CAPILLARY VISCOMETRY 351 which we are now concerned. For these all the variables except stress and rate of strain can be neglected. However, many materials exist on which viscometric investigations may be desirable, but which do not exhibit a unique relation between the two appropriate variables. The well-known viscoelastic materials are examples of materials that can be adequately described by the inclusion of only a few additional variables. Under properly selected experimental conditions these may sometimes be : adequately treated as viscous materials. The cases of thixotropy, work-hardening and degradation cited above : are different from simple viscoelasticity. The only additional variable which need be included is time, and this can never be neglected in these cases. There may be a philosophical point of whether one is dealing with a single material in these cases, or whether as the shearing proceeds one deals with a continuously changing material. Whichever position is adopted the fact remains that the complete description of the theological behavior requires only shearing stress, rate of shear and time. The above discussion does not mean that viscometry investigations are _. useless on materials for which the subject unique function does not exist. ! On the contrary, such investigations may in fact give valuable information about the complicating processes in the flow. What is difficult or im- possible for these materials is to reduce the experimentally observed data to fundamental theological quantities. If the goal of an investigation is to give qualitative information about disturbing effects, it can be highly useful. ¾.2. Laminar ?ersus Turbulent Flow Laminar flow as defined in an earlier section is only one kind of flow which can occur in a capillary. It is the only type by which viscosity can be measured. The principal other type of flow in a capillary is turbulent flow. This occurs when the orderly parallel paths of elements of the material flowing through the capillary become disturbed. These paths may become irregular and vortices eventually appear. Under these conditions no comprehensive relation is known between the experimentally observed data and the fundamental theological and other quantities. Therefore the flow data cannot be analyzed to yield viscosity data. The conditions under which turbulence may arise are partially described by the Reynolds number. This criterion indicates the relative magnitudes of the acceleration effects which tend to promote disturbances and the viscous effects which tend to damp them out. The Reynolds number R for capillary flow is calculated by the equation R- vdp (16) where v is the average linear velocity of the material flowing through

352 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS the capillary, d is the diameter of the capillary and 0 and n are, respectively, the density and viscosity of the experimental material. In capillary flow, turbulence never occurs under conditions where the Reynolds number is below the so-called critical value of 2300. At Reynolds numbers above the critical value turbulence may occur and eventually should if a given volume of fluid remained in a shear field an infinite length • of time. In practice a volume of fluid does not remain in a shear field an • infinite length of time so turbulence does not necessarily always occur: above the critical Reynolds number depending on the condition of the: capillary wall, the shape of the entrance end of the capillary, the disturbance: present in the fluid before it enters the capillary, and possibly other factors.. The exact conditions under which laminar flow does transform to turbulent :-.' flow in practice are not at all understood and are the subject of much : current investigation by hydrodynamicists. Reynolds numbers as high as 50,000 have been reached in the flow of simple liquids without the - onset of turbulence (4). The important fact for viscometry is that if one maintains conditions below the critical Reynolds number the results will not be complicated by turbulence. There have been suggestions (5) that in some viscoelastic and colloidal materials a special kind of turbulence called "structure turbulence" can occur at Reynolds numbers much lower than 2300, possibly as low as 10. However this suggestion is subject to much doubt (6) and it is highly likely that approximately the same critical number applies for viscoelastic fluids as for simple liquids. Certain hydrodynamical investigations (7) have suggested that turbulence is much more likely to occur at Reynolds numbers only slightly higher than 2300 in polymer solutions than in simple liquids. This suggestion is inconclusive and does not propose any special kind of turbulence below the usual critical Reynolds number. Although turbulent flow is not useful for the determination of funda- mental rheological quantities it is sometimes of great importance in practical deformation processes. For example, in many cases mixing and the maintenance of suspensions are dependent on turbulence. Its rejection here is limited to the pursuit of fundamental theological quantities. 4.3. Slippage at Capillary l/Fall The assumption that in capillary flow the material immediately adjacent to the capillary wall sticks to the wall is not always justified. A method of detecting and correcting for such deviation was developed by Mooney (8). Measurements are made in several capillaries of a common length to diameter ratio (to eliminate end effects, as will be mentioned later), but having different diameters. All other factors behaving properly, if the material sticks to the wall the calculated viscosity should be independent