a triangular wet fin. At first, analytical expression has been derived using HAM to calculate the local temperature field. Then using SSM, the important parameters, namely, thermal conductivity of the material, surface heat transfer coefficient and dimensions of the fin, have been estimated separately for attaining the prescribed temperature field. The transport phenomena involve simultaneous heat and mass transfers. It is found from the present study that many feasible solutions can satisfy a given thermal condition, which will offer the flexibility in selecting the fin material, adjusting the thermal conditions and regulating the fin dimensions. Further, it is determined that the allowable error in the temperature measurement should be limited within ± 15% and a good reconstruction of the temperature field is possible using the HAM–SSM combination.

Oblique wave scattering by undulating porous bottom in a two-layer fluid: Fourier transform approach

The problem involving scattering of oblique waves by small undulation on the porous ocean bed in a two-layer fluid is investigated within the framework of linearised theory of water waves where the upper layer is free to the atmosphere. In such a two-layer fluid, there exist waves with two different wave numbers (modes): wave with lower wave number propagates along the free surface whilst that with higher wave number propagates along the interface. When an oblique incident wave of a particular mode encounters the undulating bottom, it gets reflected and transmitted into waves of both modes so that some of the wave energy transferred from one mode to another mode. Perturbation analysis in conjunction with Fourier transform technique is used to derive the first-order corrections of velocity potentials, reflection and transmission coefficients at both modes due to oblique incident waves of both modes. One special type of undulating bottom topography is considered as an example to evaluate the related coefficients in detail. These coefficients are shown in graphical forms to demonstrate the transformation of water wave energy between the two modes. Comparisons between the present results with those in the literature are made for particular cases and the agreements are found to be satisfactory. In addition, energy identity, an important relation in the study of water wave theory, is derived with the help of the Green’s integral theorem.

Three-layer fluid flow over a small obstruction on the bottom of a channel

Many boundary value problems occur in a natural way while studying fluid flow problems in a channel. The solutions of two such boundary value problems are obtained and analysed in the context of flow problems involving three layers of fluids of different constant densities in a channel, associated with an impermeable bottom that has a small undulation. The top surface of the channel is either bounded by a rigid lid or free to the atmosphere. The fluid in each layer is assumed to be inviscid and incompressible, and the flow is irrotational and two-dimensional. Only waves that are stationary with respect to the bottom profile are considered in this paper. The effect of surface tension is neglected. In the process of obtaining solutions for both the problems, regular perturbation analysis along with a Fourier transform technique is employed to derive the first-order corrections of some important physical quantities. Two types of bottom topography, such as concave and convex, are considered to derive the profiles of the interfaces. We observe that the profiles are oscillatory in nature, representing waves of variable amplitude with distinct wave numbers propagating downstream and with no wave upstream. The observations are presented in tabular and graphical forms.

Wave Scattering by Small Undulation on the Porous Bottom of an Ocean in the Presence of Surface Tension

The scattering of incident surface water waves due to small bottom undulation on the porous bed of a laterally unbounded ocean in the presence of surface tension at the free surface is investigated within the framework of two-dimensional linearized water wave theory. Perturbation analysis in conjunction with the Fourier transform technique is employed to derive the first-order reflection and transmission coefficients in terms of integrals involving the shape function 𝑐(𝑥) representing the bottom undulation. One special type of bottom topography is considered as an example and the related coefficients are determined in detail. These coefficients are presented in graphical forms. The theoretical observations are validated computationally. The results for the problem involving scattering of water waves by bottom deformations on an impermeable ocean bed are obtained as a particular case.

Oblique wave diffraction by varying porous bottom in the presence of surface tension

The problem involving scattering of oblique water wave by small undulation on the porous ocean-bed in the presence of surface tension at the free surface, is investigated within the framework of linearized water wave theory. Perturbation analysis in conjunction with the Fourier transform technique is used to derive the first order corrections for the velocity potential, reflection and transmission coefficients. These coefficients are obtained in terms of integrals involving the shape function c(x) representing the bottom undulation. One special type of undulating bottom topography is considered as an example to evaluate all the related coefficients. Numerical results are plotted in graphical forms.

An alternative approach to study nonlinear inviscid flow over arbitrary bottom topography

This paper deals with a new approach to study the nonlinear inviscid flow over arbitrary bottom topography. The problem is formulated as a nonlinear boundary value problem which is reduced to a Dirichlet problem using certain transformations. The Dirichlet problem is solved by applying Plemelj-Sokhotski formulae and it is noticed that the solution of the Dirichlet problem depends on the solution of a coupled Fredholm integral equation of the second kind. These integral equations are solved numerically by using a modified method. The free-surface profile which is unknown at the outset is determined. Different kinds of bottom topographies are considered here to study the influence of bottom topography on the free-surface profile. The effects of the Froude number and the arbitrary bottom topography on the free-surface profile are demonstrated in graphical forms for the subcritical flow. Further, the nonlinear results are validated with the results available in the literature and compared with the results obtained by using linear theory.

Transformation of water-wave energy in two-layer ocean in the presence of surface tension

In the present paper, transformation of water-wave energy in a two-layer fluid having irregular bottom is studied including the effect of the surface tension. In such fluid system, time harmonic waves of two distinct modes (wave numbers) exist. When an incident wave with a particular wave number encounters the irregular bottom, it is partially reflected as well as partially transmitted into waves of both wave numbers. As a result, some of the energy is transferred from one to another wave number. Perturbation analysis followed by Fourier transformation is used to study the reflection and transmission phenomena. Reflection and transmission coefficients of both modes are estimated up to the first-order. These coefficients are obtained in integral forms and depend on the bottom profile. One special type of shape function is considered to evaluate the related coefficients. These coefficients are shown in graphical forms to demonstrate the effects of some physical parameters like number of ripples, surface tension on the transformation of wave energy between the waves of two different modes. Moreover, a comparison has been made between the present results and the experimental results available in the literature.

SCATTERING OF WATER WAVES BY FREELY FLOATING SEMI-INFINITE ELASTIC PLATES ON WATER OF FINITE DEPTH

A class of mixed boundary value problems (BVPs) arising in the study of scattering of surface water waves by the edges of floating structures comprising of elastic plates, with or without cracks, is examined for their solutions. It is observed that the simplest possible method of solution of such BVPs is the one that involves solution of an over-determined system of Linear Algebraic Equations. Such over-determined systems of equations are best solved by the method of least squares. Numerical results for useful practical quantities such as the “reflection” and “transmission” coefficients are obtained for one of the problems considered here.