Numerical method for 3D simulation of foam dynamics in the presence of surfactant Ivan B. Bazhlekov Institute of Mathematics and Informatics, BAS, Sofia, Bulgaria Abstract. 3D numerical method is presented for simulation of foam formation and dynamics in viscous flows in the presence of insoluble surfactant and under the influence of van der Waals forces. The mathematical model is based on the Stokes equations in the fluid phases, coupled with velocity and stress boundary conditions at the interfaces. A nonuniform surfactant concentration on the interfaces, governed by a convection-diffusion equation, leads to a gradient of the interfacial tension, which in turns leads to an additional tangential stress on the interfaces. The presented numerical method is a semi-implicit coupling of a boundary-integral method for the velocity in the fluid phases with a finite-volume method for the surfactant concentration on the interfaces. Additional elements of the method are: Nonsingular contour integration of the singular layer potential; Higher order approximation of the interface position; Dynamic mesh regularization. Keywords: Boundary integral method; Finite volume method; Foam dynamics; Insoluble surfactant; Van der Waals forces; Marangoni stress. PACS: 47.57.Bc; 47.55.dk; 47.15.G; 47.11.Hj; 47.11.Df

INTRODUCTION Liquid foams are multiphase structures of fluid particles at high concentration in another immiscible liquid. Their highly structured geometry (liquid films bounded by plateau borders and junctions, see figure 1) and mechanics at the film level determine their complex rheological behavior and consequently their practical importance. This is mainly due to the presence of relatively large interfacial area and correspondingly liquid films of many orders of magnitude thinner than the particle size. However, this large difference in the scales also introduces most of the difficulties that one faces during experimental and theoretical investigation of foam dynamics. In the vast majority of practical situations surfactants are present to stabilize the foam, which is related with additional difficulties. Because of that most of the theoretical investigations consider 2D [1] foams or 3D dry-film foams, see [2]. The recent advances of 3D boundaryintegral methods for simulation of drops in close approach (see [3]) have a stimulating effect on their application to investigations of more complicated problems such as foam dynamics. The work of Loewenberg and Hinch (see [3]) is, to the best of our knowledge, the first work in this direction. They have investigated emulsion up to 30% volume fraction in simple shear flow.

FIGURE 1. Foam-drop structure (4 inner drops at 95% volume fraction). The film regions (made transparent) are connected via plateau borders, joined in junctions

In the present study the application of boundary-integral/finite-volume method for simulation of polydisperse foam dynamics in Stokes flow is discussed: following section is devoted to the definition of the problem. Section 3 discusses some of the most important elements of the numerical method. Results for 3D foam-drop dynamics in a simple shear flow are presented in Section 4.

MATHEMATICAL MODEL We consider a compound drop, one which contains several smaller drops of another immiscible liquid, subjected to a simple shear flow, see figure 2. The mathematical model is based on the Stokes equations, i.e. the inertial forces are

FIGURE 2.

Schematic sketch of the problem

neglected. In dimensionless terms the governing equations are: −∇p + λi−1 ∇2 u = 0; ∇.u = 0 x ∈ Ωi , i = 0, 1, 2, ...

(1)

where p is the pressure, u the velocity and λi = λ Tfor i = 0, and λi = 1 if i > 0. Boundary conditions at the interface Si = Ωi Ω0 are stress balance boundary condition and continuity of the velocity across the interface: (Π0 − Πi ).n = f(x); (u0 − ui )(x) = 0 x ∈ Si , i = 1, 2, ...

