Rheology of a dilute suspension of spheres in a viscoelastic fluid under LAOS

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1 Rheology of a dilute suspension of spheres in a viscoelastic fluid under LAOS GAETANO D AVINO 1, MARTIEN A. HULSEN 2, FRANCESCO GRECO 3 AND PIER LUCA MAFFETTONE 1 1 Dipartimento di Ingegneria Chimica, Università degli Studi di Napoli Federico II, Piazzale Tecchio 8, 8125 Napoli, Italia 2 Department of Mechanical Engineering, Eindhoven University of Technology, PO Box 513, 56 MB Eindhoven, The Netherlands 3 Istituto di Ricerche sulla Combustione, IRC, CNR, Piazzale Tecchio 8, 8125 Napoli, Italia Abstract: Dilute suspensions of spheres in a viscoelastic liquid under oscillatory shear flow are studied via numerical simulations. In linear viscoelastic regime, the 3D finite element simulation results agree with the analytical results from Palierne (199). The analysis is extended to Large Amplitude Oscillations (LAOS). The rotation rate of the sphere as well as the bulk rheology are investigated. The moduli increase when particles are added into the fluid, in agreement with experimental findings. In the nonlinear range both moduli are found to be decreasing functions of the forcing amplitude. The non-linear response of the material is also illustrated by performing a Fourier analysis of the bulk shear stress signal which shows the appearance of higher harmonic contributions. Keywords: Suspensions, Rheology, LAOS, Oscillatory shear flow, Simulations 1. INTRODUCTION Suspensions of particles in liquids are of significant industrial interest. Polymers with fillers, ceramic pastes, biomedical materials, food, cosmetics, and detergents are examples of typical applications of such materials. New applications with nanosized fillers are also emerging, with the suspending fluid being in most cases a nonnewtonian liquid (e.g., nanocomposite corresponding author. pierluca.maffettone@unina.it 1

2 polymer melts). It is then quite obvious that understanding of the rheological and mechanical properties of these materials plays a crucial role in the processing stage. Suspensions have been intensively studied in the past. The first study on suspensions of spheres dates back to Einstein (196) who gave the prediction of the suspension viscosity under the hypotheses of Newtonian, inertialess fluid and dilute system. Since then, a great deal of works both experimental and theoretical was focused to the characterization of suspensions with rigid or deformable particles in Newtonian liquids, and even beyond the dilute limit (see e.g., Brenner 1972; Larson 1998). The problem of viscoelastic suspending liquid is in general less studied, and mainly on the experimental side (e.g., Barnes 23). From the theoretical point of view, in order to handle viscoelastic suspensions of practical interest, realistic non-newtonian constitutive equations for the suspending liquid need to be taken into account. Of course, due to the mathematical complexity of the equations involved, use of numerical simulation to tackle specific flow problems is generally required. As it is well known, a typical external flow frequently used in experiments is the oscillatory shear flow. According to the magnitude of the imposed strain, two different regimes can be distinguished. If a small forcing amplitude is applied (Small Amplitude Oscillatory Shear, SAOS), the material exhibits the so called linear viscoelastic behaviour, where the rheological quantities are linear function of the strain. By increasing the forcing amplitude (Large Amplitude Oscillatory Shear, LAOS), a non-linear relationship between stress and strain occurs. For an inertialess emulsion with non-newtonian fluids, Palierne (199) derived an analytical expression for the storage (G') and loss (G'') moduli in SAOS regime. A solid suspension with a viscoelastic matrix can be considered as a limiting case of Palierne results for the inner phase viscosity going to infinity. However, we are not aware of any prediction on the moduli of a solid viscoelastic suspension in LAOS regime. Aim of this work is to analyze dilute, inertialess, non-brownian suspensions of rigid spheres in a viscoelastic medium subjected to oscillatory shear flows, including the response to large forcing amplitudes. Simple but realistic viscoelastic constitutive equations are chosen to properly model non-linear phenomena. The analysis is carried out through a cell model for the dilute suspension, by means of 3D numerical simulations in a cell. Once the velocity and pressure fields within the cell are determined, the bulk shear stress (i.e., the stress obtained by considering the suspension as a whole) is calculated. This latter quantity allows the predictions of both moduli. The analytical prediction from Palierne (199) are used to 2

