Journal of Non-Newtonian Fluid Mechanics

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1 Journal of Non-Newtonian Fluid Mechanics 66 (20) Contents lists available at SciVerse ScienceDirect Journal of Non-Newtonian Fluid Mechanics journal homepage: Thixotropic flow of toothpaste through extrusion dies Hesam A Ardakani a, Evan Mitsoulis b, Savvas G Hatzikiriakos a, a Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, BC, V6T Z3, Canada b School of Mining Engineering and Metallurgy, National Technical University of Athens, Zografou 57 80, Greece article info abstract Article history: Received 3 April 20 Received in revised form 30 May 20 Accepted 9 August 20 Available online 4 September 20 Keywords: Toothpaste Axisymmetric contraction Paste extrusion Thixotropy Slip law Structural parameter A commercial toothpaste is investigated in this work as a model paste system to study its processing characteristics in capillary flow using various dies Its rheological behaviour has been determined as that of a yield-stress, thixotropic material with a time-dependent behaviour, and severe slip at the wall The rheological data obtained from a parallel-plate rheometer were used to formulate a constitutive equation with a structural parameter which obeys a kinetic equation, typically used to model thixotropy The predictive capabilities of this model are tested against capillary data for a variety of capillary dies having different length-to-diameter ratios (L/D), contraction angles (2a), and contraction ratios (D b /D) 2, where D b is the diameter of the barrel of the capillary rheometer The major trends are well captured by the thixotropic model and show that slip is the essential parameter in predicting the flow behaviour of toothpaste Ó 20 Elsevier BV All rights reserved Introduction Corresponding author Tel: ; fax: addresses: mitsouli@metalntuagr (E Mitsoulis), hatzikir@interchange ubcca (SG Hatzikiriakos) A paste can be defined as a mixture of a solid and a liquid phase or as a dense suspension, which is shown to demonstrate properties between liquid and solid [] Many pastes are able to retain their shapes against gravity like solids [2] At high shear stresses they start to flow, which implies the existence of a yield stress, that is, a critical stress for transition from solid-like to fluid-like behaviour [,3 5] Moreover most of these pasty materials have shown to exhibit thixotropic behaviour, ie, their viscosity decreases with time, which implies structure breakup [] Such a paste is commercial toothpaste, the subject of study of the present work Knowing the yield stress and the thixotropic behaviour of pastes is extremely important from the rheological point of view in order to develop a rheological constitutive equation capable of predicting correctly their flow behaviour in well defined flows A broad range of products in industry are in paste form or used in paste form during processing, including bricks, tiles, catalyst pellets, drugs, dental materials, pencil leads, and poly-tetra-fluoroethylene (PTFE), among others [] Many fabricating processes are available to handle pastes with paste extrusion being dominant among them During extrusion, the paste is forced through the die by a pressure difference As the paste comes out of the die land it takes the shape of the die [6] It is important to be able to predict the flow behaviour of pastes in such flows related to real processing by using constitutive equations developed from fundamental rheological measurements Such flow simulations provide additional testing for the validation of these rheological constitutive equations It is common to introduce toothpaste as an example for Bingham plastic behaviour [7] Barnes [8] listed toothpaste as a shear-thinning material and assumed that its viscosity changes almost instantaneously However, toothpaste is usually considered as a thixotropic material There are various additives to improve the rheological properties of toothpaste aiming at increasing its shear-thinning and shorten its thixotropy [9] Thixotropic behaviour of toothpaste results in a delay between instantaneous rheological behaviour and its equilibrium shear-thinning flow curve The effects of toothpaste thixotropy on its flow regime in a largegap Couette geometry have been studied in a recent paper by Potanin [] His simulations, which are verified by experimental data in simple shear, have shown that the calculated torque is greater for the thixotropic model compared to an equilibrium model His work on toothpaste and similar works on thixotropic materials [] clearly show the importance of considering thixotropy in flow analysis, and therefore the present work follows those guidelines In the present paper, we are interested in studying first the rheological behaviour of commercial toothpaste using simple rheological flows Based on these experimental results, a rheological constitutive equation is developed Furthermore, capillary extrusion experiments are performed using capillary dies of different geometrical characteristics such as die diameter, D, length-todiameter ratio, L/D, and contraction angle, 2a Using the constitutive equation developed, capillary flow simulations are performed /$ - see front matter Ó 20 Elsevier BV All rights reserved doi:6/jjnnfm208004

