Tulin 



industry and biology (blood, for example). The role of the dispersed phase de- 

 pends upon the particulate size k and upon the time scales t within the flow. 

 Thermal, viscous, and inertial forces act upon microscopic particles and depend 

 linearly on particle displacement and its time derivatives: 



thermal forces viscous forces inertial forces 



where x. is the displacement in the ensemble in reference to the state of maxi- 

 mum entropy (x. = dx./dt, etc.), fi is the liquid viscosity, p' is the mass den- 

 sity of a particle relative to the volume x^, k is Botzmann's constant, and T is 

 the absolute temperature. 



In accordance, the relative magnitudes of these forces depend upon k and t 

 in the following way: 



viscous A.V 



thermal kT t 

 inertial p'k^ 



thermal kx t^ 



inertial p'k^ 

 viscous /it 



Upon estimating the constants of proportionality implicit in relations such 

 as those above, the regimes of importance for the three forces acting on a dis- 

 persed phase may be depicted in a \- t diagram as illustrated in Fig. 1. 



The inertia of sufficiently small particles (somewhat below the line /V ~ t^^^) 

 may be neglected in comparison to viscous and thermal forces; all large mole- 

 cules fall in this category. As the figure attempts to show, the ability of these 

 inertialess particles to store thermal energy in strain increases as the time 

 scale decreases, i.e., as the frequency of impressed motion increases; for suf- 

 ficiently long time scales the molecular motions result mainly in viscous dissi- 

 pation. In this connection, time scales must be compared with the so-called 

 first relaxation time of the molecule (which is 10"^ sec for the large molecule 

 indicated by the circle on the line \ - t'-"^). 



THERMAL FORCES, ENTROPY AND STRAIN ENERGY 



What are these thermal forces ? They arise through the bombardment of 

 the dispersed particles by the agitated molecules within the solvent , i.e., 

 through Brownian motion. In the case of very large molecules or macromoie- 

 cules, the forces acting on them can be related to the entropy loss or to the 

 ordering of the molecular configuration in relation to its most disordered state, 

 which is ideally that of a spherical cloud (see Ref. 4, pages 399-494, for 