3 validate the numerical code. Nonlinear results are then obtained when large forcing amplitudes are used. Finally, a Fourier analysis is carried out in order to show the arising of higher harmonic contributions when the forcing amplitude is sufficiently large. 2. GOVERNING EQUATIONS The dilute suspension of spherical particles in a viscoelastic medium is described within the framework of a cell model (Brenner, 1972). Suspension diluteness implies that particle do not feel each other, and thus the suspension can be subdivided in regions (cells) each one containing a single particle. In this view, solving the hydrodynamic problem in a single cell is all what is needed to end up with predictions for the dilute suspension as a whole. The imposed flow field is applied at the border of the cell. Under the assumption of incompressibility, negligible inertia, buoyancy free and isothermal conditions, the hydrodynamic problem of a single sphere in an imposed flow field is governed by the following continuity (mass balance) and momentum balance equations: v = p + τ = (1) where p is the pressure, v is the velocity, and τ is the constitutive extra stress tensor. In this work, the following two constitutive equations for viscoelastic liquids are considered: λ τ+ τ = 2η D (2) αλ λ τ+ τ τ+ τ = 2ηD (3) η where τ indicates the Upper Convected (UC) time derivative of τ, λ is the relaxation time, 2D = v + v T is the rate of deformation tensor and η is the zero-shear viscosity. Equation (2) is the UC Maxwell model and Eq. (3) is the Giesekus model. The latter model contains an additional nondimensional parameter, α, that affects the shear thinning and the elongational thickening. Boundary conditions are assigned at the cell boundaries and on the sphere surface. The farfield condition corresponds to the imposed shear flow: p= p B v = v r B B (4) 3

4 with p B a constant pressure, v B the imposed velocity gradient (constant in space), and r B the position vector from the sphere center to the cell border. SAOS and LAOS regimes are achieved by imposing a co-sinusoidal deformation on the border: γ (t) = γ cos(f t) (5) where γ is the forcing strain amplitude [-] and f is the frequency [t -1 ]. From Eq. (5) it follows that the imposed velocity field is a time-dependent, sinusoidal shear flow. Calling x the flow direction, y the velocity gradient direction, and z the vorticity direction, the imposed velocity gradient is given by: vb xy = γfsin(ft) = γsin(ft) (6) the other components being nil. In Eq. (6), γ = γ f [t -1 ] is the imposed shear rate. No slip boundary conditions are imposed at the sphere surface r = R. We assume that the sphere is torque-free, or "freely rotating" (Brunn, 1976; Leal, 1979), i.e. its rotation is only due to the motion of the surrounding fluid. The torque-free boundary condition at r = R is: r T n da = (7) A S where A s is the sphere surface, n is the normal at the sphere surface, the tensor T is T= pi+ τ, and the integral of the local torque r T n da spans the sphere surface. Due to the symmetry of the imposed shearing flow, only the vorticity component z of Eq. (7) is nonzero. The velocity at the sphere surface is known, and can be written as: ( ) v R,t = ω (t) e z R (8) e z being the unit vector along z. The time-dependent velocity v(r,t) in Eq. (8) is at any point on the sphere surface the tangential velocity of that point. Hence, v(r,t) is everywhere tangential to the surface, and parallel to the shear plane xy. The unknown angular velocity ω(t) in Eq. (8) can then be determined through the z-component of the torque-free condition, Eq. (7). Within the cell description of the dilute system, the bulk tangential stress is obtained by volume averaging of the local tangential stress over the cell: τ xy n = τxydv V (9) Tot Vcell 4