2 HA Ardakani et al / Journal of Non-Newtonian Fluid Mechanics 66 (20) to predict the pressure drop in the various geometries and examine in detail the flow behaviour inside the dies with emphasis on structure evolution 2 Experimental A commercial toothpaste has been chosen for this study (Colgate) Because of the homogeneous structure of toothpaste, it is expected that liquid migration becomes negligible compared to the case of granular materials with a liquid medium, an assumption of the present work [2] The rheological properties of toothpaste, such as yield stress and thixotropy, have been studied using a rotational rheometer (Kinexus, Malvern) equipped with a cup-and-bob geometry Most experiments were performed using the cup-and-bob geometry of large gap (2 mm) Although the handling of material is harder for the cup-and-bob geometry, the advantage of this geometry over other geometries is that it provides constant shear rate, which is critical for maintaining uniform structure during the pre-shearing process and also during the steady and transient experiments However, experiments with a cup-and-bob geometry having a small gap ( mm) have been performed to check for wall slip effects (gap dependence of the flow curve) A few experiments were also performed with parallel plates (20 mm diameter and mm gap) to verify the consistency of measured rheological properties In order to obtain consistent and reproducible results, paste samples are pre-sheared (a common practice for thixotropic fluids) for a certain period of time (5 min) under a certain shear rate (50 s ) and left to rest for 2 h to recover their structure All experiments have been carried out at room temperature (23 C) The density of this particular toothpaste has been measured and found to be 300 kg/m 3 2 Equilibrium flow curve First, steady shear experiments were performed in order to determine the equilibrium viscosity (steady-state) After preshearing, steady shear experiments were performed at specified shear rate values After increasing the shear rate to a new value, a sudden increase in shear stress was observed due to thixotropy However, due to continuous structural breakdown, the resisting shear stress slowly decreased to its equilibrium value This equilibrium shear stress value corresponds to a certain value of structure that corresponds to the shear rate of pre-shearing This procedure was continued until enough points are obtained to define the complete equilibrium flow curve Fig shows the steady values of the measured shear stress as a function of the shear rate Different sets of steady-state experiments with different geometries (parallel plates, small- and large-gap cup-and-bob) have been carried out to check for slip effects It was noted that for all these three geometries (including different gaps), the results fall on the same curve at least for high shear rates, which means that slip effects are negligible, consistent with previous reports [] For shear rates beyond about 0 s, the experimental measurements were shown to be reproducible and consistent over three decades of shear rate Below 0 s, the data has shown some inconsistency, which might be due to the occurrence of shear banding It has been reported in the literature that for many materials at low shear rates, experimental measurements are not reproducible due to occurrence of shear banding, flow bifurcation and shear rejuvenation [3] The flow behaviour of some materials with shear banding behaviour has been studied with flow visualisation techniques It has been shown that for the parallel-plate geometry under some critical shear rate there is no homogeneous flow, and shear rate is localized in a plane between the plates, meanwhile the rest of the material is in solid form [3,4] Due to shear banding the flow curve exhibits a minimum and for each nominal shear stress there are at least two existing true shear rates In simple words, flow below a certain shear rate (minimum of the flow curve) is always unstable As a result of this instability it is impossible to report any equilibrium data below the critical shear rate To show the existence of shear banding, creep tests have been performed at different shear stress values at the low end of the flow curve, typically for rates below 0 s, where the data are not reproducible The same pre-shearing protocol discussed in the experimental section has been applied to ascertain that all the samples are at the same initial structural state Fig 2 shows this flow bifurcation For all shear stress values below