where 



Tulin 



(0) ( 1) 



-.2 (0) 



d u. 



T^r 



and 



Bt2 3t Bx. ^J 



^ (0) ^ (0) 

 (0) _ ^i OUj 



+ 



^i 3x . Bx . 



The driven wave equation for the shear wave field u[ '^ may be interpreted 

 to show that propagating strain waves are generated by dipoles in the strain- 

 rate s[y associated with the turbulent field. This wave equation may be solved, 

 and the elastic strain field at any point in the flow calculated as an integral of 

 the turbulent rate -of- strain field. The energy associated with the elastic strain 

 field radiated from a turbulent eddy may thus be estimated. It turns out to be of 

 the form 



radiated energy /c„ 



= f — 



usual turbulent dissipation 



where the function f has a maximum value of about 2 when c^/vg % tt, and 

 where 



3 



usual turbulent dissipation % (Cjj)/3 — ~ , 



in which Cp is the coefficient of turbulent dissipation and usually has a value of 

 about 0.8 (16). 



These rather crude considerations thus lead us to the conclusion that the 

 total dissipation, including radiation associated with a turbulent flow of given 

 intensity in a suitably elastic fluid may be as much as tripled over the amount 

 normally associated with inertial interactions alone. That is, the coefficient of 

 turbulent dissipation c^, which has the approximate value 0.8 in the previous 

 formula, may take on values as large as 2.4 in the case of polymer solutions. 

 In that case, the usual inertial transfer of the total energy production down 

 through the eddy cascade to the dissipation range may become short-circuited, 

 as indicated in Fig. 5. 



SUBLAYER THICKENING 



Let us go further and speculate how the thickness of the viscous sublayer of 

 a turbulent boundary may be affected by an increase in the coefficient of turbu- 

 lent dissipation c^. We may define (Fig. 6) the edge of the viscous sublayer as 

 the position y* where the total shear stress is equally compounded of laminar 

 and Reynolds stresses: 



14 



