CAPILLARY VISCOMETRY 349 left hand side the quantity 4•/•rR a which equation 11 equates to '• for NewtonJan flow. For non-Newtonian flow this quantity is no longer q•, but will be shown below to have useful properties. To avoid con- fusion, we shall designate it by the symbol D. Transformation of the argument of equation 9 from r to s via equation 8 and formal substitution for dr/dr byf(s) yields as a counterpart to equation 9 D = (4/swa)foS'•sy(s)ds (13) The form off(s) need not be specified here. All that is required of the function is that some such function exist for the fluid concerned, and that : it be mathematically well behaved. If D is differentiated with respect to s•, the differentiation can be formally carried under the integral sign so that the integration and the differentiation cancel each other, yielding the explicit result di/ds•: = --3i/sw q- (4/sw)f(s,o) (14) The functionf(s•), which is equal to qw, can be extracted from this result to give, after convenient rearrangement, d log D• (lS) 5•0 = (D/4) (3 + dlog s•,] The derivative d log Did log s• can be obtained graphically from the experimental data if it is measured over a range of these quantities, and from equation 15 the value of q• corresponding to a given value of s• can be calculated. For NewtonJan flow d log Did log s• is unity and the above required identity between q• and D for this case is confirmed. The foregoing is the complete simple rheological theory of capillary viscometry. The rest of this paper will be devoted to the experimental application of this theory, complications of it, and comparison of the results with those from other types of viscometry. A by-product of the simple theory is the usefulness of the quantity D in analyzing non-Newtonian flow, as mentioned incidentally above. In equation 13 s is only a dummy variable. Therefore D depends only on s•. Especially important is the fact that the relation between these two qualities is entirely independent of the dimensions of the capillary used to determine it. This means that for any non-Newtonian material for which a functional relation exists between q and s, there also exists a correspond- ing, but different, functional relation between D and s•. This latter relation appears to have only mathematical rather than physical signifi- cance. Nevertheless, measurements made at the same value of s• in capillary viscometers having different geometries should yield the same values of D, and results obtained from viscometers of different geometries in dif-

350 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS ferent regions of the variables can be combined to form an extended curve characteristic of the flow of the experimental material. Since D is obviously much easier to calculate than is % the use of D is a great convenience and is perfectly satisfactory when one desires only a mathemat- ical characterization of the flow. 4. COMPLICATIONS The simple rheological theory of capillary viscometry just presented is based on several physical assumptions, requires the elimination of several extraneous factors from the observed data, and is subject to distortion by several foreign experimental effects. The following of these complications will be discussed here. 1. Uniqueness of Relation Between Shearing Stress and Rate of Shear. 2. Laminar Versus Turbulent Flow. 3. Slippage at Capillary Wall. 4. Kinetic Energy Contribution. 5. End Contributions. 6. Surface Tension Contribution. 7. Drainage from Reservoir Walls. 8. Dissipation of Heat. ¾.! Uniqueness of Relation Between Shearing Stress and Rate of Shear The principal goal of all types of viscometry is to determine the relation between shearing stress and rate of shear for a material. Obviously this cannot be done unless a unique functional relation exists between these two quantities for the material concerned under the experimental conditions prevailing. Numerous cases have occurred where the results of arduous investigations have been rendered meaningless by the fact that no such relation existed for the material under investigation. The criterion to be applied is whether at the same temperature and pressure a material always exhibits the same rate of shear at a given shear- ing stress. Examples for which this is not so are materials exhibiting, e.g., thixotropy, work-hardening or any type of permanent degradation. In these cases a unique relation involving shearing stress and rate of shear can be obtained only if the variable of time is also included in the relation. This is only one type of situation in which no unique relation exists between the two variables concerned. In general the complete mechanical behavior of a substance will be described by a function including time, stress, strain and all the time derivatives of the latter two. Two familiar classes of materials exist for which this general function is greatly simpli- fied by the fact that all but a few of these variables can be neglected. One class is elastic materials, for which all the variables except stress and strain can be neglected. The other class is viscous materials, with