ON STREAM-LINE MOTION OF A VISCOUS FLUID. 143 



(II.) Mathematical Proof of the Identity of the Stream Lines obtained by 

 Means of a Viscous Film with those of a Perfect Fluid moving in Tivo 

 Dimensions. By Sir G. G. Stokes, F.E.S. 



The beautiful photographs obtained by Professor Hele-Shaw of the 

 stream lines in a liquid flowing between two close parallel walls are of 

 very great interest, because they afford a complete graphical solution, 

 experimentally obtained, of a problem which, from its complexity, baffles 

 the mathematician, except in a few simple cases. 



In the experimental arrangement liquid is forced between close 

 parallel plane walls past an obstacle of any form, and the conditions 

 chosen are such that whether from closeness of the walls, or slowness of 

 the motion, or high viscosity of the liquid, or from a combination of these 

 circumstances, the flow is regular, and the effects of inertia disappear, the 

 viscosity dominating everything. I propose to show that under these 

 conditions the stream lines are identical with the theoretical stream lines 

 belonging to the steady motion of a perfect (i.e., absolutely inviscid) liquid 

 flowing past an infinitely long rod, a section of which is represented by 

 the obstacle between the parallel walls which confine the viscous liquid. 



Take first the case of the steady flow of a viscous liquid between close 

 parallel walls. Refer the fluid to rectangular axes, the origin being taken 

 midway between the confining planes, and the axis of z being perpen- 

 dicular to the walls. As the effects of inertia are altogether dominated 

 by the viscosity, the terms in the equations of motion which involve 

 products of the components of the velocity and their differential coefiicients 

 may be neglected. Gravity, again, need not be introduced, as it is balanced 

 by the variation of hydrostatic pressure due to it. The equations of 

 motion, then, with the usual notation, are simply 



dp /d^u dht, d^it\ 



dx~ ^ \M- W^ di^j' 



with similar equations for y, v and z, w, /x being the coefficient of vis- 

 cosity. 



In the present case the flow takes place in a direction parallel to the 



walls, so that w^O, and the third equation of motion gives -^~ = 0, so 



dz 

 that p is constant along any line perpendicular to the walls. The velo- 

 cities u, V, vanish at the walls, and along any line perpendicular to the 

 walls are greatest in the middle. As by hypothesis the distance (2c) 

 between the walls is insignificant compared with the lateral dimensions of 

 the obstacle, the rates of variation of it and v when x and y vary may be 

 neglected compared with their variation consequent on that of z. Hence 

 the equations of motion become simply 



dp d'^u dp d^v .... 



di~^ dz^' ~dy~'^'M' ^^> 



which must be combined with 



Over an area in the plane xy, which is small compared with the 

 obstacle, though large compared with c^ the whole velocity and each 



