Direct numerical simulation of a bubble suspension in smallamplitude oscillatory shear flowCitation for published version (APA):Mitrias, C., Jaensson, N. O., Hulsen, M. A., & Anderson, P. D. (2017). Direct numerical simulation of a bubblesuspension in small amplitude oscillatory shear flow. Rheologica Acta, 56(6), 555-565.https://doi.org/10.1007/s00397-017-1009-0

DOI:10.1007/s00397-017-1009-0

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Rheol Acta (2017) 56:555–565DOI 10.1007/s00397-017-1009-0

ORIGINAL CONTRIBUTION

Direct numerical simulation of a bubble suspension in smallamplitude oscillatory shear flow

Christos Mitrias1 ·Nick O. Jaensson1 ·Martien A. Hulsen1 · Patrick D. Anderson1

Received: 21 December 2016 / Revised: 8 March 2017 / Accepted: 9 March 2017 / Published online: 20 April 2017© The Author(s) 2017. This article is an open access publication

Abstract Bubble suspensions can be found in many differ-ent fields and studying their rheology is crucial in order toimprove manufacturing processes. When bubbles are addedto a liquid, the magnitude of the viscosity changes and thebehavior of the material is modified, giving it viscoelas-tic properties. For the purpose of this work, the suspendedbubbles are considered to be monodisperse. It is assumedthat Brownian motion and inertia can be neglected andthat the fluid of the matrix is Newtonian and incompress-ible. The suspension is subject to an oscillatory strain whileremaining in the linear regime. The resulting equations aresolved in 3D with direct numerical simulation using a finiteelement discretization. Results of an ordered and randomdistribution of bubbles of volume fractions up to 40% arepresented. The presence of bubbles has an opposite effecton the rheology of the suspension depending on the appliedfrequency. When the frequency is low, bubbles act as rigidfillers giving a rise to viscosity. On the contrary, when thefrequency is high, the strain rate is being accommodatedby the gaseous phase. Hence, bubbles deform, leading to adecrease of the viscosity.

Keywords Finite element method · Small amplitudeoscillatory shear · Bubble suspension · Direct numericalsimulation

� Nick O. Jaenssonn.o.jaensson@tue.nl

1 Department of Mechanical Engineering, EindhovenUniversity of Technology, P.O. Box 513, 5600 MB,Eindhoven, The Netherlands

Introduction

Dispersions of bubbles in a liquid are very common in manydifferent fields and understanding the rheology of suchmaterials is essential in order to improve manufacturing pro-cesses. Bubble suspensions are encountered in nature asmagmas (Manga and Loewenberg 2001) and in food indus-try where the gas phase can affect the texture, spreadabilityand provide a more uniform distribution of taste (Thakuret al. 2003). In polymeric foams, the change of bubble vol-ume fraction due to bubble growth greatly affects materialproperties and has been studied extensively by many authors(Aloku and Yuan 2010; Everitt et al. 2003). Usually, theyconsist of a continuous liquid phase with bubble volumefractions up to 50% and in foams even more.

Adding bubbles to a liquid not only changes the magni-tude of the viscosity but also modifies the behavior of thematerial giving it viscoelastic properties (Llewellin et al.2002). Both Rust and Manga (2002) and Stein and Spera(2002) identify two different regimes of such a materialin flow. When the applied strain rate is small, the bubblesact as rigid fillers increasing viscosity, while at large strainrates, where the flow line distortion is smaller due to bub-ble deformation, bubbles provide free slip surfaces and thusdecreasing viscosity. Llewellin et al. (2002) highlighted theimportance of distinguishing between steady and unsteadyflow. If shear conditions remain constant for a time muchlonger than the relaxation time of the bubble, the flow isassumed to be steady.

There is a lot of work done in the past century on theshear viscosity of suspensions of particles and emulsions.The relative shear viscosity ηr is the viscosity of the sus-pension normalized by the viscosity of the matrix μ and isa function of the volume fraction of the dispersed phase,

556 Rheol Acta (2017) 56:555–565

φ. Taylor (1932) generalized Einstein’s equation for dilutesuspensions of rigid particles (Einstein 1906) to drops

ηr = 1 + μ + 52ηd

μ + ηdφ, (1)

where ηd is the viscosity of the dispersed phase.If we assume that the viscosity of the dispersed phase is

small, i.e., ηd � μ, for a dilute suspension of bubbles weget

ηr = 1 + φ. (2)

Llewellin et al. (2002) provided an extensive review ofprevious models developed so far together with their ownmodel for suspensions of bubbles under steady and oscilla-tory shear conditions.

