LX. Expansion in series of the exact solution for compressible flow past a circular or an elliptic cylinder

Abstract

Print) 1941-5990 (Online) Journal homepage: http://www.tandfonline.com/loi/tphm18 LX. Expansion in series of the exact solution for compressible flow past a circular or an elliptic cylinder A.R. Manwell To cite this article: A.R. Manwell (1945) LX. Expansion in series of the exact solution for compressible flow past a circular or an elliptic cylinder, The London, Edinburgh, View related articles On the Exact Solution for Compressible Flow past a Cylinder.499 elastic and viscous constants, since the nodes become ill-defined and are considerably displaced from the positions they would occupy in purely elastic systems. The shortest road to their elucidation in such a case is probably to plot a set of curves such as those of figs. 1 or 2 for a number of values of z., expressed as multiples of fi, and to compare the theoretical and practical distributions of amplitude, noting in particular the values at the best approximations to resonance, i. e. segmental delineation, that one can attain.The deduced values of N and ~ are shown beneath the figures. It may be objected that the physical characteristics of sols and gels should not be separated into rigidity and viscosity in the manner adopted in these experiments, that, in fact, " jelliness " is an indivisible property. Maxwell, however, regarded such systems as characterized in this way, the ratio of viscosity to elasticity being a dependent variable v characteriz ing the rate at which the system recovers equilibrium when displaced. When the liquid is subjected to simple harmonic forcing, the effective viscosity ~ should decrease from the " rest-value " V0 according to the formula 7=V o 1_t_p%2, so that the viscosity will effectively experience a rapid drop as p approaches 1/~. Evidence for these ideas cannot be derived from the present experiments, since in the systems used 1/r is a quantity probably much larger than the maximum frequency attainable. Summary.By subjecting a liquid having elasticity and viscosity to shearing forces of simple harmonic nature, it is shown how values of the two coefficients (of elasticity and viscosity) may be deduced by plotting the distribution of amplitude of oscillation throughout the system. The amplitude and the relative phase of motion are measured by means of hot-wire anemometers. Results are presented for oil, and for sols of cellulose acetate and gelatine of small concentration.

A new singularity of transonic plane flows

“…An example of this process is furnished by [13] which although, as appears on closer examination, merely an approximate solution of the problem implied by the title is a useful contribution to the problem of the behaviour of transonic flow when the boundary is kept fixed while the stream conditions are varied. (The solution of [13] satisfies correct boundary conditions on the arc BD and on the axis AB, (Fig. 3) and the flow becomes…”

“…According to [13] the image of the boundary is determined in the hodograph plane by an equation of the form cos 6 = X(g)/X(gi), (5.1) where qi is the maximum velocity on the boundary. If qi < q2 , where \'{q2) = 0, this equation may be expanded as q -q, s -i[x'(gi)rx(g,)02.…”

“…For the semi-circular body which has in fact been treated in [13] it was found that X = q/c -(5/24)(g/c)3 -[125/3456 + 35(7 -l)/576](9/c)5, (5.4) where the notation is that of Courant and Friedrichs [9]. To this approximation it appears that A'(g) vanishes for q2 = 1.03 c and that the branching streamlines in the hodograph plane make an angle arc tan (qdd/dq) = 1.86 with the line 0 = 0.…”