M. A. LAUFFER 



with the position with respect to the sphere. Thus, the velocity 

 gradient cannot be specified. In addition, there is a tendency for the 

 flow around the sphere to be somewhat turbulent — an effect which 

 cannot be eliminated entirely. 



The most commonly used method of determining viscosity is 

 that involving the flow of the liquid through a capillary tube. 

 Poiseuille (20) demonstrated empirically and Hagenbach (11) derived 

 theoretically from Newton's postulate that, when the rate of flow is 

 slow enough to exclude turbulence, the volume of liquid, V, flowing 

 through a tube in time, t, is directly proportional to the pressure 

 difference, P, across the capillary and to the fourth power of the capil- 

 lary radius, r, and inversely proportional to the length, /, of the capillary 

 and to the viscosity coefficient, rj: 



V TrPr* irrHP 



or 77 = 



t 8r?/ SIV 



If the dimensions of the capillary are known, absolute viscosity can be 

 measured; but, even if they are not known, relative viscosity can be 

 measured, for (VW = (i/h)(V(,/V){P/Po). The most usual procedure 

 is to observe the time for a definite volume of liquid to flow at a con- 

 stant pressure diff"erence, in which case v/vo = t/k, or to observe the 

 time for a definite volume to flow under the influence of gravity with 

 a constant average difference in liquid level. In this case, 77/^0 = 

 (t/to){d/do), where d and do are the densities of the solution and solvent, 

 respectively. These conditions are the ones encountered when using 

 the common Ostwald viscometer. Relative viscosities can be measured 

 with high precision by this technique. The capillary method has the 

 very considerable advantage of requiring easily constructed apparatus. 

 It has one important limitation, however. The rate of flow of a New- 

 tonian liquid at various distances from the wall of the capillary increases 

 parabolically from the wall to the center. The velocity gradient is 

 not constant throughout the cross section of the tube, but decreases 

 linearly from the wall to the center. With non-Newtonian liquids, 

 it is difficult to predict the exact nature of the variation of velocity 

 gradient with distance from the wall, but certainly the gradient varies 

 considerably. For that reason, this type of apparatus is not ideally 

 suited to the study of non-Newtonian systems. 



The method most satisfactory from the theoretical point of 



246 



