Stress Growth at Inception of Ramp-Function Velocity Gradient

Substitution of these results into the expression for the shear stress in Eq. (5.2) gives  

The expression for the shear stress contains terms which are similar to those in the expression for stress growth at the inception of steady shearing ( 11) in which an overshoot occurs. Such an overshoot phenomenon has been observed experimentally by Marsh (54) and by Warner (74). 

It should be noted that as t becomes large the lowest order term in the coefficient of the K-term is just 60, that is one half the zero-shear-rate value of the primary normal stress function. A similar result was obtained by Bird and Marsh (7) and by Carreau (14) from the slowly varying flow expansions of two continuum models. Hence the time-dependent behavior of the shear stress is related to the steady-state primary normal stress difference in the limit of vanishingly small shear rate. 

It is interesting to compare the result in Eq. (14.5) with that for the superposed flow problem discussed in 8. If in 8 we replace Km by k0, and require that k° = — ik/co, then in that section 

That is for any time t, no matter how large, there must exist an a for which sincot can be suitably approximated by cot. This equivalence enables us to interrelate Sections 8 and 14. From Eqs. (8.2), (8.3), (8.4), and (8.18) we get  

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Stress Growth at Inception of Ramp-Function Velocity Gradient... 

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