pertinent to the cavity walls, assuming that the latter could 

 be considered elastic. 



For a visco-elastic medium, relation (9) should be 

 supplemented by the viscous stresses which depend upon 

 the rate of deformation and rate of dilatation of the medium. 

 The complete stress tensor in this case is 



ZD. k .(a 



t =n D..+X ® 6.^ +n -TJ^+X f^.. (11) 



vk o vh o i,K i M i d£ l« 



which retains the symmetry property mentioned earlier. 

 The Navier-Stokes equation for a viscous fluid as given by 

 (1) employs (11) with \x = and -X ® replaced by the fluid 

 pressure P. 



As required by the second law of thermodynamics, the 

 viscous stresses can never lead to a decrease of the en- 

 tropy in a closed system. Eckart* has shown that this will 

 be assured under all conditions of deformation, if and only 

 if 



u > and JL > -f |i, (12) 



It may be remarked that the traditional stipulation regarding 

 the second viscosity coefficient (namely X 1 = -f [i ) just 

 barely satisfies condition (12). This is important from the 

 standpoint of loss of acoustic energy associated with the 

 irreversible processes within the system. The rate of 

 conversion of the dynamic (acoustic) energy to thermal 

 energy is directly related to the rate of increase of the 

 entropy due to internal processes. Clearly the latter is 

 enhanced if X exceeds the lower limit imposed by (12). 



'""Carl Eckart, unpublished class notes on Principles of 

 Hydrodynamics, Scripps Institution, University of Califor- 

 nia, 1948. 



30 



