ABSTRACT
TJR = I + kE<I> (12.1) The Einstein coefficienL. k£. is 2.5 for rigid spheres if there is no slippage of the liquid at Lhe surface of the sphere. lf there is perfect slippage of liquid at the interface. kE becomes unity [9]. ln emulsions there is circulation of fluid within the spheres in addition to displacement of the streamlines of the drop. The circulation wilhin the drops allows relative motion to take place in the neighborhood of the interface in a manner similar to what takes place during slippage at the interface. Thus. intuitively. one would expect the Einstein coefficient to be about 2.5 when the viscosity of the fluid in the drops is much greater than the viscosity of the continuous liquid, and k E should be 1.0 when Lhe viscosity of the continuous liquid is much greater than Lhat of the drops. Taylor derived Lhe following equation for Lhe Einstein coefficient of emulsions at low shear rates [ I 0]:
(12.2)
in which TJ 2 is the viscosity of the dispersed liquid and TJ 1 is the viscosity of Lhe matrix. This equation is plotted in Fig. 12.2: it has been experimentally verified by Nawab and Mason [II]. Equation ( 12.2) reduces to the Einstein equation as Lhe dispersed-phase viscosity becomes very large: at Lhe other extreme of negligible droplet phase viscosity. such as might occur with gas bubbles. it still predicts an increase in viscosity above the value of the continuous phase. It should be pointed out Lhat Lhe drops rotate in a simple shear field at the same time that Lhere is circulation within Lhe drop itself [4]: Lhe angular rotation rate is one-half Lhe shear rate.
