THE RHEOLOGY OF PASTES, SUSPENSIONS AND EMULSIONS 183 so successful. Suspensions generally require a St. Venant term. If they are pushed through a capillary tube, a certain stress is needed before they will start to flow. This leads us to Bingham's equation, which, in its modem form, is written where ,/p• is called "plastic viscosity" r is shear stress q[ •7p• = (r -- ro)/D D is shear rate [ at the tube wall ro is critical shear stress j But this is an over-simplification. What really happens is that the critical stress is reached first at the wall of the tube, where stress is highest, so the flow-curve may be divided into four parts, as pressure is increased. (1) Up to a certain pressure, there is no flow at all. (2) Later, the paste slides as a solid plug through the tube at a rate proportional to the excess of pressure over that required to start the plug sliding. (3) When the stress ro is reached, flow starts at the wall and, as pressure rises, the flowing envelope increases in width at the expense of the central plug. (4) Finally, the plug virtually disappears and all the flow is laminar. The rate of flow at still higher pressures is proportional to r -- to. With many pastes, other types of anomaly exist. When liquids which obey Newton's law flow slowly through capillary tubes, the volume of flow per second (V) is related to the pressure (P), the radius (R) and the length (L) of the capillary by the well-known equation of Poiseuille V -• P, rR•/SL,/where ,/is the viscosity. Forty years ago, Bingham and Green 5 noticed that the flow of certain paints did not follow the fourth-power relation to capillary radius. In 19306 a detailed quantitative study of such anomalies for clay and soil pastes was made, and later, similar anomalies in dairy creams, natural and recon- stituted, were reported'. Four main types of behaviour are found for pastes: (1) Some very dilute pastes are almost true fluids, i.e. show no yield- value and obey the R • law. (2) Some show a yield-value but obey the R • law. (3) Some show no yield-value but do not obey the R 4 law. (4) Some show both a yield-value and radius anomalies. With emulsions, or mixed suspension-emulsions, like cream, there are further complications. All four of the above categories must be sub-divided depending on whether the flow curves are straight or curvilinear (i.e. whether the rate of flow is eventually proportional to the stress, or to the excess of

184 JOURNAL OF THE SOCIETY OF COSMETIC CItEMISTS