a critical value (in this case about 20 Pa), a continuous decrease of shear rate is observed Meanwhile, for shear stress beyond that critical stress, the shear rate attains its steady-state value Since it takes time for the flow to develop or to reach cessation, the stability analysis is affected by the duration of the experiment Therefore, it is not simple to define a precise value for critical stress There is a window of shear rate values (roughly below 0 s ) that are shown not to be reachable by any shear stress value For example, for a shear stress value of 20 Pa, a steady shear rate 000 s does not last for more than 200 s, and flow terminates as the degree of structure is increased For some other higher values of wall shear stress, such as 25 and 30 Pa (Fig 2), apparent steady-state shear rates are obtained, which are plotted in Fig and labelled as creep test results Comparing these data with data obtained from parallel plates large-gap cup-and-bob small-gap cup-and-bob structural parameter model creep test results Shear Rate, γ (s - ) τ = Pa Shear Rate, γ (s - ) Fig The equilibrium flow curve and the fitted thixotropic-structural model Time, t (s) Fig 2 Creep tests at small values of shear stress to show shear banding or flow bifurcation

3 264 HA Ardakani et al / Journal of Non-Newtonian Fluid Mechanics 66 (20) shear-rate sweep experiments shows that the shear rates at the same shear-stress values are not necessarily the same due to the occurrence of shear banding For all practical purposes, the low shear stress part of the flow curve can be described by the existence of a yield stress, s y As pointed out by Papanastasiou [5], the downturn of the experimental data at low shear rates can be obtained by multiplying the yield stress with the following exponential function of the shear rate _c: s y! s y ½ expð m _cþš; ðþ rest γ = 2 s - γ = 5 s - γ = s - γ = 30 s - structural model (rest) structural model (γ = 30 s - ) where m is a stress-growth exponent with units of time Choosing a suitable value for m makes a yield-stress curve pass through the low-shear-rate data (see Fig ) Although, the experimental data for shear rates below 0 s are shown to be irreproducible due to the occurrence of shear banding/flow bifurcation, the simple yield stress model, modified with the Papanastasiou method, can be used to roughly capture the apparent steady-state behaviour of the material even at these small shear rates 22 Time dependency It is also necessary to characterise the time dependency and thixotropic behaviour of the toothpaste To do so, various startup experiments are carried out In each experiment, the rheometer is first loaded with a sample After pre-shearing and rest to recover the structure, the shear rate is suddenly increased from zero to a certain value As a result the shear stress increases from zero to a high value rapidly and subsequently gradually decreases with time until it reaches its steady value Essentially a shear stress overshoot is obtained in each case, also indicating time dependency Even though the thixotropic behaviour is amplified in the case of sudden step shear rate from rest, similar, although milder, thixotropic behaviour is observed when the shear rate suddenly increases from a lower to a higher value Therefore, experiments have been carried out using the same procedure with sudden shear-rate changes from a lower to a higher value The same procedure has been followed in order to design a multistep shear-rate test for model evaluation Two main methods have been suggested in the literature to study thixotropy The first method, referred to as the iso-structure flow curve, is based on the idea of measuring the flow curve of the material at a certain level of its structure [6] This method requires the determination of the flow curve with respect to a certain level of shear rate (reference rate) Each point in the flow curve is determined after the material is pre-sheared at the reference rate For a purely thixotropic material in steady shear after obtaining steady-state at the reference pre-shearing rate, application of a new higher shear rate will suddenly increase the shear-stress level The first measured shear-stress value corresponds to the iso-structure flow curve for the previous shear-rate value, ie, time needed for structural changes However, as shear continues at this higher shear rate, the structure gradually starts breaking down and the shear stress eventually reaches a new lower steady-state value, which represents the equilibrium structure for this shear-rate value Fig 3 shows flow curves with various levels of the reference shear rate We observe that increasing the pre-shearing shear rate causes the flow curve to shift to small shear stress values due to the breakdown of the structure The data also show that the sensitivity of the structure slightly increases in some cases with shear rate, which the model cannot