5 with n the number of cells, and τ xy is the shear stress in the cell containing one particle only, with the flow field imposed at the cell boundaries. As usual, the decomposition is made: τ φ φ = τ dv + R µ nµ n ds V V τ (1) xy xy x y p VFcell p SP with V Fcell the volume of fluid in the cell, and V P and S P the volume and surface of the spherical particle of radius R. It can be shown that the first term on the rhs of eq. (1) and the nv surface integral are independent of the volume fraction φ (= V implies the linearity of the bulk stress in φ. P Tot ). The diluteness assumption 3. THE NUMERICAL PROCEDURE By solving eqs. (1) with either eq. (2) or (3) and boundary conditions eqs. (4, 6, 7), the pressure and the velocity, plus the angular velocity of the sphere ω are determined. The 3D flow problem is solved by the finite element method (we used the TFEM package (Hulsen, 27)) on a cubic cell containing the sphere. Due to the symmetry of the problem, only one half (z > ) of the full domain can be considered in order to optimize the computational effort. The flow field is imposed on the boundaries of the computational domain. The cell dimension was so chosen as to give cell invariant pressure and velocity fields. A mesh with tetrahedral elements was chosen, with a higher density of elements close to the sphere, where larger gradients are expected. Convergence of the results was checked by refining the mesh. The angular velocity is monitored by varying the number of elements. A mesh size of about 6 elements suffices to obtain reliable results, whereas deviations of about 1% is observed when the number of elements is decreased to 25. Stabilization techniques are used for the discretization of equations to improve numerical convergence. The momentum equation is discretized using elastic viscous stress splitting (DEVSS-G) formulation (Guénette and Fortin, 1995) combined with the streamline upwind Petrov Galerkin (SUPG) method (Brooks and Hughes, 1982) for the constitutive equation. Moreover, the log-conformation representation is also used, leading to a significant improvement of numerical stability (Fattal and Kupferman, 24; Hulsen et al., 25). The procedure implemented allows to perform transient calculations. In particular, the 5

6 momentum and continuity equations are decoupled from the constitutive equation, and an implicit stress formulation is used. In this formulation the time-discretized constitutive equation is substituted into the momentum balance in order to obtain a Stokes like system. The torque-free condition (eq. (7)) is imposed through constraints on the sphere surface, by means of Lagrange multipliers. In this way, the sphere rotation is automatically calculated by solving the augmented system of equations. At each time step, the solution vector for v, p, and G is computed, with G the projected velocity gradient (see Guénette and Fortin, 1995; Bogaerds et al., 22). Subsequently, the solution for the stress τ (in terms of the conformation tensor) is determined, through a combined Euler forward/backward formula of the constitutive equation with convection taken implicitly. A continuous quadratic interpolation for the velocity, a continuous linear interpolation for pressure and a continuous bilinear interpolation for the stress tensor and auxiliary velocity gradient are used. Finally, at each time step, we need to solve two linear nonsymmetric sparse systems. We use the parallel direct solver PARDISO (Schenk and Gartner, 24). Finally, integrals appearing in eq. (1) were calculated with fourth order quadrature formulas. 4. LINEAR VISCOELASTICITY The dynamic viscoelasticity of the dilute suspension under SAOS is of course affected by the presence of the rigid spherical inclusions. Palierne (199) determined the moduli for an emulsion of non-newtonian liquids in the linear viscoelastic regime. The solid particle suspension is recovered as a special case, and his prediction for the complex modulus, G* ( f) = G' ( f) + ig'' ( f) is: ( ) ( f) * G f 1 3/2 G * + φ = 1 φ (11) with * G the complex modulus of the suspending liquid. Equation (11) states that the complex modulus of the suspension is merely a vertical shift of that of the pure liquid. Once expanded to first order in φ, eq. (11) gives in terms of moduli: ( ) G' f /G' (f) 1 2 ( ) 5 = + φ 5 = + φ G" f /G'' (f) 1 2 (12) 6

7 One can note that the normalized storage and loss moduli are independent of frequency forcing. Both moduli are shifted by the same factor, and the equation for G is a generalization of the Einstein result to the linear viscoelastic case. We now validate our numerical procedure by comparing the results of the simulations with Palierne predictions. The linear viscoelasticity condition is satisfied by choosing a sufficienly small amplitude of the forcing (i.e. γ = γ f, see Eq. (6)). In our simulations, γ was chosen 1-3 s -1 : this value was verified to be sufficiently small so that linear regime was attained. It should be pointed out that in linear viscoelasticity the choice of the viscoelastic model, Eq. (2) or (3), does not affect the moduli because Maxwell behaviour is always recovered in such a regime. Figure 1 shows simulated bulk values for G' and G'' (symbols) for a dilute suspension reported as a function of the frequency, for φ =.5. 1 G''.1 G', G''.1.1 G' unfilled fluid suspension (φ =.5) f Fig. 1. G' and G'' as functions of forcing frequency (f), for a dilute suspension with φ =.5 (symbols) and unfilled fluid (solid lines). The constant vertical shift at all frequencies with respect to the pure liquid behaviour (solid lines in Fig. 1) is clearly observed. A quantitative comparison of the code predictions with prediction by Palierne (199) is reported in Fig. 2, where the relative errors ( ε ', ε " ) are reported as a function of the forcing frequency. In the explored frequency range, quantitative agreement with theory is found. r r 7