(2)

where Π is the stress tensor Π = −pI + λi−1 (∇u + (∇u)T ), I being the unit tensor, n(x) is the unit vector normal to S. In equation (2) f(x) is the interfacial force and will be specified later. Simple shear flow is considered as a boundary condition at infinity, which in dimensionless form is: u∞ = (x2 , 0, 0) The position of the interface

Si (x,t)

(3)

is given by the kinematic condition: dx = u(x,t) + w(x,t), x ∈ Si dt

(4)

where w(x,t) can be an arbitrary velocity tangential to the interface. In the presence of surfactants on the interface the interfacial tension σ is not constant, but depends on the local surfactant concentration Γ. Thus the interfacial force f(x) has three components: The capillary pressure, which is normal to the interface and proportional to the local curvature and interfacial tension; The Marangoni stress, the surface gradient of the interfacial tension; the disjoining pressure due to the van der Waals intermolecular forces. In dimensionless form the interfacial force reads: f(x) =

A σ (x)2k(x)n − ∇s σ (Γ) − 3 n, x ∈ S Ca h (x)

(5)

where where k(x) = 0.5∇s · n is the dimensionless mean curvature of the interface, σ is the dimensionless interfacial tension scaled with σeq and Ca = Rγ˙ µ /σeq is the capillary number, where σeq is the interfacial tension at uniform

surfactant distribution on an equivalent spherical drop. The Marangoni stress is proportional to the surface gradient of the interfacial tension, ∇s σ (Γ). The term P = −A/h3 (x) is the disjoining pressure, where A is a parameter proportional to the Hamaker constant and h(x) is the distance to the closest interface, see [2]. The surface gradient ∇s is defined as: ∇s = (I − nn) · ∇. Due to deformation of the drop, the interfacial velocity and surface diffusion, the concentration of the surfactant Γ on the interface will, in general, be nonuniform. The equation governing the concentration of insoluble surfactant expresses the mass balance of the surfactant on the interface, known as convection- diffusion equation. In dimensionless terms it is (see for instance [4]):

∂Γ 1 − (w + us ) · ∇s Γ + ∇s · (Γus ) + 2k(x)Γun − ∇2s Γ = 0, x ∈ S (6) ∂t Pe where Γ is the dimensionless surfactant concentration scaled with Γeq , the uniform concentration on an equivalent spherical drop. The surface Peclet number is Pe = γ˙R2 /Ds , where Ds is surface diffusivity. In order to close the mathematical model (1)-(6) we need a relation between the interfacial tension and the surfactant concentration, σ (Γ), which is given by the equation of state. In the present work we use a linear equation of state given by: σ=

1 (1 − β Γ) , 1−β

(7)

More realistic non-linear relation between the interfacial tension and the surfactant concentration can also be used in the present method. However, their disadvantage is the larger number of parameters.

NUMERICAL METHOD The method described in the present section is an extension of our previous works [5, 6], where the applicability of boundary integral method for simulation of foam dynamics in the absence of surfactant and drop dynamics in the presence of surfactant is demonstrated, respectively. A new element in the present study is the mutual effect of disjoining pressure and insoluble surfactant. Both, the disjoining pressure and the Marangoni stress act to stabilize the foams, by suppressing the film drainage. Some of the elements of the numerical methods are discussed in the following subsections. More information can be found in [5, 6, 7].

Integral Formulation The solution of the mathematical model (1-3) at a given point x0 can be obtained by means of boundary integral formulation, see for instance [3] and [7]: (λ + 1)u(x0 ) = 2.u∞ −

1 4π

Z

S

f (x) · G(x0 , x).n(x)dx +

λ −1 4π

Z

S

u(x).T(x0 , x).n(x)dx

(8)

where S = ∪i Si ; G(x0 , x) = I/r + xˆ xˆ /r3 is the Stokeslet, T(x0 , x) = −6ˆxxˆ xˆ /r5 is the stresslet, xˆ = x − x0 and r = |ˆx| (see [3] and [7]). In order to pay more attention on the interfacial forces (capillary pressure, Marangoni stress and disjoining pressure) the present study considers the case in which all phases have equal viscosities, λ = 1. In this case the second integral in (8) disappears and it reduces to: u(x0 ) = u∞ −

1 8π

Z

S

f(x)G(x0 , x).n(x)dx.

