The equation of continuity and the stress equations of motion are not sufficient in themselves to describe the motion of matter under given boundary conditions. In addition one must describe the behavior of the particular material to be considered by stating the relation between stress and deformation, the constitutive equation, A number of properly invariant constitutive equations have been proposed. Reiner (17) and Ridin (18, 19; see also 21) present the most general isotropic relation between stress and rate-of-deformation for homogeneous material. Oldroyd (13, 1 4 ) gives an excellent discussion of the formulation of properly invariant constitutive equations; he examines in detail the simplest constitutive equations which reduce to experimentally observed viscoelastic behavior at small rates of strain. Rivlin and Ericksen (20) assume the stress to be a function of the gradients of velocity, acceleration, second acceleration, Noll (12, 5 ) and Green and RivIin (6, 7) consider memory effects; that is, the state of stress in the fluid is assumed to depend upon its past history as well as its current kinematic state. All of these proposals appear to be qualitatively consistent with observed viscometric flows (for example, flow in a cone-plate viscometer, flow between rotating concentric cylinders, flow through a circular cylindrical tube) for some materials with the exception that the Stokesian fluid model proposed by Reiner and by Rivlin does not properly predict observed normal stresses (1 1 ) .The Stokesian fluid model for an incompressible fluid" ti = -pai, + J&+~ + yd",:is by far the simplest of all the constitutive equations proposed, It includes as special cases the simple empirical models long used in engineering work to represent the results of viscometric experiments such as the power model Ti = ti + F3 = gm(2II) (n--1)f2 & While the power model cannot fully represent the behavior of any real fluid presently known (an immediate objection is that it does not predict nonzero and finite limiting viscosities at very low and a t very high rates of deformation), over a limited range of stress it often gives a satisfactory approximation for the stress vs. rate-ofdefomation curve from a viscometric study [or in terms of Noll's theory for simple fluids, it gives a satisfactory approximation for the first material function in a viscometric flow ( 5 ) 1. In practice engineers have been using the power model [also the Ellis model ( 2 2 ) , the Sisko model ( 2 3 ) , and many others] as an approximate description for the behavior of real fluids which on the basis of viscometer studies appear to be approximately powermodel fluids over a limited range of stress, but which probably exhibit normal stress effects (1 1 ) (and according to popular usage would be called v8coelastic"). It seems reasonable to ask about the error incurred in using this model to represent the behavior of such a fluid in a more complicated geometry, where according to Noll's theory it is entirely possible that a description based upon viscometri...

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The effects of a trace quantity of a surface-active agent on creeping flow past a bubble or droplet are investigated. The equations describing mass and momentum transfer are simultaneously solved by a perturbation technique, consistent with the jump mass and momentum batances a t the phase interface. The stream function for the velocity distribution is evaluated as an infinite series of spherical harmonics. Golerkin's method, which reduces the partial differential equation of continuity to a set of ordinary differential equations, is used to evaluate the concentration distribution of surfactant.A sample calculation is carried out for relative motion between an air bubble and an infinite body of water which contains a trace of isoamyl alcohol. The relative velocity of the water a t an infinite distance from the bubble is found to be highly sensitive to small changes in surfactant concentration from zero, although the bubble varies imperceptibly from a spherical shape.Consider a fluid globule which moves at a constant velocity under the action of gravity through a second immiscible phase. Hadamard and Rybczynski (1, p. 395) independently presented a solution to this problem which neglects interfacial effects, but their results failed to explain available experimental data for settling velocities.This led Boussinesq (2) to hypothesize that a skin which inhibits the internal circulation can form around a moving droplet. He described this quantitatively by a constitutive equation which expresses the stress in the interface as a linear function of the rate of deformation of the interface. This relationship, often called the Newtonian surface fluid model ( 3 to 6 ) , has two parameters in addition to surface tension: surface shear viscosity and surface dilatational viscosity. Acceptable laboratory measurements of these surface viscosities have not been obtained to date ( 7 ) . Boussinesq (2) obtained an exact solution for creeping flow past a droplet under the assumptions that interfacial behavior could be described by the Newtonian surface fluid model and that surface tension and the two surface viscosities were independent of position on the phase interface. The results of Boussinesq's analysis do not appear to be in any better agreement with available experimental data than those of Hadamard andNeither the analysis of Boussinesq nor the theory of Hadamard-Rybczynski considers the possible presence of surfactants (materials which have an affinity for a phase interface and which consequently alter the surface tension of a system). In practice it is very difficult to eliminate trace quantities of surface-active materials from an experiment. The observation of a fairly immobile cap on the trailing surface of a droplet ( 8 ) suggests that any surfaceactive agents present are not uniformly distributed around the surface, but rather that they are continuously swept toward the rear of the droplet by the fluid motion. If we express surface stress in terms of surface tension alone, the resultant accumulation of surfactants ...