The lift and moment on a flat plate in a stream 
of finite width 
By T. H. Havetock, F.R.S. 
(Received 8 February 1938) 
1. The problem of the lift on a flat plate in a stream between parallel rigid 
walls has been solved in an exact form, by using a suitable conformal trans- 
formation, by Tomotika (1934), who also gives an expansion for the lift 
in the particular case when the mid-point of the plate is midway between the 
walls; a similar solution for the moment on the plate does not seem to have 
been given. The method used in the following paper is quite different and 
is, perhaps, of sufficient interest to justify further examination of the 
problem. The flat plate is treated as the limiting case of an elliptic cylinder, 
and the method of solution leads directly to expansions for the lift and for the 
moment suitable for any position of the plate subject to the parameters 
being within the range necessary for convergence. Moreover, by a simple 
modification, expansions for lift and moment are obtained when the stream 
is bounded by parallel free surfaces, taking the boundary condition in an 
approximate form; and a further modification gives the corresponding 
results when one surface is rigid and the other free. A brief examination is 
also made of the moment for an elliptic cylinder. 
GENERAL EXPRESSIONS 
2. Consider the two-dimensional motion due to a cylinder placed in a 
uniform stream bounded by plane parallel walls, including circulation 
round the cylinder. Let C be the contour of the cross-section of the cylinder, 
and take the origin O so that the parallel walls are given by y =a, and 
y = —b, respectively. To simplify the argument, we assume that a position 
can be found for O such that a circle can be drawn, with centre O, entirely 
in the liquid and enclosing the contour C. 
With w for the complex potential function, we take 
dw amtiniA, dw, 
—=C a hs 
dz o entl dz 
(1) 
Tn (1), ¢ is the velocity of the stream, in the negative direction of Oz, 
the series is a suitable expansion for the singularities of the potential 
[else 
439 