capture adequately Two possible reasons are that (i) almost no material shows purely thixotropic behaviour and typically thixotropic behaviour is combined with viscoelasticity Therefore, the overshoot due to viscoelasticity interferes with the sudden increase of the stress due to thixotropy Shear Rate, γ (s - ) Fig 3 Iso-structure flow curves at the new shear-rate value, although these two effects can be distinguished as discussed below; and (ii) technical limitations due to the fact that it takes time for the rheometer to collect the first point after changing to the new shear rate Our experiments are at relatively small shear-rate values, and the latter reason might not be a problem Another method for the characterisation of thixotropy is the observation of the material s rheological response to any change in the flow regime like the shear rate In this work, shear-rate step tests have been performed It was observed that as the shear rate suddenly increased, the shear stress started from zero and quickly rose to a higher value This initial part is related to the viscoelastic response of the material After this viscoelastic response, thixotropic behaviour can be observed Fig 4 shows both viscoelastic and thixotropic effects The viscoelastic effect can only be tracked at low shear rates where the thixotropic effect is limited Therefore for model development, the elastic response is neglected and the material is assumed to be purely thixotropic Figs 5 and 6 show two different sets of shear-rate jump tests; the first from a material at rest to a high shear rate and the second from an equilibrium state of shear rate at 5 s to a new higher shear rate As can be seen the overall description of the changes are satisfactory Deviations at short times are related to the viscoelastic nature of the material; as such effects are neglected such deviations are expected Time, t (s) step shear rate γ = 0 s - γ = s - γ = 5 s - structural model (from rest) Fig 4 Rheological response of toothpaste to different values of shear rate

4 HA Ardakani et al / Journal of Non-Newtonian Fluid Mechanics 66 (20) Time, t (s) 23 Slip at the wall γ = 25 s - γ = 55 s - γ = 0 s - structural model Fig 5 Thixotropic behaviour of paste in three start-up tests at different shear rates Time, t (s) Experiments were performed with capillary dies having different diameters to detect effects of slip Fig 7 depicts the results, where a diameter dependence of the flow curve is obvious γ = 40 s - γ = 30 s - γ = 20 s - γ = s - structural model Fig 6 Thixotropic behaviour of toothpaste at various shear-rate jumps from an initial shear rate of 5 s Moreover the flow curves determined from capillary flow deviate significantly from the one obtained from the parallel-plate rheometer (data of Fig ) The Mooney analysis has been used to determine the slip velocity of toothpaste as a function of the wall shear stress [7 9] This analysis gives: _c A ¼ _c A;s þ 8V s D ; where _c A is the apparent shear rate, _c A;s is the slip-corrected apparent shear rate, which corresponds to the parallel-plate flow curve of Fig 7, and V s is the slip velocity Fig 8 plots the slip velocity versus the wall shear stress, s w A linear relationship for slip is obtained, which can be written as: V s ¼ bs w ; where b is the slip coefficient Its value was found to be 8 5 m/ (Pa s) For the flow conditions used in the capillary experiments and the rheological properties at hand, this value amounts to massive slip, as will become evident in the simulations further down It should be noted that our previous papers on PTFE paste extrusion [20 22] also showed severe slip occurring at the die walls, which is the norm rather than the exception for pastes in die flows 24 Capillary extrusion results Capillary extrusion experiments were also performed using a piston-driven constant-speed capillary rheometer (Bohlin RH 2000) Various dies were used in order to study the effects of geometrical parameters of the die on the capillary flow (essentially pressure drop) of toothpaste Important parameters include the die diameter, D, the length-to-diameter ratio, L/D, the reduction ratio, RR ðd 2 b =D2 Þ, where D b is the barrel diameter, and the entrance contraction angle, 2a Table lists all capillary dies used in this study along with their geometrical characteristics The barrel diameter, D b, is 5 mm These results will be presented along with the flow simulations below 3 Constitutive modeling A constitutive model for a typical paste should possess at least two elements First, a yield stress which is the stress below no flow is observed Then, thixotropy should be included through an appropriate kinetic equation involving a structure parameter It has been claimed by many authors that the existence of yield stress ð2þ ð3þ D=085 mm D=27 mm D=223 mm large gap cup-and-bob Slip Velocity, V s (m/s) RR=3 RR=40 RR=45 slip model V s = 8x -5 τ w (m/s) Shear Rate, γ Α (s - ) Fig 7 The effect of die diameter, D, on the flow curve of toothpaste Slip effects cause the diameter dependence Wall Shear Stress, τ w (Pa) Fig 8 Slip velocity of toothpaste as a function of wall shear stress at 23 C

