On the non-absolute summability of a Fourier series and the conjugate of a Fourier series by a Nörlund method

(Proc. Camb. Phil. Soc. 63 (1967), 407–411)

By R. Mohanty and B. K. Ray

In the proof of Theorem 2 there is the tacit assumption that χ(t) is of bounded variation in (½π, π)—an assumption altogether lost sight of in its statement.

Consider an inviscid incompressible fluid confined within an infinite tube of radius b and assume that its motion initially consists of a uniform translation with velocity U and a uniform rotation, with angular velocity ω, about the tube axis. Suppose now that at time t = 0 a unit source of fluid situated at a point O of the axis begins to emit fluid at a uniform rate and that ultimately the flow pattern so modified becomes steady (at a fixed point as t → ∞). Under the further assumption that far upstream flow conditions are undisturbed by the source the stream function ψ of the disturbance satisfies

where z denotes distance along the axis from O, r distance from the axis and k = 2ω/U.

Consideration is given to the flow of a micropolar liquid down an inclined plane. The steady state is analysed and Yih's technique is employed in an investigation of the stability of this flow with respect to long waves. Detailed calculations are given for thin films and it is shown that the micropolar properties of the liquid play an important role in the stability criterion.

The study of electrostatic effects on the motion of fluids goes back to the work of Rayleigh (l) who discussed the effect of surface charges on the vibration of spherical drops, and Bassett (2) who studied the effect of a radial electric field on the stability of a jet. The subject has recently been taken up again by Melcher (3) and Crowley (4), and the author became interested in the subject through the monograph of the former author in which a wide variety of wave problems are described.

The paper studies the motion of an electro-conductive non-viscous incompressible fluid in a generator of rectangular section having finite electrodes and the rest of its walls being insulating. It is assumed that in the electrodes region a homogeneous magnetic field is applied, whose orientation is normal to the plane of the motion. If the induced magnetic field is neglected, the current function ψ(x, y) and the electrical potential ϕ(x, y) satisfy a system of equations with partial derivatives of the second order, with distributions. Since the system depends upon the parameter λ = RM RH a method of successive approximations is developed, which permits one to obtain the exact solution of the problem. In the last paragraph the results are extended to the case of a generator with N pairs of symmetrical electrodes.

The paper studies the influence of the electrical conductivity tensor upon the motion of a fluid in a MHD generator, under the conditions shown in our preceding work. The problem is also formulated in terms of the theory of distributions and the system of equations which determines the functions Ψ = ψ + y and φ = ϕ + ϕe depends on two parameters λ and v (= ωτ). Due to the Hall current, the problem is no longer asymmetrical in the variable y, as in the first case, it therefore has to be separated into an asymmetric and a symmetric problem. Developing a method of successive approximations as against two parameters, it can be shown that the exact solution of the problem may be obtained.

In a recent paper the symmetrical bifurcation of a parallel-plate waveguide by a unidirectionally conducting half screen was considered. In the present paper the analysis is both extended to cover the more general case of an asymmetrical bifurcation, and simplified by using the straightforward version of the Wiener-Hopf technique due to D. S. Jones. Some numerical values are given for the modulus of the reflexion coefficient as a function of the position of the half screen between the parallel plates.

The theory of large elastic deformation of reinforced elastic materials is applied to discuss the rotation of a circular tube of incompressible elastic material, reinforced with inextensible cords lying along helical paths.