CAPILLARY VISCOMETRY 353 of capillary diameter. However if the material slips along the wall, there will be a dependence on diameter. Since the perimeter of the capillary cross section increases as the radius while its area increases as the square of the radius, if there is slippage it will become relatively less important as the capillary diameter is increased. Therefore the calculated viscosity will increase with increasing capillary , diameter eventually reaching a correct asymptotic value when the wall . effect becomes relatively insignificant. If it is assumed that the disturbing effect is a clean slippage at the wall and does not extend any distance into the capillary, the velocity of this slippage can be calculated from the increase of calculated viscosity with capillary radius. Using this velocity : the calculated viscosity may be corrected for the slippage to yield a correct value. A modification of the above procedure has been designed by Oldroyd (9) to deal with slippage at the capillary wall in turbulent flow. Cases have been known (10) where, in the above test, the calculated viscosity decreased instead of increased with increasing capillary diameter. This effect might be interpreted as something the opposite of slippage, ! but is more likely a compositional than a rheological effect. • 4.4. Kinetic Energy Contributions So far in this paper it has been implicitly assumed that in capillary viscometry the pressure drop through the capillary is associated entirely ß with overcoming viscous resistance to flow. This is not always true. Other effects can contribute to this pressure drop. Under some circum- stances these other contributions are significant and cannot be disregarded. One sometimes important other contribution is associated with the kinetic energy which must be imparted to the liquid to make it flow through the capillary. When a unit volume of liquid of density 0 is accelerated from rest to a uniform velocity v an amount of kinetic energy equal (1/2)0v2 must be supplied. Furthermore if the liquid is accelerated to a nonuniform velocity averaging to the value v, there will be an additional requirement proportional to ov 2 because the kinetic energy is a quadratic rather than a linear function of the velocity. When the liquid decelerates to rest after emerging from the capillary, it gives up the kinetic energy originally imparted to it. If this kinetic energy were all dissipated at the exit end of the capillary in such a way as to raise the pressure there, it would serve only to raise the pressure at both ends of the capillary by an equal amount and make no contribution to the difference. However an undetermined amount of this kinetic energy, depending partly on the shape of the capillary exit, is instead dissipated immediately as heat at the exit end of the capillary, in which case it does not raise the pressure at the exit end but does make a contribu- tion to the pressure drop through the capillary.

354 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS This contribution can be evaluated as follows. Transposition of equation 12 gives as the pressure drop due to viscous resistance P• = 8•L•i/•'R 4 (17) If kinetic energy effects are also included in the total pressure drop through . the capillary this total drop will be P = 8•OL,i/•rR 4 + mtv • (18) where m is an unknown coefficient. Now v = •/•rR 2 and • = lZ/t where /z is the volume of fluid which flows through the capillary in time t. Using these relations, equation 18 transforms to Pt = 8L/•4• mp/• 1 (19) ß -R • +'•-•R 4 t If measurements are made in a capillary over a range of values of P, then a plot of Pt versus 1It at fixed F will give as its intercept 8LFrl//rR 4. Since everything is this quantity except r/ is known, the correct value of r/, free of kinetic energy effects, can be extracted from it. If the value of the coe•cient m is desired,- it can be obtained from the slope of the above plot, which is mpF•/•r2R4 in which everything but m is known. If none of the kinetic energy imparted to the fluid at the capillary entrance is reconverted to hydrostatic pressure at the capillary exit, the value of rn to be expected theoretically is unity. In practice at very low rates of capillary flow, apparently all the kinetic energy imparted at the entrance is reconverted to hydrostatic pressure at the exit and the experi- mental value of m is zero. At higher rates of flow the kinetic energy is not reconverted and as the rate increases rn approaches an asymptotic experimental value in the neighborhood of 1.1. Why this value is greater than unity is not exactly understood. The importance of the kinetic energy contribution in any case is easily evaluated from the relative magnitude of the two terms of the right hand side of equation 19, and is readily subject to the control of the investigator through the proper choice of the experimental conditions. ¾.5. End Contributions Another contribution besides that expressed in equation 17 to the observed pressure drop arises just inside the entrance and just outside the entrance and exit ends of the capillary. The flow laminae in these regions are, respectively, converging to and diverging from the right circular cylindrical shells which prevail within the capillary. These processes require extra energy which is reflected in a contribution to the observed pressure drop. This contribution is often referred to as the Couette effect. This additional pressure drop can be taken into account in the calcula-