5 266 HA Ardakani et al / Journal of Non-Newtonian Fluid Mechanics 66 (20) Table List of capillary dies used in the present work The barrel diameter, D b, is 5 mm Die number RR L/D 2a D # ( ) ( ) ( ) (mm) Table 2 Fitted parameters for the structural model k ( ) k 2 (s ) s y (Pa) g (Pa s) m (s) indicates the existence of structure that requires initial energy to decay and let the material flow, which implies that flow might alter the structure and thus thixotropy is expected [6,23,24] To represent toothpaste rheology, the proposed model includes two terms and can be written as follows: s ¼ s y ð expð m _cþþþð þ nþg _c: The first term represents the yield stress behaviour of the material It is often assumed that the yield stress is related to the structure of the material through a structural parameter, n However, in the present case the apparent yield stress shows a weak dependence on n, and we have neglected this dependence Papanastasiou s exponential modification (Eq ()) is applied on this term for fitting the data of Fig at the lower shear-rate range The second term includes the viscous part of the response, and it is a function of the structural parameter n, which can be described by a kinetic equation The rate of change in structure can be described by []: dn dt ¼ k _ cn þ k 2 ð nþ: This way the structural parameter n is a normalized quantity that varies between 0 and and indicates the integrity of the network (n = 0: no network or structure; n = : fully developed network or ð4þ ð5þ structure) The first term on the RHS of Eq (5) indicates breakdown of the network due to material deformation; the second term is responsible for build-up of the network with a time constant /k 2 associated to it (here found to be equal to s) Structure formation occurs due to Brownian motion and is partially due to imposition of shear rate [6] According to the above Eq (5), the shear contribution in structure build-up is neglected, and the rate of formation is set proportional to ( n) [24,25] It has been assumed that the shear rate may break down structure The rate of structural break-down is proportional to the shear rate and also to the degree of structure According to this kinetic equation, the structural parameter approaches a steady-state value, n eq, at a given value of shear rate, _c, that is: k 2 n eq ¼ : k _c þ k 2 Therefore the equilibrium flow curve is given by: s eq ¼ s y ð expð m _cþþþð þ n eq Þg _c: Then the equilibrium apparent viscosity g eq is given by: g eq ¼ s y _c ð expð m _cþþþð þ n eq Þg : Optimized values for the proposed models are summarized in Table 2 The ratio k /k 2 and m were calculated by fitting Eq (7) to the data of Fig by writing n eq ¼ =½ þðk =k 2 Þ _cš from Eq (6) Subsequently, using the data of Figs 5 and 6, the individual values of k and k 2 were calculated Each flow curve at a given level of the structural parameter (or apparent viscosity) should look like a Bingham model and cross the equilibrium flow curve at its corresponding initial shear rate (see Fig 3) Figs 5 and 6 show two different sets of shear-rate jump tests; the first from a material at rest and the second from an equilibrium state of shear rate at 5 s The thixotropic behaviour of the material has been adequately captured by the proposed model 4 Governing equations We consider the conservation equations of mass and momentum for incompressible fluids under isothermal, creeping, steady flow conditions These are written as [7]: r u ¼ 0; 0 ¼ rp þ r s; ðþ ð6þ ð7þ ð8þ ð9þ n E D u n =0, u t =βτ w, ξ n =0 u z =f(r) u r =0 ξ= A R res u r = ξ n =τ rz =0 n C S F R z =0 r u r =ξn=0 0 z B p =0 L res L c L Fig 9 Field domain and boundary conditions for capillary flow of toothpaste

6 HA Ardakani et al / Journal of Non-Newtonian Fluid Mechanics 66 (20) where u is the velocity vector, p is the pressure and s is the extra stress tensor The viscous stresses are given for inelastic incompressible fluids with structure by the relation [7]: s ¼ ngðj _cjþ _c; ðþ where gðj_cjþ is the apparent viscosity of Eq (8), in which the shear rate _c is replaced by the magnitude j _cj of the rate-of-strain tensor _c ¼ ru þ ru T, which is given by: rffiffiffiffiffiffiffiffiffi j_cj ¼ II _c ¼ 2 2 ð _c : =2 _cþ ; ð2þ where II _c is the second invariant of _c II _c ¼ð_c : _cþ ¼ X X _c ij _c ij ; i j Thus, the apparent viscosity is written as: ð3þ gðj _cj; nþ ¼ s y j_cj ð expð mj _cjþþ þ ð þ nþg : ð4þ In the above, the apparent viscosity is a function of n, which obeys the kinetic Eq (5) In general flows, Eq (5) becomes the convectivetransport þ u rn ¼ k j_cjn þ k 2 ð nþ: ð5þ For steady-state conditions, on/ot = 0 The above rheological model (Eqs () and (4)) is introduced into the conservation of momentum (Eq ()) and closes the system of equations Boundary conditions are necessary for the solution of the above system of equations Fig 9 shows the solution domain and boundary conditions for the tapered contraction geometry of capillary flow It should be noted that for the structural parameter, n, the entry boundary condition is n = (ie, it is assumed that the network is fully established) ¼ 0at the walls, where n is the unit outward normal vector The last Fig Yielded/unyielded (shaded) regions in toothpaste extrusion at 23 C for three different apparent shear rates Incompressible flow with slip at the wall for Die #3, L/ D = 20, 2a =45, RR = 3

