~ 
— = 
= 
’ 
SECTIONAL TRANSACTIONS.—A*. 347 
Prof. R. V. SournweE tt, F.R.S., and Mr. H. B. Squiru.—A Modification of 
Oseen’s Approximation to the Equations of Motion for a Viscous In- 
compressible Flucd. 
The problem discussed is the motion in two dimensions, past a fixed cylindrical 
body, of a fluid which has negligible compressibility but finite viscosity. Its import- 
ance for Aerodynamics is evident, since a solution would render possible a theoretical 
prediction of ‘ profile drag.’ 
The exact hydrodynamic equations have so far proved intractable,—mainly on 
account of the terms representing convection of vorticity, which render them non- 
linear. Accordingly, Oseen has proposed, as an approximation, to replace the actual 
velocities w and v, in these convectional terms, by U and V, the undisturbed velocities 
of the fluid. The resulting equations are linear, and have been solved in relation to 
a circular profile. Oseen’s method of approximation is not restricted to problems 
in two dimensions. 
This paper suggests an alternative method of approximation, leading to an 
equation (linear in form) which would seem to be a closer approximation than Oscen’s. 
Tt can be solved exactly in relation to a point source of vorticity, and the authors are 
endeavouring, on the basis of that solution, to construct approximate solutions for 
specified profiles. Their procedure, which involves graphical and numerical methods, 
is being applied in the first instance to a circular profile, in order that the results may 
be compared with those derived from Oseen’s equation. 
Results are not yet available. The aim of the paper is to bring to the notice of 
mathematicians a modification of the governing equation for motion in two 
dimensions, which would seem to be a close approximation, and which may prove 
to be tractable by purely analytical methods. 
Dr. 8. Gotpstein.—The Stability of Viscous Fluid Flow between Rotating 
Cylinders. 
Tt is assumed that an incompressible viscous fluid is flowing, under the influence 
of a pressure gradient parallel to the axis, along a narrow annular space between two 
infinitely long coaxial rotating cylinders. The stability of the system is investigated 
mathematically by the method of small oscillations, with the assumptions that the 
disturbance is symmetrical about the axis and periodic alongit. There is no steady 
disturbance possible, and for any given Reynolds number of the flow and ratio of 
the velocities of rotation of the inner and outer cylinders it is possible to calculate 
the lowest angular velocity for which instability occurs, the time-period of the 
critical flow, and its wave-length along the axis. Numerical results have so far 
been worked out for small Reynolds numbers of the flow with the outer cylinder 
stationary. 
Prof. A, Rosensiatr.—The Stability of Laminary Movements. 
The problem of the stability of laminary and other movements of incompressible 
viscous fluids has been treated in this country by Reynolds, Kelvin, Rayleigh, and 
others, but not exactly, that is, for finite disturbances by consideration of non-linear 
equations. For laminary motion between two parallel walls certain disturbances 
which vanish exponentially at infinity and exponentially also with the time, will 
be described. 
Tf ¥, (x, y) is the stream function of the undisturbed motion, the stream function 
W (a, y, t) of the disturbed motion is of the form 
co 
‘ Wo (2, y) +> W(x, y, t). 
es 
In the case of exponentially vanishing perturbations between the walls y= +H 
we can write 
Vi: (a, yt) = e-kAxtut) fy (y), k=1,2,... « 
It can be shown that suitable real numbers 1, wu exist, and that within every 
_ interval of values of there is an interval for which the convergence of ¥ can 
be established by the method of dominant functions. In general, there is a 
