Mixtures of Liquids and of Solutions. 



129 



The fundamental equation of the theory of viscosity o£ 



a fluid is F = ?/ . -y-, where dv is the difference of the parallel 



velocities of two plane layers of fluid situated at a distance 

 dx apart, F is the tangential force transmitted through an 

 area of 1 sq. cm. between the layers and parallel to them, 

 and 7) is the viscosity of the fluid. 



If the motion throughout the fluid is steady and parallel 

 to the y axis, and there is no variation of it along that axis, 

 the equation 



? 2+ d* 2 



:0 



(1) 



must hold throughout the fluid. At a surface of separation 

 of two fluids of viscosities rj l and rj 2 , if there is neither 

 chemical action nor slipping, we must have 



«a=r 2 (2) 



and "dvi _ ~dv 2 



Vl Tn- V2 ^i (3) 



where V\ and v 2 are the velocities close to the surface in the 

 two fluids, and dn is an element of the normal to the surface. 

 At a solid surface touched by a fluid there must be no relative 

 motion of solid and fluid. 



These are the necessary and sufficient conditions for the 

 solution of any "plane" problem on the motion of viscous 

 fluids, and the difficulty of finding a solution in any special 

 case is a mathematical one which increases with the number of 

 different fluids present in the space considered, and the com- 

 plexity of their lines of separation. We proceed to consider 

 a few simple stable arrangements of the fluids for which the 

 solutions can be readily obtained. 



I. Taking the velocity throughout parallel to they axis, let 

 us assume that it is independent of z, equal to v throughout 

 the plane x = x , and to v, throughout the plane x—x u and that 

 the surfaces of separation of the various 

 fluids present between those planes, 

 are the planes x = x 2l <r 3 , &c. 



If there are only two fluids of vis- 

 cosities rj l and 7? 2 present in equal 

 volumes (tig. 1), the equations (1), (2), 

 and (3) above, give for the viscosity -n 

 of the mixture, 



1 = VL L\ 



If each c. c. of the mixture contains 

 Phil. Mag. S. 6. Vol. 1. No. 1. Jan. 1901. 