Problem definition

For the purpose of this work, the suspended bubbles are con-sidered to be monodisperse of radius R and they are notsubject to Brownian motion. The suspension is subject toan oscillatory strain γ (t) = γ0 sin(ωt) while keeping γ0small, so that we remain in the linear regime. The fluid ofthe matrix will be Newtonian. The initial geometry of sus-pended spherical bubbles inside a cubic domain is shown inFig. 1. The bubble interfaces are described by the surfacesSi and periodic boundaries of the box ∂�j for j = 1, ..., 6.The periodic domain will be considered as a representativevolume element of the bulk and the effect of the volumefraction on the relative viscosity will be investigated.

Governing equations

To describe the flow dynamics, we assume that inertia canbe neglected and that the fluid is incompressible. As a resultthe momentum and the mass balance reduce to

− ∇ · σ = 0 in �, (3)

∇ · u = 0 in �, (4)

where u is the velocity vector and σ the stress tensor.For incompressible Newtonian fluids, the stress tensor σ iswritten as:

σ = −pI + 2μD, (5)

where p is the pressure, I is the identity tensor, μ is theviscosity of the matrix fluid, and D = (∇u + (∇u)T )/2 isthe symmetric part of the velocity gradient tensor.

The gas inside the bubbles is assumed to be incom-pressible. For that reason, we impose a constant volume

Fig. 1 Initial geometry of spherical bubbles suspended in fluid

constraint on N − 1 bubble interfaces, where N is thenumber of bubbles:∫

Si

u · n dS = 0, i = 1, . . . , N − 1, (6)

with Si the ith bubble boundary on which we define n asthe outwardly directed unit normal vector. The N th bubbleconstraint is fulfilled automatically because of the incom-pressibility of the matrix fluid. Thus inside bubble N,

we have zero pressure and for the remaining bubbles thepressure is such that the volume remains constant.

Boundary conditions

To close the balance equations, the interface of the bubblesis assumed to be a material interface and triperiodic bound-ary conditions are imposed for the velocity and traction. Theinterfacial boundary conditions for a sharp interface can beexpressed as

σ · n|m = �κn, (7)

where the subscript m denotes the matrix fluid, � is thesurface tension coefficient and κ is the curvature. Theperiodicity of the domain can be formulated as

u|∂�3 = u|∂�1 + γ Lex,

u|∂�4 = u|∂�2, u|∂�6 = u|∂�5,(8)

where γ is the imposed shear rate and is time dependent,L is the length of the cubic computational domain, andadditionally we have imposed

t |∂�3 = −t |∂�1, t |∂�4 = −t |∂�2, t |∂�6 = −t |∂�5, (9)

where t = σ · m is the traction vector with m the outwardlydirected unit normal vector on the outer boundaries of theperiodic domain. It is required to indicate a reference valuefor the velocity in a single point in the periodic domain toremove the rigid body motion of the domain. The periodic

Rheol Acta (2017) 56:555–565 557

boundary conditions and the bubble volume constraints areimposed using Lagrange multipliers.

Interface tracking

The boundary of the sharp interface between two phases canbe described explicitly by a surface mesh. We can track theinterface by following the motion of the surface mesh. Aninterface in 3D can be described by a moving curvilinearcoordinate system given by

x = x(˜ξ, t), (10)

where˜ξ = (ξ1, ξ2) are the curvilinear coordinates and x

is the function that maps the coordinates˜ξ onto the spatial

coordinates x.For a material interface, the velocity of the interface x

must be such that:

x · n = u · n, (11)

where u is the material velocity at the interface and n is thenormal vector of the surface. The interface velocity is givenby

x = ∂x

∂t

∣∣∣∣˜ξ is constant

. (12)

Note, that x is equal the velocity of the material in thenormal direction only.

The following equation for the motion of an interface hasbeen implemented:

x = (u − uc) · nn + uc, (13)

where uc is the velocity of the center of volume of the bub-ble. The main purpose for introducing the field uc is to makethe movement of the nodes on the interface Galilean frameindependent.

