Response of an elastic Bingham fluid to oscillatory shear

Transcription

1 Rheologica Acta Rheol Acta 26: (987) Response of an elastic Bingham fluid to oscillatory shear A. S. Yoshimura and R. K. Prud'homme Department of Chemical Engineering, Princeton University Abstract: The response of an elastic Bingham fluid to oscillatory strain has been modeled and compared with experiments on an oil-in-water emulsion. The newly developed model includes elastic solid deformation below the yield stress (or strain), and Newtonian flow above the yield stress. In sinusoidal oscillatory deformations at low strain amplitudes the stress response is sinusoidal and in phase with the strain. At large strain amplitudes, above the yield stress, the stress response is non-linear and is out of phase with strain because of the storage and release of elastic recoverable strain. In oscillatory deformation between parallel disks the non-uniform strain in the radial direction causes the location of the yield surface to move in-and-out during each oscillation. The radial location of the yield surface is calculated and the resulting torque on the stationary disk is determined. Torque waveforms are calculated for various strains and frequencies and compared to experiments on a model oil-in-water emulsion. Model parameters are evaluated independently: the elastic modulus of the emulsion is determined from data at low strains, the yield strain is determined from the phase shift between torque and strain, and the Bingham viscosity is determined from the frequency dependence of the torque at high strains. Using these parameters the torque waveforms are predicted quantitatively for all strains and frequencies. In accord with the model predictions the phase shift is found to depend on strain but to be independent of frequency. Key words: Elastic B_ingham fluid, yield stress, oscillatory shear, oil-in-water emulsion, _nonlinear rheological behavigr Notation A AR G m M N r m f - R - t -- 7E -- )~.0 -- YPR F E -- Fp - F.0 R plate strain amplitude (parallel plates) plate strain amplitude at disk edge (parallel disks) elastic modulus torque (parallel disks) normalized torque (parallel disks) = 2 m/~ R 3 Zo ratio of viscous to elastic stresses (parallel plates) = k~ ~ A/z o ratio of viscous to elastic stresses (parallel disks) =/ o3 AR/ro normalized radial position (parallel disks) = g/r radial position (parallel disks) disk radius (parallel disks) normalized time = co t- ~/2 time elastic strain plate strain (displacement of top plate or disk divided by distance between plates or disks) plate strain at disk edge (parallel disks) yield strain normalized elastic strain = ye/~, 0 normalized plate strain = 9.0/70 normalized plate strain at disk edge (parallel disks) = 7.0R/?'o F 0 - normalized plate strain amplitude (parallel plates) =A/70 - normalized plate strain amplitude at disk edge (parallel disks) = AR/?~ o 6 - phase shift between F.0 and T (parallel plates) - phase shift between F.0R and M (parallel disks) p - Bingham viscosity - stress z0 - yield stress T - normalized stress = z/z0 ~o - frequency. Introduction Dynamic mechanical measurements are useful probes of the structure of complex emulsions, dispersions, gels, and slurries which may display yield stresses. The analysis of the oscillating stress response in terms of linear viscoelasticity requires that the stress signal be sinusoidal which is often not the case for these systems. No general theory exists to relate non-linear waveforms

2 Yoshimura and Prud'homme, Response of an Elastic Bingham Fluid to Oscillatory Shear 429 to fundamental microscopic material properties. However, often viscoelastic moduli are obtained without ensuring that the stress signals are sinusoidal. Our interest is in understanding how complex theology influences dynamic oscillatory measurements, and the converse question, how non-linear stress waveforms can be analyzed to provide fundamental information on fluid rheology. In this paper we analyze the response of an elastic Bingham fluid to an oscillatory deformation between infinite parallel plates and between parallel disks. In Section2. the elastic Bingham fluid model is introduced. The elastic Bingham fluid model is a new extension of the classical Bingham fluid model that allows elastic-solid deformation below the yield stress and Newtonian fluid flow above the yield stress. In Section 2.2 the response of the elastic Bingham fluid model to oscillatory strain between infinite parallel plates is analyzed. At low oscillatory strain the yield stress is never exceeded