Volume 91, Issue 3
November 1982, pages 167-234
pp 167-182 November 1982
This paper is devoted to a study and classification ofG-invariant convex cones ing, whereG is a lie group andg its Lie algebra which is simple. It is proved that any such cone is characterized by its intersection withh-a fixed compact Cartan subalgebra which exists by the very virtue of existence of properG-invariant cones. In fact the pair (g,k) is necessarily Hermitian symmetric.
pp 183-193 November 1982
The minimum error solutions of boundary layer equations in the least square sense have been studied by employing the Euler-Lagrange equations. To test the method a class of problems,i.e., boundary layer on a flat plate, Hiemenz flow, boundary layer on a moving sheet and boundary layer in non-Newtonian fluids have been studied. The comparison of the results with approximate methods, like Karman-Pohlhuasen, local potential and other variational methods, shows that the present predictions are invariably better.
pp 195-199 November 1982
The complementary variational principles are given for the poiseuille flow of an Oldroyd fluid by taking the pressure gradient to be exponentially increasing with time and the bounds on the flux are obtained.
pp 201-210 November 1982
Torsional wave motion in a finite hollow cylinder of piezoelectric material of (622) crystal class, under a time-dependent mechanical boundary condition is investigated. The inhomogeneity is restricted to the variations of density and other physical constants of the medium as a certain power of the radial distance. The expressions for the displacement and the electric potential of the present solution are compared with those under time-dependent electric boundary condition. Numerical values of the roots of the frequency equation for β-quartz are presented.
pp 211-216 November 1982
The paper examines the slow motion of a micropolar fluid produced by the relative motion of a solid sphere to an inside porous sphere. The result extends the Cunningham’s problem to micropolar fluid when the inner sphere is porous with prescribed radial suction/injection velocity at the surface of the sphere. The result can also be taken as an extension of the work of Ramkissoon and Majumdar when the fluid is bounded at a radiusr=b (b>a) but the solid sphere is replaced by a porous sphere. The force experienced by the inner sphere has been calculated and particular cases of interest have been deduced.
pp 217-221 November 1982
Zero-free regions of thekth derivative of the Riemann zeta function ζ(k)(s) are investigated. It is proved that fork≥3, ζ(k)(s) has no zero in the region Res≥(1·1358826...)k+2. This result is an improvement upon the hitherto known zero-free region Res≥(7/4)k+2 on the right of the imaginary axis. The known zero-free region on the left of the imaginary axis is also improved by proving that ζk)(s) may have at the most a finite number of non-real zeros on the left of the imaginary axis which are confined to a semicircle of finite radiusrk centred at the origin.
pp 223-234 November 1982
In the theory of precessional motion of the rolling missile,  observed that the aspidal angle is always bounded below and above by π/2 and π respectively, provided the roots of the libration polynomial satisfy certain launching conditions. In this paper, we have obtained the same results without these conditions. We have also proved that the bounds are sharp, so far as the purely retrograde and direct motions are concerned.