For implementation, Eq. 13 is rewritten as (Villone et al.2014)

x + (u − uc) · (I − nn) = u. (14)

By substituting the expression for the surface identity tensorI − nn = gαgα , where gα , α = 1, 2 the dual base vectorsand gα = ∂x/∂ξα , α = 1, 2 are the covariant base vectors.For the term gαgα , we use the Einstein notational conven-tion for simplifying the expression of summation. We canrewrite Eq. 14 to

x + (u − uc) · ∇sx = u, (15)

where ∇s = gα ∂

∂ξαis the surface gradient operator.

Techniques to obtain rheology

The stress is one of the most important rheological prop-erties and we would like to know the stress generated byimposing a certain strain history to the bulk. However, the

stress response will vary with position of the bulk andin time. Thus, the “bulk stress” only exists in an aver-age sense (Batchelor 1970). To calculate the average stress,we assume that the triperiodic domain is a representativevolume element which is small compared to macroscopicvariation of the suspension but still large enough comparedto the distances between bubbles. We describe the macro-scopic problem by a representative fully periodic domain.The box over which we calculate the stresses is the periodicdomain. By integrating the stress over the box and divid-ing by its volume, we can obtain the volume-average stress.The volume-average stress is given by Batchelor’s formula(Batchelor 1970)

σ ≡ 1

V

∫Vf

σdV − 1

V

∫S

�(nn − 1

3I )dS, (16)

where σ indicates the volume average stress in the volume

of the box V , Vf is the volume of the fluid and S =N⋃

i=1Si

are the surfaces of the bubble interfaces.One common way to probe the rheological properties of

a fluid is the use of oscillatory measurements (Macosko1994). In oscillatory measurements, a sinusoidal straindeformation of magnitude γ0 is applied at a frequency ω.

γ (t) = γ0 sin(ωt). (17)

For a linear response, the resulting stress response remainssinusoidal with a phase shift and amplitude that depends onthe material and the frequency. Thus, the stress response canbe split in two parts, the in-phase response and the out-of-phase response, as follows:

σxy(t) = G′(ω)γ0 sin(ωt) + G′′(ω)γ0 cos(ωt), (18)

where G′(ω) and G′′(ω) are the elastic and viscous modu-lus respectively. To ensure that the response is linear for agiven strain amplitude γ0, we check if by doubling the strainamplitude, the stress response doubles as well

〈σxy(t)〉|2γ0 = 2〈σxy(t)〉|γ0 . (19)

If we write the applied oscillatory strain deformation in theform of a complex function γ (t) = γ0e

iωt , then the tworesponses would be determined respectively by the real andimaginary part of the stress response, written as

σ(t) = G∗(ω)γ0eiωt , (20)

where G∗(ω) is the complex shear modulus

G∗(ω) = G′(ω) + iG′′(ω). (21)

From the complex shear modulus, the complex viscosity canbe obtained from

η∗(ω) = G∗(ω)

ωi. (22)

558 Rheol Acta (2017) 56:555–565

Numerical description

Weak form

The weak form can be obtained by multiplying Eqs. 3, 4,and 15 with test functions v, q and w: Find u, p, x such that

(v, −∇ · σ ) = 0 for all v, (23)

(q, ∇ · u) = 0 for all q, (24)

(w, x + (u − uc) · ∇sx − u) = 0 for all w, (25)

where the inner products are defined as

(a, b) =∫

�

ab dV, (26)

(a, b) =∫

�

a · b dV, (27)

(A, B) =∫

�

A : B dV . (28)

Using partial integration and Gauss’ theorem, we obtain thefollowing weak form: Find u, q, and x such that(

Dv, 2ηD) − (∇ · v, p) = −〈v, �κn〉S

for all v, (29)

(q, ∇ · u) =0

for all q, (30)

〈w, x〉S + 〈w, (u − uc) · ∇sx〉S = 〈w, u〉Sfor all w, (31)

using appropriate spaces for u, p, x, v, q, and w. Further-more, similar to Glowinski et al. (1999), we use the discreteinner product on the bubble interface 〈· , ·〉S which will bedefined in “Discretization” section. In the weak form of themomentum balance, Eq. 29, Dv is defined as

Dv = (∇v + ∇vT )/2.

