64 REPORT — 1882. 



lias publislied fartlier investigations on tlio same subject, but more from 

 the point of view of tlie liydraulic engineer. 



In 1876 Bntcher, in Lis paper before referred to,^ also touches on 

 the question of the motion in straiglit tubes, but without adducing new 

 results. Besides this he finds the general form for Stokes's stream func- 

 tion for a motion taking jjlace in planes through an axis. Graetz ^ has 

 worked out precisely the same questions as Boussinesq, to whose work 

 he does not refer. He has taken the trouble to calculate numerical 

 results for a tube with a square section, and has also followed out St. 

 Venant's idea by taking algebraical solutions of the equations, and finding 

 for what shaped tubes tbey are the solutions. 



Greenhill ^ has made a valuable remark by which the motion of a 

 viscous fluid in a straight tube of any section, without gliding at the 

 surface, may be deduced at once from the solution for the case of the 

 motion of a perfect fluid in a cylinder of the same section rotating about 

 its axis. This is seen at once when it is noticed that the difierential 

 equation for the velocity parallel to the axis in the first case has precisely 

 the same form as that for the stream function relative to the boundary, 

 rotating with an angular velocity 21c ; and that the bounding condition for 

 the two functions is the same, if in the first case the fluid is supposed to 

 stick fast to the boundary. In this way the velocities in tubes whose 

 sections are a circle, an ellipse, an equilateral triangle, two hyperbolas, 

 a sector of a circle, and a rectangle, are written down at once. 



Cylinders. — If two co-axial cylinders rotate with difiereut velocities, 

 wi, m, the velocity of the fluid between at a distance r from the common 

 axis, when steady, has been given by Stokes,'* with a single cylinder in 

 an infinite fluid as a particular case. When the motion takes place 

 between two co-axial cylinders at rest — being produced and kept up by 

 pressures across two plane sections through the axis — the expression for 

 the velocity is not algebraical as in the previous case. The solution for 

 this is due to Boussinesq, and is given in his paper in Liouville just 

 referred to. Rohrs ^ treats a similar question, taking account of non- 

 permanent motion. The determination of the motion of the fluid when a 

 cylinder oscillates in a direction perpendicular to its axis, forms one of 

 the chief problems considered by Stokes ^ in the second of his classical 

 papers on viscosity. The method employed is precisely similar to that 

 adopted for the corresponding problein for the sphere (noticed below) ; 

 but, unfortunately, the solution of the differential equations occurring 

 cannot be represented in finite forms, as in that case. The functions 

 entei'ing are cyliudric harmonics, and this introduces a diSiculty in ap- 

 plying the condition of finite motion at an infinite distance, to determine 

 the arbitrary constants appearing in the solution, but that is surmounted. 



• ' On Viscous Fluids,' Proe. Lond. Math. Soc. viii. p. 120. (See first part of 

 report, p. 79.) His analysis is wrono; where he considers the determination of the 

 arbitrary constants in the solution from the bounding conditions. 



- ' Ueber die Bewegung von Fliissigkeiten in Rohren,' Zeits. f. Math. u. Phys. 

 XXV. pp. 316, 375 (1880). 



^ ' On the flow of viscous fluid in a pipe or channel,' Proc. Lond. Math. Soc, xiii. 

 p. 43 (1881). 



* ' On the friction of fluids, &;c.' (1845). 



* « Spherical and cylindric motion in viscous fluids,' Proc. Lond. Math, Soc. v, 

 p. 133 (1874), 



« ' On the effect of the internal friction of fluids on the motion of pendulums,' 

 Camh. Phil. Trans, ix. part ii. p. 35. 



