550 XL HYDRODYNAMICS. 



depend on the time -rates of change, that is on the velocities of the 

 shears, stretches, and dilatations. The simplest assumption that we 

 can make is that the stress -components are linear functions of the 

 strain -velocities. The fluid being isotropic, considerations regarding 

 invariance bring us to precisely similar conclusions to those we 

 reached in 175, so that to the stresses of equations 229) for a 

 perfect fluid are added stresses given by equations 142), 175, A and p 

 being constants for the fluid, and u, v, tv, 6 now denoting velocities, 

 instead of displacements, returning to the notation of this chapter. 

 (We put P = 0, since these additional terms vanish with the velocities.) 

 We thus obtain 



Z 2 = -p + ^ + 2ti~ z 

 230) 



which are of the same form, with a different meaning, as 142), 175. 

 If the fluid is incompressible we find, putting 6 = 0, 



X x + Y y + Z,= - 3p, 



and assuming that this holds also for compressible fluids we must have 

 231) 3^ + 2^ = 0. 



a 



Replacing A by its value - jt, we find for the forces, as in 175, 144), 

 v dp . 1 



232) 



r, 3p . 1 



e Z -U+ 3 



which are to be introduced into the equations of hydrodynamics 6). 

 Thus we obtain the general equations, putting = v, 



du , du , du , cu v ds ^ 1 dp 



^rr-h^K -- h^o -- h^o -- -^ o -- V^U = X -- K^-J 



dt dx ' dy ' dz 3 0x Q dx 



rtoo\ ^^ dv dv t dv v da T;r 1 dp 



233) - H-MO h^^-f^o- - vdv = Y -- 7p-> 



' ot dx dy dz 3 dy Q dy 



dw . dw . dw . dw v ds ^,1 dp 



-r-\-U~ -- h^o -- h^o -- ^ - vAw = Z -- r~? 

 dt ' dx ' dy ] dz 3 dz Q dz 