Discretization

The Eqs. 29–31 are discretized using the finite elementmethod employing a mesh of quadratic tetrahedra. Theinterface mesh consists of quadratic triangles which are con-forming with the volume mesh. Quadratic interpolation forthe velocity and linear interpolation for the pressure (Taylor-Hood) is used. We solve the resulting equations coupled sothat our timestep is not limited by the characteristic meshsize time limit. In a decoupled approach, the timestep is lim-ited by the capillary time which is t = xμ/� where x

is the element size, μ is the viscosity of the matrix and � thesurface tension (Courant et al. 1967). A second-order timeintegration scheme (BDF) is used for the time derivative ofthe position of the bubble interface. Equations 29–31 form anon-linear system of equations which needs to be linearizedso that it can be solved with a Newton-like iterative scheme.

It is important to note here that since the position x as anunknown exists only on the bubble interface the method isnot fully Newton-Raphson because the domain � is not per-turbed. To linearize velocity and position, u and x can bewritten as follows

u = un + δu, (32)

x = xn + δx, (33)

where subscript n indicates the current iteration and δu, δx

are the differences of velocity and position after one iter-ation respectively. For implementation the surface tensionterm after manipulation is rewritten taking into account thechange in position and area as

〈υ, �κn〉S = 〈 ∂υ

∂ξα, �gα∇s · δx〉S

−〈∇sυ, �∇sδx〉S+〈 ∂υ

∂ξα, �gα〉S. (34)

The discrete inner product 〈· , ·〉S is analogous to imposinga constraint using a collocation method and is given by

〈υ, u〉S =ncoll∑k−1

υk · uk, (35)

where ncoll is the number of collocation points. The colloca-tion points coincide with the nodes on the bubble interface.Furthermore, since the pressure term is linear, we chooseto solve for p. Substituting in Eqs. 29–31 and dropping thesubscript n leads to

((∇υ)T , η(∇δu + (∇δu)T ) − (∇ · υ, p)

+〈 ∂υ

∂ξα, �gα∇s · δx〉S − 〈∇sυ, �∇sδx〉S =

−((∇υ)T , η(∇u + (∇u)T ) − 〈 ∂υ

∂ξα, �gα〉S, (36)

(q, ∇ · δu) = −(q, ∇ · un), (37)

〈w,δx

t〉S + 〈w,

3x − 4x(i) + x(i−1)

2 t〉S

+〈w, (u − uc) · gα ∂x

∂ξα〉S

+〈w, (u − uc) · gα ∂δx

∂ξα〉S

−〈w, (u − uc) · gα · ∂δx

∂ξβgβ ∂x

∂ξα〉S

+〈w, δu · gα ∂x

∂ξα〉S

−〈w, δu〉S − 〈w, u〉S = 0. (38)

To take the change of area into account, Eq. 31 is multipliedby ∇s · δx and added to Eq. 38, while keeping only linearterms. The converged solution is obtained when the maxi-mum difference of rx,k = xk −xk−1, ru,k = uk −uk−1 andrp,k = pk − pk−1 of two consecutive iterations is less thanrmax < ε‖b‖2 where ε = 10−8 and ‖b‖2 the 2-norm of the

Rheol Acta (2017) 56:555–565 559

right hand side of the linear system of equations. By numer-ical experimentation, we found that multiplying the matrixterms containing the unknown δx in Eq. 34 by −1 greatlyincreases convergence. Since the resulting system of equa-tions can be large, the necessary physical memory for usingdirect solvers can grow fast. For that reason in most of thesimulations presented in this paper, we use iterative solvers,such as GMRES, from the SPARSKIT library (Saad 2001)in combination with an ILUT preconditioner. Since the con-nectivity of the nodes does not change, the structure of thesystem matrix remains the same and thus it is possible tore-use the ILUT preconditioner for many timesteps or evenfor the whole simulation. To increase the convergence rateby reducing the fill-in of the preconditioner, the mesh nodesare renumber usingMeTiS (Karypis and Kumar 1998). Nev-ertheless, selecting the proper parameters for the iterativesolver can be challenging. In those cases, we used a directsolver from the HSL library (HSL 2013).