and the stress is sinusoidal. The storage modulus measured below the yield stress gives the modulus of the elastic Bingham solid. At larger strains the yield stress is exceeded and the material flows, the result is that the stress wave becomes flattened at the top. When the oscillatory strain reverses direction the stored elastic strain in the elastic solid is recovered. For the oscillating parallel disk case analyzed in Section 2.3 the situation is more complicated since the strain is non-uniform - it increases with radial position. Therefore the location of the yielding surface changes radial position during a single oscillation and is not known a priori. The analysis gives the location of this yield surface and the torque on the disk during oscillatory deformation. Finally, experiments have been performed on a model oil-water emulsion which displays a yield stress. The stress signals show quantitative agreement with predictions of the model as to the shape of the waveform and the phase shift between the torque and strain signals. 2. Elastic Bingham fluid model 2. Description The conventional Bingham fluid model does not consider elastic deformation of the material below the yield stress. We modify the Bingham model to include elastic strain below the yield stress and stored recoverable strain (i.e. stored elastic energy) once the material has yielded. The stress-strain relationships for this modified Bingham fluid are plotted in figure and can be expressed as ~=G~ (ITEI < 7ot, () r= GTe+/~ ~ (!Tel = 70), (2) where r designates the stress, G the elastic modulus, t the Bingham viscosity, 7e the elastic (recoverable) strain, )~ the time derivative of the total strain (strain rate) and 70 the yield strain. All strain and stress variables throughout this paper refer to shear flow and as such this model is not a full constitutive equation that can be applied to arbitrary flow fields. The yield strain, 70, is the maximum strain the fluid can support elastically. It is related to the yield stress, z0, through r0 = G 70. (3) At low strains, the fluid behaves as an elastic solid and the stress is given by r = G 7e. (4) If the strain is increased beyond Y0, the fluid flows; viscous dissipation occurs and the additional strain is not recoverable. Under theses conditions, the stress is given by r= GyE+~t ~ (Te = 70) =r0+~. (5) O9 "~ o ~ i. Elastic Strain 09 ~ ;o O9 Strain Rate "Co Fig.. Elastic Bingham fluid model. Below the yield stress (or strain), the fluid behaves as an elastic solid. Above the yield stress, the stress increases linearly with strain rate

3 430 Rheologica Acta, Vol. 26, No. 5 (987) During this viscous shearing, the elastic strain, 7e, maintains its maximum value of 70. Thus, elastic energy is stored during the viscous flow. If the shearing occurs in the opposite direction, the elastic strain attains a value of- 70, and the stress is then given by r= GTe+/~ ~ (ye = -- )'0) = - "Co +/ ;~. (6) 2.2 Oscillatory shear between infinite parallel plates We first consider an elastic Bingham fluid contained between two infinite parallel plates as shown in figure 2. The lower plate is fixed while the motion of the upper plate is described by 7p=Asincot (t>0) (7) where 7e is the horizontal displacement of the plate divided by the distance between the plates. For brevity, we refer to 7~ as the "plate strain". Its amplitude and frequency are denoted by A and co respectively. Time is represented by the variable i. We now define two dimensionless parameters; A F0 =- (strain parameter), (8) 70 N - /~coa (stress parameter). (9) "CO The parameter F0 is the plate strain amplitude normalized by the yield strain of the fluid, and N is a measure of the ratio of viscous to elastic stresses. For convenience, we define a dimensionless time t as 7C t - co t-- ~- (time) (0) and we normalize the stress and strain variables as follows: 2" T- =- (stress), () "C o 03 Fig. 2. Infinite parallel plates. The top plate oscillates with frequency co and strain amplitude A Fe -= 7! (elastic strain), (2) Y0 7P Fp (plate strain). (3) 7o The plate motion can now be described by F~ = F0 cos t (t _-> - g/2), (4) and the fluid stress-strain relationship, eqs. () and (2), can be written as T=FE (]Fe]<l), (5) N T=Fe+~ol~p=FE-Nsint (]FE] = ). (6) In the following analyses we assume that inertial effects are negligible and that the fluid is initially in a stress free state (T = 0 at t = - ~/2) Strains below the yield strain: Fo <- If the plate strain amplitude is less than the yield strain, the fluid does not yield or flow but responds simply as