8 1e-2 ε' r ε'' r ε' r, ε'' r 1e-3 1e f Fig. 2. Absolute values of the relative error of G' (full circles) and G'' (open circles) as function of forcing frequency (f), for a dilute suspension with φ = LARGE AMPLITUDE OSCILLATIONS In this section, the results of our simulations for LAOS regime are presented and discussed. First, the time-dependent rotation rate of the sphere is investigated. Then, we focus the attention on the bulk properties of the suspension, i.e., its bulk shear stress and storage and loss moduli. Let us first consider how the particle rotation rate is influenced by a large forcing amplitude. As it is well known (Einstein, 196; 1911), an isolated sphere immersed in a Newtonian fluid without inertia subjected to a steady shear flow follows without any lag in time the imposed flow. Indeed, the local fields are independent of time and Newtonianness implies that the stress istantaneously develops (no transient occurs). Einstein's analysis showed a particle rotation rate ω = γ /2, where γ is the externally imposed shear rate. If an oscillatory shear flow is applied, an analogous behaviour is expected. When imposing a sinusoidal shear rate (see eq.(6)), the rotation rate is ω = γ f sin(f t) / 2. D'Avino (27) showed a slowing down of the sphere rotation rate under steady shear flow when a viscoelastic suspending matrix is considered. An analogous result is found here for the case of an imposed sinusoidal shear rate in LAOS. Figure 3 shows the rotation rate of the sphere immersed in viscoelastic liquids (Maxwell and Giesekus respectively) as a function of dimensionless time for a forcing frequency f = 1. and at different strain amplitudes. The dashed line refers to the linear viscoelastic curve whereas the curves with symbols are for different strain amplitudes. 8

9 .5.5 γ =.5 γ = 1. γ = 1. γ = γ = 2. linear viscoelasticity.25 γ = 3. linear viscoleasticity ω/γ. ω/γ t/t t/t Fig. 3. Particle rotation rate in a Maxwell fluid (left) and a Giesekus fluid with α=.2 (right) as a function of time. Linear viscoelastic regime (dashed line), and three strains (solid lines with symbols), at f = 1.. From both figures, it appears that the lowering of angular sphere velocities becomes more pronounced as the strain increases. Passing now to the bulk stress calculation in LAOS, one notes that by increasing the forcing strain, the linear viscoelastic behaviour is lost. For a Maxwell suspending fluid, results for bulk shear stress divided by γ at two different strains for f = 1. are reported in Fig. 4. After a transient, a periodic regime is achieved. In the figure (as in all subsequent similar figures), only a period is reported, after the transient has extinguished. 1. linear viscoelastic γ = 1. γ = 2..5 σ xy /γ t/t Fig. 4. Shear stress divided by γ = γ f for a dilute suspension of spheres in a Maxwell fluid as a function of dimensionless time (with T the oscillation period) for linear viscoelastic 9

10 regime (dashed line) and two strains (solid lines with symbols). For low strains (up to.2), the linear viscoelastic behaviour (dashed line) is found. Well above this value of the forcing amplitude, a slight modification of bulk shear stress is observed. Specifically, maxima and minima reduce in absolute values, and are anticipated with respect to the linear viscoelastic curves, see inset. The effect of the solid spheres on the bulk behaviour of the suspension is even clearer if a Giesekus suspending fluid is considered. The bulk shear stress for such a suspension is reported in Fig. 5, for f = 1. and three different strains, with α = linear viscoelasticity γ = 1..5 γ = 2. γ = 3. σ xy /γ t/t Fig. 5. Shear stress divided by γ = γ f for a dilute suspension of spheres in a Giesekus fluid as a function of time for linear viscoelastic regime (dashed line) and three strains (solid lines with symbols). Again, deviations from linear viscoelastic curves are more and more pronounced as higher strain amplitudes are applied. Furthermore, if the forcing amplitude is sufficiently high, the resulting stress is patently no longer sinusoidal as the response stress appears to be strongly distorted. To better show the impact of the presence of the spherical particle on the bulk shear stress, a comparison with the unfilled fluid for f = 1. and γ = 3. is reported in Fig. 6. In the case of a pure Giesekus liquid, the LAOS response shows a distortion with respect to the linear case. The presence of a filler further deepens the difference. Notice that for a Maxwell fluid, the 1

