Mr Harrison, The pressure in a viscous liquid etc. 307 



The ^pressure in a viscous liquid moving through a channel unth 

 diverging boundaries. By W. J. Harrison, M.A., Fellow of Clare 

 College, Cambridge. 



[Read 24 November 1919.] 



If non- viscous liquid is flowing along a tube having a cross- 

 section which is increasing in area in the direction of flow, the 

 pressure will also increase, in general, in the same direction. On 

 the basis of this remark an explanation has been given of the 

 secretory action of the kidneys. The author's attention was drawn 

 to this explanation by Dr Ffrangcon Roberts. The physiological 

 aspect of the question and a more detailed numerical consideration 

 will be dealt with by Dr Roberts and the author in a separate 

 paper. 



In the present paper two problems are considered, viz. the flow 

 of liquid in two and three dimensions when the stream lines are 

 straight lines diverging from a point. 



Two-dhnensional 'problem. 



Let the boundaries of the channel be ^ = ± a, where (r, d) are 

 two-dimensional polar coordinates. The motion in which the stream 

 lines are straight lines passing through the origin has been ob- 

 tained by G. B. Jeffery^. With a slight change of notation the 

 results of his solution are as follows. 



Let the velocity at any point be ujr, where m is a function of d 

 only. Then 



2 A ^^** , 



U^ = — 4:VU — V -77i5- + a, 



dO^ 



where v is the kinematic coefiicient of viscosity, and a is a constant 

 of integration. Whence 



u = — 2i^ (l — m^ — m^k^) — Qvkhn^ sn^ {md, k), 



where k and m are constants, which may be determined from the 

 conditions that u must vanish at ^ == ± a, and that the total rate 

 of flux may have a given value. Instead of the latter condition it 

 is simpler to assume that the velocity is given for ^ = 0, i.e. u = Uq 

 for 6 = 0. 



Thus the conditions are 



— 2v {1 — m^ — m^k^) = Uq, 



(1 — ni^ — m^k^) + 3kh)i^ sn^ {ma, k) = 0. 



1 Phil. Hag. (6), vol. xxix, p. 459. 