an elastic solid. The elastic strain is then equal to the plate strain, and the stress is obtained through eq. (5), T= FE = Fp = F0 cos t (t _-> - ~/2). (7) Since the stress response is in phase with the plate strain, the phase shift between them, 6, is equal to zero, and it follows that tan 6 = 0. (8) Strains above the yield strain: Fo > If the plate strain amplitude is greater than the yield strain of the fluid, the fluid yields and flows periodically. Plots of the imposed plate strain and the resulting elastic strain and stress curves are shown in figure 3. Since the curves become periodic after a "start-up" time of one quarter of a cycle (-7r/2 = t-< 0), the steady state response can be represented completely by the strain and stress curves during the period 0 _-< t _-< 2 7r. During the first part of this period, the stored elastic strain, Fe, decreases from its maximum value of at time t=0, to its minimum value of- at some unknown time t = tl (to be determined in the following analysis). Time tl marks the onset of viscous flow in the negative direction. This viscous shearing continues and the elastic strain maintains its value of- until time t = ~ when the plate motion direction changes. The stored elastic strain then increases from - at

4 Yoshimura and Prud'homme, Response of an Elastic Bingham Fluid to Oscillatory Shear 43 calculated through eq. (6), E.0 03._o -.0 LLI - Fo /,, / / \ / \,,, / X / I \,~ tl\'l I ~, ;,. \ I \\ / T = - - N sin t (t I =< t =< 7r). (22) The strain and stress conditions of the period ~r < t _-< 2 ~ are similar to those of the period 0 < t = except opposite in sign. These conditions can be expressed as FE = -- + Fo( + cos t) (~ N t N t2), (23) T =-l+fo(l+cost) (~<t<t2), (24) F E = (t 2 _-- t -- 2 ~), (25) T =l-nsint (t2--<t-2~). (26) I- -.0 co F s \ I \ ;"-.. '.,..., \ t,,,, [, /, %, Fig. 3. Large strain response for infinite parallel plates. Elastic strain and stress waveforms are non-linear because the yield stress has been exceeded. The imposed sinusoidal plate strain is represented by broken lines We derive expressions for t I and t2 by noting the strain requirements at these times. From eq. (2), FE it=t, = --. (27) Combining with the expression for FE from eq. (9) gives t, = cos-i [ ]. (28) Similarly, we obtain from eqs. (25) and (23) t2 = cos - - = t~ + ~ (~ =< t2 =< 2~). (29) t = ~z to its maximum value of at some unknown time t = t2 (also to be determined). At t = t2, viscous flow in the positive direction begins. This viscous flow continues and the elastic strain maintains its value of until time t= 2~, when the plate motion direction again changes. At this time, the fluid strain and stress conditions are identical to those at time t = 0. At time t = 0, the plate strain is at its maximum value (Fp = F0) and the elastic strain of the fluid is equal to its yield strain (FE = ). During the period 0 < t < tl, this stored elastic strain is released as the plate strain decreases: FE = -- (F0 - Fe) =l-f0(-cost) (0 --- t=< tl). (9) Since Fe]<, the fluid behaves as an elastic solid and the stress is calculated through eq. (5), T=FE=l-Fo(-cost) (0<t<h). (20) At time t = ta, the elastic strain is equal to the yield strain in the negative direction, Fe=- I (q <- t <- ~), (2) and it maintains this value until time t = ~. During this period, viscous dissipation occurs and the stress is As seen in figure 3, there is a phase shift between the plate strain and stress response when F0 >. From eq. (20), we find that the stress is zero when t = cos-' [ -~0 ]. (30) tan6=tan -cos - I --~0]} ' (3) It follows that 2.30scillatoryshearbetweenparalleldisks We now consider an elastic Bingham fluid contained between two parallel disks of radius R. The disk geometry and the coordinate system are shown in figure 4. The lower disk is fixed while the motion of the upper disk is described by 7eR = AR sin co'i (i ) (32) where YPR is the plate strain at the disk edge, and AR and co denote its amplitude and frequency respectively. This geometry is more complex than the previous example because the strain is now a function of radial position and the yield surface is at an unknown radial position. Since the disks are solid, the radial depen-