7 268 HA Ardakani et al / Journal of Non-Newtonian Fluid Mechanics 66 (20) boundary condition means that the structural parameter is free to take its values at the wall such that normal to the wall there are no changes in n Simulations with different initial n-values have shown that the results are not affected much, a finding which has been observed experimentally for thixotropic fluids [26,27] Values of n = 0 at entry affected only the n-values in the reservoir, not in the die, and only at the upper end of apparent shear rates (_c A > 2000 s ) Because of symmetry only one half of the flow domain is considered, as was done in our previous works [2,22] All lengths are scaled by the die radius R, all velocities by the average velocity U at the die exit, all pressures and stresses by g (U/R) 5 Method of solution The numerical solution is obtained with the Finite Element Method (FEM), employing as primary variables the two velocities, pressure, and structural parameter (u v p n formulation) Noting the similarity of the kinetic equation with the energy equation, we have substituted in our previous formulation the temperature, T, with the structural parameter, n [2,22] We use Lagrangian quadrilateral elements with biquadratic interpolation for the velocities and the structural parameter, and bilinear interpolation for the pressures The solution process is similar to that employed by Mitsoulis and Hatzikiriakos [2,22], using the same grids as before As in the previous results with PTFE paste modeling, the solution process was based on incrementing the apparent shear rate from low to high values Due to the massive slip at the wall present in both cases (PTFE and toothpaste extrusion), convergence was fast and easy and could be obtained within few iterations to satisfy criteria for the norm-of-the-error and the norm-ofthe-residuals below 5 6 Flow simulations 6 Flow field The flow simulations have been performed with the above conservation, constitutive and convective-transport equations, their boundary conditions, and the parameters of Table 2 Reference results are given for the highest apparent shear rate (_c A ¼ 4Q=pR 3 ¼ 4U=R = 2560 s ) and one die design (Die #3, 2a =45, Fig Contours of the structural parameter n in toothpaste extrusion at 23 C for three different apparent shear rates Incompressible flow with slip at the wall for Die #3, L/ D = 20, 2a =45, RR = 3

8 HA Ardakani et al / Journal of Non-Newtonian Fluid Mechanics 66 (20) L/D = 20) The purpose is to find out the effect of the structural parameter n on the results, and how it affects the relative importance of the various forces at play in toothpaste extrusion Results are shown as contours of two important variables of the model, namely, the yield stress, s y, and the structural parameter, n First, Fig shows the yielded/unyielded (shaded) zones for different apparent shear rates The dividing line between the two regions is the contour of the magnitude of the stress tensor s =s y For the lowest apparent shear rate of 256 s, the yielded region is contained in the tapered region just before the die entry At intermediate shear rates, this region is extended inside the die, leaving a small unyielded plug in the die core At the highest apparent shear rate, the yielded region is further extended both upstream and downstream to encompass the whole tapered region and die The corresponding results for contours of the structural parameter n are given in Fig It becomes apparent that structure changes are small, and occur mainly in the die at high levels of shear rate This is not surprising because the flow is fast as evidenced by the inverse of the apparent shear rates, and no time is given for the material to break down its structure It should be noted that n starts with at entry (n not being or 0 at the wall when slip is allowed), and then most of the changes occur in a cone near the die entry and inside the die The effect of L/D on the structural parameter for a given geometry (2a =45 ) and the highest apparent flow rate (_c A = 2560 s )is shown in Fig 3 A longer die allows the material more time to break down its network and to reach lower values of n The effect of the apparent shear rate for a given geometry (2a =45 ) is shown in Fig 4 Increasing the apparent shear rate decreases the n-parameter, hence it leads to network breakdown, but again these changes are mild, hardly reaching n = 088 Again the effects are more significant at high shear rates and at the die walls At this point it is instructive to examine what causes this small change in n The way to study this is to keep the ratio k / k 2 = 002 s = constant (since this is dictated by the steady-state flow curve) and change the individual values of k and k 2 This effect on the structural parameter for a given geometry (2a =45 ) and apparent flow rate ( _c A = 2560 s ) is shown in Fig 5 A larger k 2 parameter (k 2 = s, hence a time constant /k 2 = s) brings down the structural parameter on the die wall to almost zero, while a smaller k 2 parameter (k 2 = 00 s, hence a time constant /k 2 = 0 s) keeps the structural parameter everywhere close to Thus, thixotropic effects are well captured by the model but for the present toothpaste and its value of k 2 = 0 s the effect on the breakdown of the network is small and the values of n are between and 08 in the range of experiments 62 Effect of flow and design parameters The effect of the die entrance angle on the structural parameter n is depicted in Fig 2, where the axial distribution is presented along the centreline and the wall for the highest apparent shear rate _c A = 2560 s We observe that for all three die designs the structural parameter decreases continuously from the inlet to the die exit on the die wall, while it remains constant within the die along the centreline where there is no shearing The continuous linear decrease at the die wall is expected from the kinetic Eq (5) However, all the values are close to, meaning that the structure is mildly affected by the flow kinematics The angles have a small effect on n These effects originate from differences in the extensional components of the flow conditions at the entry, which are functions of the entrance angle Structural Parameter, ξ (-) =2560 s -, L/D=20, RR=3 2α = 30 o 2α = 45 o 2α = 60 o die wall 63 Comparison with experiments As mentioned in the experimental part, experiments were performed in capillary dies of different designs (see Table ) The simulations have been carried out for all dies and shear rates and below they are compared with experimental results The first geometrical property studied is L/D, while 2a =45 and RR = 3 are kept constant Because of the smaller diameter of the die land compared to the die entrance, it is assumed that the main pressure drop occurs in the die land Fig 6 shows the extrusion pressure for three different L/D ratios of 0,, and 20 For the case of L/D = 0, we have used in the simulations a small value L/D = 02 For the other L/D ratios, a linear relation between L/D and pressure Structural Parameter, ξ (-) =2560 s -, 2α=45 o, RR=3 tapered wall die wall L/D = 5 L/D = L/D = Axial Distance, z / R Fig 2 The effect of die entrance angle, 2a, on the structural parameter, n Axial distribution of n along the centreline and the wall for toothpaste extrusion at 23 C Incompressible flow with slip at the wall for _c A = 2560 s, L/D = 20, RR = Axial Distance, z / R Fig 3 The effect of die length, L/D, on the structural parameter, n Axial distribution of n along the centreline and the wall for toothpaste extrusion at 23 C Incompressible flow with slip at the wall for _c A = 2560 s,2a =45, RR = 3

9 270 HA Ardakani et al / Journal of Non-Newtonian Fluid Mechanics 66 (20) Structural Parameter, ξ (-) Die #3, 2α=45 o, L/D=20, RR=3 tapered wall = 256 s - = 2 s - = 3 s - = 27 s - = 2560 s - die wall Pressure, P (kpa) L/D = 20 L/D = L/D = 0 L/D = 20 (sim) L/D = (sim) L/D = 02 (sim) 0 2α = 45 o RR = 3 Apparent Shear Rate, γ α (s - ) Axial Distance, z / R Fig 4 The effect of apparent shear rate, _c A; on the structural parameter, n Axial distribution of n along the centreline and the wall for toothpaste extrusion at 23 C Incompressible flow with slip at the wall for 2a =45, L/D = 20, RR = 3 Structural Parameter, ξ (-) Die #3, 2α=45 o, L/D=20, RR=3 tapered wall k /k 2 =002 s k 2 = s - k 2 = 0 s - k 2 = 00 s Axial Distance, z / R die wall die wall Fig 5 The effect of k and k 2 on the structural parameter, n, keeping the same ratio k /k 2 = 002 s Axial distribution of n along the centreline and the wall for toothpaste extrusion at 23 C Incompressible flow with slip at the wall for 2a =45, L/D = 20, RR = 3 drop is expected However, as illustrated in Fig 6, the pressure drop for L/D = 20 is much greater than the pressure drop for L/ D = The origin of this phenomenon could be the formation of an unyielded area inside the die land, which grows because of structural formation as the material relaxes through the die land The growth of the unyielded area increases the shear rate inside the yielded area Therefore, the shear rate increases along the die land wall and causes higher shear stresses