Seyer and Metzner 



steady flows, the experiments recorded by Johnson (31) using high-speed pho- 

 tography are of interest. If a flat object is dropped, slowly, into a puddle of a 

 dilute polymeric solution it simply moves aside like a conventional fluid would. 

 If, on the other hand, the impact is intense and rapid the sheet of fluid created 

 does not "splash" but instead stretches and recoils just as a suddenly stretched 

 elastic solid would. On the time scale involved (a few milliseconds) only the 

 elastic response is felt. Quantitative analyses of this effect have been published 

 (39,40,60) in which it is shown that Eq. (1) as well as the correct asymptotic 

 forms from simple fluid theory (though not Eq. (2)) predict such purely elastic 

 responses for the short time scales involved. 



These considerations suggest that the kind of material response observed, 

 and the correct choice of asymptotic approximations such as Eq. (2) which may 

 be used to describe the material properties, are dependent on an additional di- 

 mensionless parameter in which the time scales of the material and the defor- 

 mational process are considered. This dimensionless parameter, the Deborah 

 number, may be defined (40) by 



"Deb 



characteristic time of fluid /.> 



(4) 



characteristic time of process 



For flows that are steady in a Lagrangian sense, for example steady lami- 

 nar shearing flow in a tube, the Deborah number is identically zero. It is in 

 this limit (and only in this limit) that Eq. (2) may be used to portray the mate- 

 rial properties of simple fluids. In this asymptotic, limiting flow process it 

 may be shown that for ducted flows through round tubes neither the velocity 

 profile nor the viscous drag are influenced by the presence or absence of elastic 

 forces. That is, the fluid is behaving as a purely viscous material even though 

 one may have present large elastic forces as described by the Weissenberg 

 number in Eq. (3). (Experimental results discussed in the next section show the 

 elastic forces may be 10 to 40 times larger than viscous forces at the shear 

 rate levels of interest.) 



On the other extreme if we consider fluid elements subject to deformations 

 of a duration smaller than the relaxation time of the fluid, as in the case of the 

 Johnson experiments in which a puddle of fluid is suddenly deformed, the Debo- 

 rah number is large and only an elastic response is observed. Intermediate 

 cases, and typical magnitudes of the Deborah number in flows of pragmatic in- 

 terest, have been described elsewhere (40,48,51). 



The intuitive notions embodied in Eq. (4) have been formalized mathemati- 

 cally by Astarita (2) as 



N -,l/H^ (5) 



Equation (5) represents a properly invariant definition of the Deborah number 

 and gives a process time scale that is a measure of how long an element of fluid 

 is subjected to a particular state of deformation. 



24 