5 432 Rheologica Acta, Vol. 26, No. 5 (987) and we normalize the stress, strain, and torque variables as follows: T = 7 (stress), (39) 2"0 FE ~oe (elastic strain), (40) 70 Fig. 4. Parallel disk geometry. The upper disk oscillates with frequency co and strain amplitude at disk edge AR Fp = 7p (plate strain), (4) 70 FpR -- 7PR 7o (plate strain at disk edge), (42) 2m M (torque). (43) rc R 3 2"o dence of the plate strain, yp, is linear and is given by?e =-~- 7eR = ~-AR sin co t (t == 0). (33) In experiments with this type of geometry, the measured variable is the torque, m, experienced by the upper disk. We can calculate this torque by integrating over the disk, R m = 2 Jr ~/~2 2" d? (34) 0 m where 2" is the z0-component of stress at the upper disk. As before, we define two dimensionless parameters, AR F0 - (strain parameter), (35) 70 # coar N (stress parameter). (36) 2" 0 The parameter F0 is the plate strain amplitude at the disk edge normalized by the yield strain of the fluid, and N is a measure of the ratio of viscous to elastic stresses. These parameters are the same as those defined for the parallel plate geometry except that A is replaced by AR. We also define dimensionless time and spatial variable as 7Z t - co F- ~- (time), (37) 7: r --- (radial position), (38) R In terms of dimensionless variables the plate strain is given by Fe = r FeR = r F0 cos t (t >= - ~r/2), (44) and the fluid stress-strain relationships are r=r (Ir l < ), (45) N T=Fe+ ol =Fe-rNsint (Ir l = ). (46) The expression for torque is now M= 4 ~ r 2 r dr. (47). 0 In the following analyses we assume that inertial effects are negligible and that the fluid is initially in a stress free state (T = 0 for all r at t = - ~/2) Strains below the yield strain: Fo <= If the plate strain amplitude at the disk edge does not exceed the yield strain, the fluid behaves as an elastic solid over the entire disk at all times. Therefore, FE = Fe = r Fo cos t (48) and the corresponding stress is calculated through eq. (45), T = Fe = rfo cos t. (49) From eq. (47) the torque is then M = 4 S r3 F0 cos t dr = Fo cos t. (50) 0 Since the torque is in phase with the plate strain, the phase shift between them, 6, is equal to zero and tan 8 = 0. (5)

6 Yoshimura and Prud'homme, Response of an Elastic Bingham Fluid to Oscillatory Shear Strains above the yield strain: Fo > If the plate strain amplitude at the disk edge is greater than the yield strain, the fluid near the edge yields and flows. The fluid near the disk center, where the plate strain amplitude is smaller, always behaves as an elastic solid. The critical radius which divides these two regions, r, is given by r =--. (52) F0 At this radius, the plate strain amplitude is equal to the yield strain of the material. The elastic strain is depicted in figure 5 as a function of position during the time period 0 ~ t _-< n. At t = 0, when the plate strain is at its maximum value (Fe = rfo), the elastic strain in the edge region is equal to the yield strain, Felt=0=l (rl---r-<-l) (53) while the elastic strain in the center region is equal to the plate strain, Fe[t=o=Fe[t=o=rFo (O<=r<=r O. (54) As the plate strain decreases during the period 0 < t < tl, the elastic strain in both regions decreases accordingly as re=r l,=o-(rpl,=o-r ) (O<=t<-q). (55) Combining with eqs. (53), (54) and (44) yields Fe = rio cos t (0 <= r <= q, 0 <= t <- tl), (56) Fe= l-rfo(-cost) (rl <-r <- l,o--< t=< to. (57) During the period 0 < t < tl, no viscous dissipation occurs because the elastic strain is less than the yield strain at all radial positions. Therefore, the stress at any position is given by T= Fe (O < t < tl). (58) Combining with eqs. (56) and (57) gives T = rfo cos t (0 - r - ri, 0 < t < tl), (59) T= - rf0( - cos t) (r --< r _-<, 0 < t < tl). (60) The torque can then be calculated from eq. (47), rl M=4 S r3fo c stdr+4 ~ r2[ - rr0( -cost)]dr 0 r~ = F0 [ cos t - -~04 +34F0] (0 < t < Is). (6) Z rr t-- 03 _o t-- ~0 5 tla.0 ~ ~ t t~ 0 At t = q, the elastic strain at the disk edge is equal to the yield strain in the negative direction, FE It= 2, t=a = -- l. (62) Inserting the expression for Fe from eq. (57) and solving for tl gives tl=cos-l[-@0 ]. (63) -.0 g rr ,.<, t=2n t=t2 i- F Fig. 5. Elastic strain as a function of radial position. When i Fe ] <, the fluid behaves as an elastic solid. When I Fe] =, the fluid flows. At any given time, the smallest radial position at which I Fe ] = defines the yield surface We now label the smallest radius at which ]FE] = as ry. Since the fluid is solid-like (]FE[ < ) inside of this radius and is yielded (]Fe[=l) beyond this radius ry is referred to as the yield surface position. From figure 5 we see that ry decreases from r e = at t = tl to ry = rl at t = ~. The elastic strain during this period is given by FE = rfo cos t Fe = - rf0( - cos t) Fe =- (0_-<r_-<rl,tl_-<t_-<~), (64) (rl--<r--<ry, tl--<t--<~), (65) (ry _-< r -, tl - t ~ ~). (66) Evaluating and comparing the expressions for Fe in eqs. (65) and (66) at r = ry leads to an expression for ry, 2 ry - Fo( - cos t) (67)