11 distortion at large amplitudes is not observed at all in the unfilled system, hence it is completely due to the presence of solid inclusions. From the calculated bulk shear stresses, loss and storage moduli of the suspension can be derived. Figure 7 shows the normalized moduli for a Maxwell suspending fluid with φ =.5 reported as function of strain for f =.5 and f = 1.. At vanishing strains, the curves start from the linear viscoelastic value (the Palierne prediction) equal to for the volume fraction considered..4 unfilled φ =.5.2 σ xy /γ t/t Fig. 6. Shear stress divided by γ = γ f as a function of time for γ = 3. and f = 1.. Dashed line is the unfilled Giesekus fluid (α=.2); solid line is the dilute suspension of spheres. A decreasing trend for both moduli is found, more pronounced with increasing frequencies. As just remarked, such a trend is completely due to the presence of the solid particles, as for a pure Maxwell fluid no effect of amplitudes is foreseen. 11

12 f = f = G'/G', G''/G'' G'/G', G''/G'' G'/G' 1.8 G'/G' 1.7 G''/G'' 1.7 G''/G'' γ γ Fig. 7. G'/G' and G''/G'' as function of forcing strain, for a dilute suspension of spheres in a Maxwell fluid with φ = f = f = G'/G', G''/G'' G'/G' G''/G'' G'/G', G''/G'' G'/G' G''/G''.2 unfilled G'/G' unfilled G''/G''.2 unfilled G'/G' unfilled G''/G'' γ γ Fig. 8. G'/G' and G''/G'' (lines with symbols) as function of forcing strain, for a dilute suspension of spheres in a Giesekus fluid with φ =.5. The same quantities for an unfilled fluid are reported with lines without symbols. The normalized loss and storage bulk moduli for a suspending Giesekus fluid are reported in Fig. 8. Again, a decreasing trend from the linear viscoelastic limiting values is found. In this case, however, the effect of the inclusions is superimposed on the characteristic non-linear response of a Giesekus fluid in the LAOS regime. The relative impact of solid particles on the bulk response can be captured by comparing the normalized moduli with those of the unfilled Giesekus fluid (grey lines). As expected, the presence of inclusions leads to higher moduli with respect to the unfilled fluid. However, both quantities are decreasing with strain, and the separation between the filled and unfilled fluid curves becomes progressively smaller at high strains. 12

13 It is now worth showing how the presence of an inclusion affects the loss of linearity with increasing the strain. Figure 9 shows the behaviour of the quantities (G' L - G')/G' L and (G'' L -G'')/G'' L as a function of strain. The pedices L indicate the linear value of the moduli, thus these quantities are nil for vanishing amplitudes of the strain. The curves show that a suspension abandons the linear regime at a lower strain amplitude with respect the pure fluid. Moreover, the curves for a suspension are always higher than those for the unfilled fluid, and the deviations from the linear viscoelastic values are more and more pronounced as γ increases..7.3 (G' L - G')/G' L suspension unfilled fluid (G'' L - G'')/G'' L suspension unfilled fluid γ γ Fig. 9. (G' L - G')/G' L and (G'' L - G'')/G'' L as function of forcing strain γ, for a dilute suspension of spheres in a Giesekus fluid with φ =.5 (black lines). The same quantities for an unfilled fluid are reported with grey lines. The distorsion of the tangential stress waveform from the sinusoidal one, already reported in Fig. 5 for a Giesekus fluid, can be better appreciated by considering the power spectrum as reported in Fig. 1a. Indeed, in the Fourier transformed shear stress signal, higher order odd harmonics appear when the strain amplitude is sufficiently high, as shown in Fig. 1a. Nonlinearities in the response are signalled by the appearance of a nonzero third harmonic (note that even harmonics are nil by symmetry of the tangential stress). Even for dilute systems, one observes a third harmonic around 1% of the fundamental one. In the zoomed inset, also hints of nonnegligible 5 th harmonic are visible. Figure 1b shows the trend of the amplitude of the fundamental and of the third harmonics as a function of the strain. The limiting slope of the fundamental harmonic for vanishing strain indicates, of course, the linear viscoelastic limit, and is correctly captured by the simulations. The curve of the fundamental 13

