THE LAWS OF ELASTICO-VISCOUS FLOW 19 



For small stresses A^ = A2 = i, and if n^ equal n^ this expression 

 takes a form resembling that given by Jeffreys. 



It is important to note, however, that this formula is based on 



the assumption that the viscosity is "external," that is, it acts as 



ds 

 though the viscous resistance were due to an absolute velocity -y. 



at 



But this is by no means evident; and indeed the probabihty is 

 that a considerable part if not the major part of the viscous resist- 

 ance may be "internal," that is, due to the relative motion of parts. 

 Thus if an element consists of two parts y and 2, y being coupled 

 to the next adjacent elemenf by an elastic coupling «i and with 2 

 by an elastic coupling W2, together with a viscous coupling e^, while e[ 

 and e'2 represent the "external" viscosities, the equations of motion 

 will be 



Pi'z =e2{z—y)-{-n2{z—y)+e[z 



P2j=eXz-)i'+«2(z-y)+e^>'+«i^ 



If p2e[ and e[ be considered negligible, the solution, for not too 

 rapid extinction, is 



in which 



i8 = - 



z = ae~^'^ cos p(t—vx), 



p n2in2-pp')-{-p'e^ 

 p^^2p^e 



2 1/ Wi[w2 {n2—pp^)+ p^e^] 

 If pe is large compared with Hz 



2ev 



so that in this case the higher the viscosity the less rapid the decay 

 of the oscillations — quite the reverse of the conclusions on the former 