Mesh movement

Since our problem contains moving boundaries, i.e., bubbleinterfaces and boundaries of the domain, it is essential todefine the motion of the mesh by using a smooth displace-ment field. We do this by solving a Laplace equation in aperiodic domain, similar to Hu et al. (2001). Firstly, we haveto solve the Laplace problem to obtain the position of theperiodic domain in the next time step

∇ · (ke∇( z1)) = 0 in �, (39)∂ z1

∂n= 0 on Si, (40)

where z1, are the nodal mesh displacements and ke is aconstant diffusion coefficient. The periodicity of the domaincan be formulated as

z1|∂�3 = z1|∂�1 + γLex,

z1|∂�4 = z1|∂�2, z1|∂�6 = z1|∂�5, (41)

where γ is the strain difference between two consecutivetime steps

γ = γ (t + t) − γ (t). (42)

The mesh displacement can be then used to find the newmesh coordinates as

zn+11 = zn + z1. (43)

Then, the bubble interfaces are moved during the New-ton iterations according to the solution δx of the interfacemotion. The rest of the periodical boundaries remain con-stant during the iterations. This leads to the following set ofequations

∇ · (ke∇( z2)) = 0 in �, (44)

z2 = 0 on ∂�i, (45)

z2 = δx on Si, (46)

Once more, we obtain the new mesh coordinates using themesh displacements

zn+1 = zn+11 + z2. (47)

Mesh generation

For a given bubble distribution, a triperiodic mesh has tobe generated. For numerical reasons, we want to do thissuch that there are no intersections between the bubbles andthe outer surfaces. However, due to the bubbles being ran-domly distributed, it is difficult (or even impossible) to findstraight planes that do not intersect any of the bubbles. Wehave therefore designed an algorithm that generates surfacesthat curve around the bubbles. The algorithm is based on anapproach used in Jaensson et al. (2015), where a convection-diffusion problem is solved for the nodes on the outer curvesof a 2D mesh to obtain smooth boundaries that do not inter-sect particles. We start the mesh generation by finding thecorner points of the mesh. Note, that by choosing one cor-ner point, the remaining seven corner points are fixed dueto the periodicity of the problem. The corner points aretypically chosen far away from any bubble boundary, toavoid problems with meshing the volume. Next, coarse sur-face meshes are generated of three mutually perpendicularstraight surfaces (the remaining three surfaces are periodiccopies). A number of nodes on these surfaces are likely to belocated inside the bubbles, therefore we solve the followingconvection-diffusion equation on these surfaces

∂z

∂t− ∇s · (α∇sz) = (ua · m)m, (48)

where z are the nodal coordinates, ∇s is the surface gradi-ent operator, α is a diffusion coefficient, ua is an artificialvelocity field, and m is the normal to the surface. The dif-fusion coefficient α is chosen such that the resulting surfaceis smooth. The velocity field us points away from the centerof the bubbles, with a magnitude that quickly decays to zerooutside the bubbles. This ensures that nodes are “pushed-out” of the bubbles, but not into adjacent bubbles. Similarto Jaensson et al. (2015), a tanh-function is employed todetermine the magnitude of ua. Equation 48 is integrated intime using an explicit Euler scheme. During the time inte-gration, the planes are iteratively refined in regions wherebubbles are close, ensuring that the element size on the sur-faces matches the required element size of the volume mesh.An example of the three surfaces after running the algorithmis shown in Fig. 2. Finally, the periodic surfaces are used togenerate a triperiodic volume mesh using Gmsh (Geuzaineand Remacle 2009).

560 Rheol Acta (2017) 56:555–565

Fig. 2 Three planes that curve around the bubbles in an arbitrarybubble distribution. For clarity, not all bubbles are shown

Convergence test

Convergence is tested for a problem of a single bubble in aperiodic domain with length L = 1, which is equivalent toregularly stacked distribution of bubbles. For the constantparameters, we choose φ = 0.3, μ = 1, � = 0.1, γ0 =0.04. Three different meshes are used with varying num-ber of elements on the equator of the bubble, neq = 10, 20and 40. In Fig. 3, the error er of the viscosity is plotted forthe cases of neq = 10, 20, which is relative to the case ofneq = 40. We can see that the error quickly converges forneq = 20 giving a maximum relative error of less than 0.5%.Choosing now the number of elements on the equator to beneq = 20, we perform simulations with the same constantparameters. This time, we use different number of steps ntin each period. In Fig. 4, we plot the error for nt = 20, 40relative to the case of nt = 80. This clearly indicates that

