Volume 96, Issue 1
August 1987, pages 1-69
pp 1-25 August 1987
The problem of thermal instability in a fluid saturated porous spherical shell heated internally, due to uniform internal heat sources and in equilibrium under its own radial gravitational field is studied theoretically. A general disturbance is analysed into modes in terms of spherical harmonics of various orders,l, for different values of the thickness of the mantle and the criteria for the onset of convection for the first fifteen modes is obtained in four different cases when the outer and inner bounding surfaces are either impermeable or permeable. It is shown that as the thickness of the shell decreases, the pattern of convection which sets in at marginal stability shifts progressively to harmonics of higher order for all the three cases except when both the bounding surfaces are permeable, in which case the onset of convection occurs at a harmonic of order 1. A comparison of some representative results of these cases is made with that of continuous fluid shell with rigid or free boundary surfaces. The neutral stability plots for various thickness of the mantle, for five different models of the mantle, are plotted for the different types of boundary surfaces.
pp 27-33 August 1987
A specific problem concerning the vertical vibrating response of a composite layer (elastic-porous and saturation fluid) under rigid circular stamp is studied. The motion of the medium is governed byBiot’s dynamic potential equations and the problem is solved by the method of Fourier-Bessel transformation. Formulae are obtained for the vertical displacement of the rigid phase. Graphs are presented for the amplitude of vibration.
pp 35-48 August 1987
A theoretical formulation to study the problem of scattering of Rayleigh waves due to the presence of a rigid plane strip in a deep ocean is presented. A rigid plane strip (0 ≤z ≤ H, 0 ≤x ≤ l) is fixed in the surface of the ocean occupyingz ≥ 0. Fourier transformation and Wiener-Hopf technique are used to arrive at the solution. The scattered Rayleigh waves behave as cylindrical waves emerging out of the corner of the strip and its image in the free surface of the ocean. The scattered waves are obtained in terms of Bessel functions whose behaviour near and far from the strip is well-known. The numerical calculations for the scattered waves show that their amplitude increases rapidly for a small increase in the value of the wave number. Scattering of Rayleigh waves due to a thin plane vertical barrier and a thin barrier in the free surface of the ocean has been considered as the special cases.
pp 49-59 August 1987
A necessary and sufficient condition is shown in order that a linear system of quadrics inSr, joined of first and second kind, has a Jacobian with rankr— k.
pp 61-69 August 1987
The aligned magnetic effects on a steady laminar, viscous, incompressible, conducting fluid down an open inclined channel bounded below by a bed of varying permeability has been studied when the free surface is exposed to atmospheric pressure. Beavers and Joseph slip condition at the interface of the free flow region and the fluid flow in the porous bed and the Darcy’s law in the porous medium have been used. The expressions for velocity, magnetic strength and the mass flow across the cross-section of the channel are obtained.