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IV. Notes on the Motion of Viscous Liquids in Channels. 

 By J. Proudman *. 



1. TN a recent communication to this Journal f Messrs. 



X Deeley and Parr remark that the conditions of the 

 steady flow o£ a viscous liquid in a parabolic channel, under 

 a constant force parallel to the length of the channel, have 

 not yet been ascertained. It is implied that the results 

 might be of interest in connexion with the motion of 

 glaciers. 



In the present communication the problem is solved for 

 the special case in which the free surface of the liquid passes 

 through the focus of the parabolic section, and also for a 

 particular triangular section. Some remarks are also added 

 in connexion with the mathematical expansions used. 



The general problem % for a channel of any section may 

 be reduced to that of finding a function % which satisfies 



2u" Br "' ( 



over the section, which vanishes over the sides of the section, 

 and for which ^^/^w = over the free surface. Here #, y 

 are rectangular Cartesian coordinates in the plane of the 

 section, and "dfon denotes differentiation along the normal 

 to the free surface. 



The velocity of the liquid, which is parallel to the length 

 of the channel, is given by F^/2/jl, where P is the pressure 

 gradient along the channel, and fi is the coefficient of 

 viscosity. In applications, the function 



where the integral is taken over the area of the section, is 

 required. 



Particular Parabolic Section. 



2. For convenience, take the length of the latus-rectum of 

 the parabola to be 4-7T 2 . Then if we take polar coordinates 

 r, } having for pole the focus S, and for initial line the axis 



* Communicated bv the Author. 



t "The Hintereis Glacier," Phil. Mag. (6) xxvii. p. 153 (1914). 



X See Lamb, Hydrodynamics, 3rd ed., p. 545. 