Fig. 3 Relative error er of the viscosity using different meshes withthe number of elements on the equator of the bubble being neq =10, 20. As a reference value, we use the result obtained from the caseof neq = 40

Fig. 4 Relative error er of the viscosity using different number ofsteps in each period of the oscillations. The error is plotted for the casesof nt = 20, 40 relative to the one obtained with nt = 80

our numerical method is converging and from now on wewill use neq = 20 and nt = 40, which is a good trade offbetween accuracy and numerical efficiency.

In the case of multiple bubbles and for higher volumefractions, one should take into account the thin layers thatare present between bubble interfaces. In order to accuratelydescribe the flow dynamics between bubbles, it is essentialto perform mesh refinement. Figure 5 shows amplitude ofthe viscous stress tensor in the thin layer with use of 1, 3,and 6 elements between bubbles surfaces. It can be seen thatthree elements are enough to accurately describe the solu-tion in the thin layer between the bubble interface. Hence,in the remainder of this paper, we will use at least threeelements between bubbles.

Results

To study the effect of suspending bubbles in a Newto-nian fluid, many simulations were performed using ourmodel with regularly distributed, as well as randomly dis-tributed bubbles. The problem is described by the followingdimensionless parameters, the relative viscosity

ηr = |η∗(ω)|μ

, (49)

as the ratio of the magnitude of the complex viscosity η∗(ω)

over the fluid viscosity μ and the frequency scaled with therelaxation time λω. The relaxation time is defined as

λ = μR

�, (50)

which means that λω represents the ratio of viscous forcesversus surface tension forces. The results are presented for

Rheol Acta (2017) 56:555–565 561

Fig. 5 Two interacting bubbles with a thin fluid layer forming in between. The fluid layer consists of 1, 3, and 6 elements between the bubbles,respectively. The color indicates the amplitude of the viscous stress tensor

a wide range of volume fractions φ. Starting from the diluteregime, it is possible to validate our results, by compar-ing with analytical results that already exist in the literature(Llewellin et al. 2002). We then present results from sim-ulations that were performed with a volume fraction wellabove the dilute limit up to φ = 0.4. This can be used tostudy the effect of the volume fraction on the rheology ofthe suspension.

Validating against an analytical model

As mentioned earlier in “Techniques to obtain rheology”section, one convenient way to obtain rheological propertiesis to apply an oscillating strain γ and measure the resul-tant stress response. In Llewellin et al. (2002), the linearizedFrankel and Acrivos model was used to describe a systemwhich is subjected to forced oscillations. They give the fol-lowing expression for the dynamic viscosity η′ and lossviscosity η′′

η′ = β1 + α1β2ω2

1 + α21ω

2, (51)

η′′ = (β1α1 − β2)ω

1 + α21ω

2, (52)

where ω is the angular frequency and α1, β1, β2 are relatedto physical properties of the bubble suspension

α1 = 6

5λ, (53)

β1 = μ(1 + φ), (54)

β2 = μα1(1 − 5

3φ). (55)

We are going to use this model, which is the exact asymp-totic result under the assumptions of the dilute limit andsmall bubble deformation, to verify our numerical results.For the constant parameters, we choose φ = 0.005, μ = 1,� = 0.1, γ0 = 0.04 and we perform simulations for a rangeof frequencies ω scaled by relaxation time λ from 10−2 upto 102. To validate our method, the result of our simulationfor the dynamic η′ and elastic η′′ components of the com-plex viscosity are compared with the expression given bythe analytical model. The results are presented in Fig. 6 andboth the components show a good match for the whole rangeof the scaled frequency λω. The small difference betweensimulations and analytical results can be explained by thefact that the analytical model has been derived for the caseof a single bubble in an infinitely large domain, while inour numerical model there is a finite distance between thebubble and the periodic boundaries. In Fig. 7, the relativeviscosity ηr = √

η′2 + η′′2/μ over the scaled frequencyλω is compared with the analytical results from the lin-earized Frankel and Acrivos model. As we can see, thereis an excellent agreement between the two under the sameassumptions of volume fraction in the dilute limit and smallbubble deformations.

