206] VISCOUS FLUIDS. 551 



which reduce to 6) when ^ = 0. The coefficient p is called the 

 viscosity of the fluid, and its quotient by the density, v, is called by 

 Maxwell the kinematical coefficient of viscosity. 



The equations 233) are too complicated to be used in all their 

 generality. We shall here consider only the case of incompressible 

 fluids, for which the terms in 6 vanish. If we form the equation 

 of activity as in 188, we obtain beside the terms in the first integral 

 of 29) the additional terms 



~ P I I I 



4- 



which by Green's theorem may be converted into 



dv dw 



u \ 2 , /2\ 2 , / 3s A 3 , /M 2 , 



w + u + y + u + 



If the integration be extended to a region where the liquid is at 

 rest, say the surface of a containing solid, where the liquid does not 

 slip, the surface integrals vanish, and the volume integrals give a 

 positive addition. That is to say, the applied forces have to do an 

 amount of work over and above that going into kinetic and potential 

 energy, and this work is dissipated into heat. If there are no applied 

 forces, the energy of the fluid is dissipated, and it will eventually 

 come to rest. 



In order to find simple solutions of our equations, we may deal 

 either with steady motion, or with motions so slow that we may 

 neglect the terms of the second order in u, v, w and their derivatives. 

 Let us first consider steady motion. The simplest case is uniplanar 

 flow parallel to a single direction, or as we may call it, laminar flow. 

 If we take 



noA\ du dp 



234) = v = w= -75- = -^-9 

 } dz dz 



the equation of continuity gives 



235) * = 0. 



If there are no applied forces, equations 233) reduce to 



OQ \ d*u dp dp A 



236) ^Q-2=:r^ ^ = ^ 



^ cy* ox oy 



Since u depends only on y and p only on x, this equation cannot 

 hold unless each side is constant. Accordingly 



d z u dp 1 a 9 



^7 2==a== ^' M = c + &/-f- -y 2 , p = d + ax, 