7 434 Rheologica Acta, Vol. 26, No. 5 (987) No viscous dissipation occurs in the region r < ry since here I Fel<. Thus, the stress is due completely to the elastic strain in the fluid and can be obtained through eq. (45), T=FE (O<=r <ry,q <=t<=~r). (68) Combining with eqs. (64) and (65) gives T = rfo cos t (0 _-< r _-< rl, tl --< t =< ~z), (69) T=l-rFo(-cost) (q<-r<ry, tl<=t<-~). (70) o=6.0 N=.003 \ /, \ J \ \,, \ / Fo = 3.0 N =.005 '\ / "~ Fo=6.0 N=0.3 \,,, \ \ \ Fo = 3.0 N = 0.5 At radial positions outside the yield surface (r >- ry), viscous flow occurs and the stress is calculated from eqs. (46) and (66), T=-l-rNsint (ry<-_r<-_l, tl<-t<-~). (7) The torque can now be calculated through eq. (47) as r ry M= 4 ~ r 3 F ocost dr + 4 ~ r2[ - rfo( - cos t)] dr 0 rl + 4 ~ r 2 [-- -- rn sin t] dr ry F03 ( - cos t) 3 3F 3 3 -Nsint - F4(l_cos0 4. (q --<t-<~z)" (72) The strain and torque conditions of the period z~ _-< t _-< 2 ~r are similar to those of the period 0 _-< t _-< z~ except opposite in sign. The elastic strains during this period are shown in figure 5. The torque can be expressed as [ '4] M=F0 cost+l+3f~ 3I'0 (~ <t<t2) (73) (74) M- 3r0~( + cos t)3 +~03+ 7 where -Nsint [ - F4(l+cost),6 4 l (t2nt=<2~z) t2 = tl + ~. (75) Eqs. (6) and (72-74) can be used to predict the entire torque waveform as a function of time. Plots of the torque for various values of F0 and N are shown in figure 6. We see that the phase shift between the torque and plate strain curves, 5, increases with increasing amplitude F0 and is independent of N. From \ / \ \, / L/k "\ /' ", \ / Fo=0.3 N=.0005 Fo=0.3 N=.05," "-, Fig. 6. Torque waveforms for an elastic Bingham fluid between oscillating parallel disks. Parameters F 0 and N can be interpreted as the normalized strain amplitude (at disk edge) and normalized frequency, respectively. Imposed strain is represented by broken lines eq. (6) we find that the torque is zero when [ 4] t=cos - 4 3~ 4 3FO" ' (76) It follows that tan 5= tan {2 - cos-' [ ]} 3 Fo 4 3 Fo " 3. Experimental 3. Materials The fluid chosen for this study is an oil-in-water emulsion. The oil phase consists of paraffin oil while the aqueous phase consists of a 20% (by weight) solution of Alipal CD-28 surfactant (GAF Corporation). The volume fraction of oil is 92%, and the mean drop size is 7 gin, as measured with a Particle Data (Elmhurst, IL) particle counter.

8 Yoshimura and Prud'homme, Response of an Elastic Bingham Fluid to Oscillatory Shear Methods and results Rheological data were taken with a Rheometrics System IV Rheometer using the parallel disk geometry. The disks were made of fritted glass to prevent slippage of the emulsion. Sinusoidal deformations of various amplitude and frequency were imposed on the emulsion sample and the resulting torque was recorded. Figure7 shows samples of strain (at disk edge) and torque curves as a function of time for strain amplitudes of 0.,.0, and 2.0 and frequencies of 0. and 0 radians per second. The time scale has been normalized with the experiment frequency. The magnitudes of the curves are not scaled consistently. What is of interest here is the characteristic shapes of the torque curves. The phase shift, b, was measured as the difference between the zero crossing of the torque and strain curves. Phase shifts were found to be independent of frequency and are plotted as a function of strain amplitude in figure 8. AR = =0. AR = = 0 "k\ / f "\\ \ \\ 60..i...a t.- I Yo = 20% 30 ~ ~ Experiment Model Strain (%) Fig. 8. Phase shift vs. strain amplitude. Symbols are the measured data for an oil-in-water emulsion. Solid lines are the phase shifts predicted with the elastic Bingham model, with various assumed values of the material yield strain, 70 E o 0 5 G) t- "~ v 0 U o= Experimental 4o L 2 Aa=l.0 03=0. \ / \,-J/ \ / \ AR =0. 03 =0. \\ I \ - "4 AR=I.0 03=0 \ i '\,, E AR=0. o3=0 ~- _ 0, -0 5 = 2.0 Time (normalized) Model Fig. 7. Torque waveforms for an oil-in-water emulsion between oscillating parallel disks. Parameters A R and co are the strain amplitude (at disk edge) and frequency of the experiment respectively. Frequency is in rad/s. Imposed strain is represented by broken lines -0 5 Time (normalized) Fig. 9. Frequency dependence of torque waveforms. Experimental curves are for an oil-in-water emulsion. Model curves are based on the following parameters: 70 = 33%, G= 3000 dyn/cm 2, /z = 5 poise. Frequency is in rad/s