Regularly stacked bubbles

Having verified our numerical model, we continue withmore simulations and investigate cases where the assump-tion of the dilute limit does not apply any more. We willbegin by using only one sphere in a periodic domain asshown in Fig. 8, which represents a regularly stacked dis-tribution of bubbles. All of the following simulations for

562 Rheol Acta (2017) 56:555–565

Fig. 6 In the left figure, thedynamic component η′ iscompared for the case of onebubble of φ = 0.005 betweenanalytical results andsimulations. While on the rightthe elastic component η′′ isplotted

the case of regularly stacked bubbles were performed withconstant parameters μ = 1, � = 0.1, γ0 = 0.04 for awide range of volume fractions and frequencies. In Fig. 9,we can see the effect of the increasing volume fraction tothe relative viscosity ηr. When the volume fraction increasesthe relative viscosity diverges more and more from whatwould be a constant line at ηr = 1 for the case of a Newto-nian fluid without any suspended bubbles. As mentioned inLlewellin et al. (2002), that behavior can be explained as fol-lows. For low λω, the change in the bubble deformation ismuch smaller in comparison with the applied strain rate andthe bubbles can be seen as rigid fillers, thus the suspensionmedium takes most of the strain. When φ increases, flowlines are increasingly distorted, which causes viscosity toincrease with φ. On the other hand, for very high λω, mostof the strain is taken by the bubbles and since their viscosityis negligible an increase of the volume fraction would leadto an decrease of the relative viscosity. To quantify that, it

Fig. 7 Comparison between simulations and analytical results of therelative viscosity over a range of scaled frequency of regularly stackedbubbles of φ = 0.005

is possible to calculate the strain rate accommodated by thefluid similarly to the bulk stress calculation using Eq. 56

γf = 1

V

∫Vf

(∂ux

∂y+ ∂uy

∂x

)dV, (56)

where γf is the average of the maximum strain rates for alloscillations. The ratio of γf/γ , as depicted in Fig. 10, is inagreement with the claim previously stated.

Since the matrix viscosity μ is known as a direct inputof the model it is interesting to compare the results aftersubtracting it, as shown in Fig. 11. Considering that theanalytical model contains only the linear terms of φ, it ispossible to check what is the effect of the higher order terms(Fig. 12). Using a function for η′ that contains the higherorder terms φ2 and φ3

η′

μ= 1 + a(λω)φ + b(λω)φ2 + c(λω)φ3, (57)

Fig. 8 Geometry of a single bubble in a periodic domain

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Fig. 9 Relative viscosity of regularly stacked bubbles suspended inNewtonian fluid for volume fractions 0.1 ≤ φ ≤ 0.4

and fitting the parameters a, b and c which are all functionsof frequency λω, it is possible to create the master curves asshown in Fig. 13. These master curves indicate the impor-tance of the higher order terms. The fit was performed for0.005 ≤ φ ≤ 0.4 and the square of the correlation betweenthe actual values and the predicted ones, r2 is greater than0.99. As expected the parameter a approximates the valuesobtained from the analytical expressions.

Random distribution of bubbles

Ordered structures as the one used in the previous sectionare rarely encountered in nature. For that reason, studyingthe change in rheology due the increase of volume fraction

Fig. 10 The ratio of the average of the maximum strain rates forall oscillation over the applied strain rate γf/γ , versus the scaledfrequency λω

Fig. 11 The dynamic viscosity η′ minus the matrix viscosity μ as afunction of volume fraction φ, for different scaled frequencies λω

of a random distribution of bubbles is essential. The rep-resentative volume element (Fig. 14) consists of a periodicmesh where the periodic boundaries curve around the bub-bles as explained in “Mesh generation” section. To avoidextreme cases where bubbles nearly overlap, the randomdistribution is constraint such that the minimum distancebetween bubble interfaces can not be less than 0.1R. Tocomplete our simulations in a feasible time, we had toconstrain the number of bubbles to 6. For all the simula-tions of the multiple bubble problem, the following constantparameters where chosen. The viscosity of the matrix andsurface tension are the same as in the case of regularlystacked bubbles (μ = 1, � = 0.1), where a lower strainamplitude is preferred (γ0 = 0.001) to ensure numerical