The numerical simulations with the structural parameter model show that the general trends are captured by the model, although quantitatively the L/D = results are over-predicted The simulations show that doubling the die length leads to doubling the pressure in the system, while experimentally there is a quadrupling effect! Pressure, P (kpa) Pressure, P (kpa) Fig 6 Effect of L/D on extrusion pressure RR = 45 RR = 39 RR = 3 RR = 45 (sim) RR = 39 (sim) RR = 3 (sim) L/D = 20 2α = 45 o 0 00 Apparent Shear Rate, γ α (s - ) Fig 7 Effect of reduction ratio RR on extrusion pressure 2α = 30 o 2α = 45 o 2α = 60 o 2α = 80 o 2α = 30 o (sim) 2α = 45 o (sim) 2α = 60 o (sim) 0 L/D = 20 RR = 3 00 Apparent Shear Rate, γ Α (s - ) Fig 8 Effect of contraction angle 2a on extrusion pressure 000 The second geometrical property studied is RR, while 2a = 45 and L/D = 20 are kept constant Typically the material would tolerate higher shear rates through the conical zone when the geometry has a high reduction ratio As it can be seen in Fig 7,

10 HA Ardakani et al / Journal of Non-Newtonian Fluid Mechanics 66 (20) the pressure drop is almost the same for the two lower reduction ratios of 45 and 39, and slightly lower for the highest reduction ratio of 3 This can be explained by a greater structure breakdown for this case, which results in a lower apparent viscosity The numerical simulations are in general agreement with these findings but they distinguish more clearly among the three cases Thus the model can adequately take into account the effect of RR on toothpaste extrusion The third geometrical property studied is 2a, while RR = 3 and L/D = 20 are kept constant The extrusion pressure could be affected by the die entrance angle Since most of the pressure drop occurs in the die land, and the pressure drop in the conical zone is almost negligible, it is hard to detect the effect of die contraction angle on the pressure drop However, Fig 8 shows that for lower contraction angles, the extrusion pressure is slightly higher due to an increased effect of shear In the case of polymer melts, it has been observed that the extrusion pressure is independent of the die entrance angle for angles greater than about 30 The same conclusion can be made for thixotropic toothpaste The numerical simulations, in general, corroborate these findings as they show that for all cases the pressure drop is more or less the same, except for the lower angle of 30, where the results are slightly higher Thus the model underestimates the pressures for the lower angles, but can differentiate the effect of the contraction angle 2a on toothpaste extrusion 7 Conclusions Experiments and simulations were performed in this work for commercial toothpaste The experiments showed the existence of a yield stress and time-dependent phenomena associated with thixotropy, hence structure Severe slip at the wall was found to occur in capillary flow with different die designs A simple thixotropic model was formulated and the parameters of the model were found by matching experimental data The model accounts for yield stress and structure through a kinetic-type convectivetransport equation Simulations based on the proposed model for capillary flow showed that the model essentially captures the pressure drop in the system for the majority of the cases investigated Slip in massive and responsible for the lower-than-expected pressures at elevated shear rates reaching 2500 s The simulations also provide details about structure evolution in the die and show that due to slip and fast flows, structure breakdown is not strong Some discrepancies between theory and experiments may be attributed to possible viscoelastic effects not accounted for in the model Such work to include viscoelasticity together with thixotropy is currently under way by the authors Acknowledgements Financial assistance from the Natural Sciences and Engineering Research Council (NSERC) of Canada and the programme PEBE for basic research from NTUA are gratefully acknowledged References [] P Coussot, Rheometry of Pastes, Suspensions, and Granular Materials: Applications in Industry and Environment, Wiley, New York, 2005 [2] HA Barnes, A Handbook of Elementary Rheology, University of Wales Institute of Non-Newtonian Fluid Mechanics, Aberystwyth, Wales, 2000 [3] EC Bingham, Plasticity and elasticity, J Franklin Inst 97 (924) 99 5 [4] AW Blair, AL Prince, Influence of organic matter on crop yield and on carbon nitrogen ratio and nitrate formation in the soil, Soil Sci 35 (933) [5] HA Barnes, The myth of yield stress fluids, Prog Trends Rheol V (998) [6] J Benbow, J Bridgwater, Paste Flow and Extrusion, Clarendon Press, Oxford University Press, Oxford, 993 [7] RI Tanner, Engineering Rheology, second ed, Oxford University Press, Oxford, 2000 [8] HA Barnes, The yield stress a review or panta rei everything flows? 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