9 436 Rheologica Acta, Vol. 26, No. 5 (987) 4. Comparison of data with model predictions Good agreement is found between the shapes of the measured torque curves (figure 7) and of those predicted with the elastic Bingham model (figure 6). At low strains, the torques are sinusoidal at all frequencies. At higher strains, the curves flatten out as the yield stress is reached, and a viscous hump increases in size with increasing frequency. Figure 8 shows the measured phase shifts (symbols) and predicted phase shifts (solid lines) calculated from eq. (77) with various assumed values of the material yield strain. Excellent agreement is shown for a yield strain value of 33%. The frequency dependence of the measured and predicted torque curves are compared in figure 9. A yield strain of 33%, elastic modulus of 3000 dyn/cm 2 (determined from the sinusoidal torques at low strains), and Bingham viscosity of 5 poise have been assumed for the predicted curves. 5. Conclusions An elastic Bingham fluid model has been developed that includes solid-like elastic deformation below the yield stress and flow above the yield stress. The response of the elastic Bingham fluid to oscillatory deformation has been analyzed to understand the linear and non-linear response of materials with yield stresses in dynamic mechanical tests. First, the analysis of oscillatory strain between parallel plates shows sinusoidal stress below the yield stress and nonsinusoidal stress above the yield stress (or strain). The stress waveform above the yield stress depends on both the strain amplitude and frequency. Next, the torque produced by oscillatory motion between parallel disks is analyzed which has the added complexity that the strain goes from zero at the center of the disks to a maximum value at the outer edge of the disk. The calculated torque on the stationary disk is sinusoidal when the strain everywhere between the disks is below the yield strain (or stress). Above the yield strain the torque is non-linear. The radial location of the yield surface moves in-and-out during each cycle of the oscillation of the rotating disk. The non-linear torque signals are determined as a function of strain and frequency. The strain dependence arises when the applied strain exceeds the yield stress; the frequency dependence (at constant strain above the yield strain) arises because the viscous stress is proportional to shear rate which is related to the product of strain times frequency. The phase angle between the strain and torque is independent of frequency but depends on strain. Experiments were conducted on a model oil-in-water emulsion which has a yield stress. The torque response to dynamic oscillatory deformation between parallel disks shows sinusoidal response at low strains for all frequencies and non-linear responses at high strains. In the non-linear regime the waveforms are functions of frequency. The yield strain of the emulsion is determined from the phase shift between stress and strain, the elastic modulus is determined from low strain data, and the Bingham viscosity is determined from the frequency dependence at high strains. Using these parameters the non-linear torque waveforms can be fit quantitatively. The elastic Bingham fluid model we have developed is a useful model for describing materials with yield stresses in transient flows or in flows which reverse direction. The model quantitatively predicted the torque waveforms for the emulsion we investigated. The model, like the classical Bingham model, does not include time dependent effects and is, therefore, unable to duplicate thixotropic fluid behavior. The fact that the model emulsion shows a completely time-independent rheology makes it an interesting test fluid for rheological studies. Our conclusion is that dynamic-mechanical tests in the linear viscoelastic regime are useful for obtaining fundamental properties of structured fluids. However the complexity of the torque waveform makes determination of the yield stress by dynamic mechanical testing relatively difficult. Alternate techniques, such as we have reported on earlier [], are more attractive. Acknowledgements We would like to acknowledge financial support from the National Science Foundation, Exxon Research and Engineering, and Chevron Exploration and Production Company. Reference. Yoshimura AS, Prud'homme RK, Princen HM, Kiss AD (986) A Comparison of Techniques for Measuring Yield Stresses (to appear in J Rheol, 987) Authors' address: A. S. Yoshimura, Prof. R. K. Prud'homme Department of Chemical Engineering Princeton University Princeton, NJ (U.S.A.) (Received February 2, 987)