RECENT rnOGRES^ IN HYDRODYNAMICS. 63 



fluids througli tubes and along canals. Stokes,' in Lis fii'sfc paper on vis- 

 cosity, worked out the case of watei' flowing down an inclined circular 

 cylinder under the action of gravity, as an example of the methods 

 developed in the paper. In a cylinder of radius a, inclined at an angle a, 

 the velocity at a distance r from the axis is h(a'^ — r'-) + U, U being 

 the velocity at the surface, and lc = ^g sin a jfj. In 1860 Helmholtz - 

 considered the analogous question where the motion is caused by a dif- 

 ference of pressure at the two ends, allowing also for a certain amount of 

 slipping at the surface of the tube. Here the velocity is given by 

 A;(a* -1- 2Xa — r^), where h = (diff. of pressures) /4/iZ, which gives a 

 flow of ^Trkp(a'*' + Xa^), agreeing well with experiments. The same 

 question has also been treated by Stefan, Boussinesq, Butcher, Graetz, and 

 Greenhill. Stefan ^ takes into consideration a motion of rotation of the 

 vessel as well. Boussinesq's * investigations are more general than the 

 others, and extend to tubes of non-circular sections. Considering the 



equation — - + -4r + 4Z.- = 0, which gives the velocity parallel to the axis, 



he shows, from the principle of similitude, that in tubes of similar sections 

 the velocities at corresponding points are proportional to h and the areas 

 of sections, and that the flows in the same times are proportional to the 

 fourth powers of the constant of similarity — the forces acting being the 

 same. He then solves for the j^articular cases where the sections of the 

 tubes are — (1) elliptic, (2) rectangular, and (3), in a note at the end of 

 the paper, where the section is an equilateral triangle. If A be the area 

 of the section, the flow for the elliptic tube is kAa-b"^ l(a^ +■ h'^) for the 

 triangular (sides = 2a) is IcLa^jh, whilst for the rectangular the expres- 

 sion is naturally more complicated. After noticing the case where in 

 a straight tube the section gradually changes, he passes on to consider 

 the motion where the axis of the tube is circular. This is interesting, as 

 a steady motion in circles is impossible if the boundary is at rest. Treat- 

 ing the velocities in any section of the tube as small compared with the 

 velocity across it, he solves the equation when the section is a rectangle 

 whose height is small compared with its breadth. The motion consists 

 of two circulations combined with a translatory motion along the tube of 

 greater magnitude. If the rectangular section be divided by a medial 

 line in the plane of the tube, the circulations may be represented by 

 supposing the particles of fluid near it to move outwards, increasing 

 their distance from it, and at last, on nearing the outer boundary, revers- 

 ing their direction, and coming back nearer the longer sides. A similar 

 result was also subsequently (1875) arrived at by Oberbeck. Boussinesq^ 



' 'On the theories of the internal friction of fluids in motion, &C.' Camh. Phil. 

 Trans, viii. p. 287. 



■■* ' Ueber die Rcibung tropfbarer Fliissigkeiten,' Sitzhcr. d. li. Ahad. Wins. 

 Wien. xl. p. 652, and Collected Warhs, Bd. I. p. 215. 



' ' Ueber die Bewegung fliissiger Korper,' Sitzber. d. h. Ahul. Wicn. xlvi. p. 495. 

 The dimensions of the coefficient of viscosity are wrongly given. He seems to regard 

 the vortex rotations as if the small elements of fluid turned round as rigid bodies. 



* ' Memoire sur I'influence des frottements dans les mouvements reguliers des 

 fluides,' Ziour. (2) xiii. p. 377 (1868). 



* ' Essai sur la theorie des eaux courantes,' Acad, des Soienees. Paris. Mem. 

 par divers Savants, xxiii. xxiv. 1877. This is a long memoir, consisting of 680 quarto 

 pages of printing. The latter part is devoted to wave motion, and this contains 

 some results of value — especially the theory of the solitary wave ; but this Lad been 

 published before in Liouville. (See first part of this report.) 