Fig. 12 The difference of the dynamic viscosity η′ between numericaland analytical results as a function of volume fraction φ, for differentscaled frequencies λω

564 Rheol Acta (2017) 56:555–565

Fig. 13 Parameters a, b, c of Eq. 57 are plotted over the whole rangeof scaled frequencies

stability. That is especially more imperative for the casesof high volume fractions. The range of volume fractions isagain 0.1 ≤ φ ≤ 0.4 and we perform simulations for awide range of frequencies. To obtain the correct rheologi-cal response when using representative volume elements, itis important to average over multiple realizations. Since oursimulations are rather time consuming, it would be impos-sible to verify that for the whole range of volume fractionsand frequencies. We instead calculated the standard devi-ation for volume fraction φ = 0.3 and ω = 50 whichwas 0.0112. Hence, we are confident that our results rep-resent accurately the true rheological response. In Fig. 15,

Fig. 14 Geometry of a random distribution of bubbles with a bound-ary fitted mesh

Fig. 15 The difference of the relative viscosity ηr between regularlyand randomly distributed bubbles as a function of scaled frequency λω,for different volume fractions φ

the two different cases, of regularly and randomly dis-tributed bubbles are plotted together. As one might expect,the relative viscosity of the randomly distributed bubblesshow an increase of viscosity in the low frequency plateaucompared with the case of an ordered distribution. This canbe explained if we keep in mind the explanation given byLlewellin et al. (2002) and illustrated in “Regularly stackedbubbles” section. Since the strain is taken by the viscousphase, a random distribution of bubbles would cause big-ger disturbance in the flow lines than an ordered structureand thus would lead to an increase of viscosity. On the otherhand, in the high-frequency regime, this increase is smallersince most of the strain is accommodated by the inviscidphase. Note that the vertical axis in Fig. 15 has a logarithmicscale.

Conclusion

We have presented 3D direct numerical simulations of bub-bles suspended in a Newtonian fluid subject to oscillatorystrain. The influence on the rheology of the suspension dueto the increase of the volume fraction was investigated forthe cases of regularly stacked bubbles and a random distri-bution of bubbles in a fully periodic domain. The resultingstress, as an output of our simulation, was used to calcu-late the viscosity of the bubbly fluid. Using the linearizedFrankel and Acrivos model which is derived for the dilutelimit and small strain amplitudes, we were able to validateour results. Comparing the analytical results, where only lin-ear terms of volume fraction are included, with our simu-lations we were able to show the significance of higher orderterms. The increase in volume fraction has an opposite effect

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on viscosity in the high and low frequency regimes. Whenthe frequency is small, the bubbles act as rigid fillers, thusincreasing the viscosity. In contrast, in the high-frequencyregime the bubble deformation rate compared to the bulkstrain rate is much larger, hence, most of the strain is takenby the inviscid phase leading to a decrease in viscosity.

We ran preliminary tests to investigate the effect of thenumber of bubbles in the domain. The first results showthat the effect is minor, especially for low volume frac-tions. Further optimization of the numerical technique willallow us to study more bubbles and higher volume fractions.For this work, we only consider bubbles as the suspendedphase. However, the model can be easily extended to sim-ulate droplets, and the results could be compared to thetheory of Oldroyd (1953). Studying large amplitude oscilla-tory shear in bubble suspensions is part of ongoing research.To achieve this, it is essential to apply a remeshing techniqueto avoid highly deformed elements. Furthermore, a multi-level adaptive refinement method is necessary to describeflow dynamics between two bubble interfaces that approacheach other. In this paper, we studied only the case of aNewtonian matrix. It should be noted that, it would be rel-atively easy to extend our numerical model to the case ofviscoelastic matrices.

Acknowledgments The research leading to these results hasreceived funding from the European Commission under the grantagreement number 604271 (Project acronym: MoDeNa; call identifier:FP7-NMP-2013-SMALL-7)

Open Access This article is distributed under the terms of theCreative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricteduse, distribution, and reproduction in any medium, provided you giveappropriate credit to the original author(s) and the source, provide alink to the Creative Commons license, and indicate if changes weremade.

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