14 harmonic is a cubic. The third harmonic vanishes at small strain, and increases quadratically with γ γ G* ω 5ω γ G*.8.6 ω 3ω Linear viscoelasticity.4.2 a γ =.5 γ = 1. γ = 2. γ = b. ω 3ω 5ω 7ω γ Fig. 1. Complex modulus (times the strain amplitude) versus frequency for different strain amplitudes (a) and versus the strain amplitude (b)for the first and third harmonic contribution. Data are for a dilute suspension of spheres in a Giesekus (α=.2) fluid with f = 1.. See Fig CONCLUSIONS In this paper, we studied a dilute, non-brownian, inertialess suspension of spheres in a viscoelastic fluid, subjected to an oscillatory shear flow. The motion of the sphere as well as the bulk rheology are investigated. The numerical analysis encompasses both small and large amplitude oscillation regimes. The governing equations are discretized by the finite element method over an unstructured grid surrounding the particle. The rigid-body motion is imposed on the sphere surface through Lagrange multipliers constraints. The bulk stress is obtained by resorting to a cell model. The 3D numerical simulations are validated by comparison with the analytical predictions by Palierne (199). The trends of G' and G'' are investigated for different forcing frequencies and amplitudes. The behaviour is found to be strongly dependent on the constitutive model. However, we found that the presence of the inclusion always increases both moduli. Moreover, for high strains, when the linear viscoelastic regime is abandoned, a decreasing trend of G' and G'' is found in 14

15 agreement with experimental results (Barnes, 23). It is important to point out that such an effect is evident even if a Maxwell suspending fluid is considered, in spite of the fact that the corresponding pure fluid does not exhibit any thinning of both moduli. Finally, a Fourier analysis is performed for the Giesekus model, showing a clear nonzero third harmonic contribution, more and more pronounced with increasing the forcing amplitude. BIBLIOGRAPHY Barnes H. A. in Rheology Reviews, 1, British Society of Rheology (23). Batchelor G.K., The stress system in a suspension of force-free particles, J. Fluid Mech. 41, (197). Bogaerds A.C.B., A.M. Grillet, G.W.M. Peters, F.P.T. Baaijens, Stability analysis of polymer shear flows using the extended pom-pom constitutive equations, J. Non- Newtonian Fluid Mech. 18, (22). Brenner, H. Suspension rheology Proc. Heat and Mass Transfer , (1972). Brooks A.N., T.J.R. Hughes, Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier- Stokes equations, Comp. Meth. Appl. Mech. Eng. 32, (1982). Brunn P.O., Slow motion of a sphere in a second-order fluid, Rheol. Acta 15, (1976). Einstein A., Eine neue bestimmung der moleküldimensionen, Ann. Physik 19, (196); ibidem 34, (1911). Fattal R., R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech., 123, (24). D Avino G., Computational rheology of solid suspensions, PhD Thesis, University of Naples, (27) Guénette R., M. Fortin, A new mixed finite element method for computing viscoelastic flows, J. Non-Newtonian Fluid Mech. 6, (1995). Hulsen M.A., R. Fattal, R. Kupferman, Flow of viscoelastic fluids past a cylinder at high Weissenberg number: stabilized simulations using matrix logarithms, J. Non- Newtonian Fluid Mech., 127, 27-39, (25). Hulsen M.A., TFEM: A toolkit for the finite element method, Userguide. (27). Larson R. G. The Structure and Rheology of Complex Fluids Oxford University Press. (1998) Leal L. G., The motion of small particles in nonnewtonian fluids, J. Non-Newtonian Fluid 15

16 Mech., 5, (1979). Palierne J.F., Linear rheology of viscoelastic emulsions with interfacial tension, Rheol. Acta, 29, (199). Schenk O., K. Gartner, Solving unsymmetric sparse systems of linear equations with PARDISO, J. of Future Generation Computer Systems, 2, (24). 16