(9)

The case λ 6= 1 can also be handled by the present method, see [6]. The accurate calculation of the integrals in (8) is very important. The main problem is related with the singularity at x = x0 . Different approaches exist in the literature to overcome this, see for instance [3] and [4]. In the present study

the non-singular contour integration of the single layer potential proposed in [7]1 is used: Z

Sj

I

G(x0 , x).n(x)dx =

Cj

(x − x0 ) × {n(x) × b(x)} dx, |x − x0 |

(10)

where × is vector product. More information about the application of the contour integration (10) for the velocity calculation (9) is given in [7], where the advantages of (10) are also discussed. A finite-volume form of equation (6) is obtained (see for instance [4]) by taking the surface integral over an element of the interface S: Z

Sj

∂Γ ds = ∂t

Z

Sj

(w + us ) · ∇s Γ ds −

Z

Cj

Γ us · b dl −

Z

Sj

Γk u · n ds +

1 Pe

Z

Cj

b · ∇s Γ dl

(11)

where the second and the last term in the r.h.s. are converted into contour integrals, using the divergence theorem. In the following sections, we discuss approximations of the spatial terms and the time derivative of formulation (9) and (11).

Spatial Discretization and Approximation First order triangular elements are used for a initial interface discretization, see figure 3. Based on this triangulation surface elements S j around the collocation points x j (•) are constructed. Thus the surface element S j is bounded by a polygon C j , connecting the centers of mass of the triangles () and element sides () to which x j belongs. The

FIGURE 3. Interface discretization by triangles with vortices ◦. The surface element S j is defined by the centers of mass of the triangles () and element sides ()

integrals in the boundary integral formulation (9) are approximated using the non-singular representation (10), while the values of the interfacial force f(x) in (9) is considered constant in every surface element S j . The curvature k(x j ) at the vortices x j as well as the normal vector, n(x j ), are calculated by the formula, see also [3]: k(x j )n(x j )

Z

Sj

ds = −

I

bdl

(12)

Cj

where S j is a part of the discretized interface around the collocation point x j (•) and b is the unit vector tangential to S j and perpendicular to C j , see figure 3. For the surfactant concentration Γ, and respectively the interfacial tension σ a linear approximation is used in each triangle, defined by the corresponding values in the three vertices of the triangle. Thus, ∇s (Γ) is a constant vector inside each triangle. The approximations of the spatial terms in the convection-diffusion equation (11) as well as the Marangoni stress in the interfacial force f(x) in (9) are based on this approximation of Γ and ∇s (Γ). More information

1

Similar contour integration for the double layer potential is also given.

about the approximations of the spatial terms in (11) can be found in [6], where also numerical test are performed. The tests show that the discussed approximations are of second order accuracy. An accurate calculation of the interface-to-interface distance h(x j ) in the film region, where h(x j ) is of order 10−3 , is not only important for the accuracy, but is also very essential for the numerical stability. This is due to the fact that the disjoining pressure in the interfacial force f(x) in (9) is proportional to h(x)−3 . The main difficulties are due to the fact that the element size in the film regions is about two orders of magnitude larger than h(x). In the case of monodisperse foams, where typically the film regions are flat, an approximation of the interfaces by linear elements could be sufficient for an accurate calculation of h(x). However, in the case of polydisperse foam the films may have significant curvature and then a first order approximation of the interfaces is insufficient for the calculation of the interface-to-interface distance h(x j ). In the present work a higher order approximation of the interfaces is used. It is based on the initial discretization by triangles and information about the curvature and the normal vector in the nodes of the mesh, see also [5]. To construct this approximation the steps below are followed: (i) - A sphere C j (O j , R j ) is associated with every nodal point x j . The sphere is determined using the mean curvature k(x j ) and the normal vector n(x j ): 2n(x j ) 2 ; Oj = xj − (13) Rj = |k(x j )| k(x j ) (ii) - Let x be an arbitrary point from the initial discretization of S and the triangle to which x belongs has vortices x1 , x2 , x3 . The radial projection of x on the three spheres, C j , j = 1, 2, 3 are denoted by x pj respectively. (iii) - The last step is to define the projection of x by a proper linear combination: 3

x p = ∑ wi .xip ; i=1

3

∑ wi = 1

(14)

i=1

The coefficients wi in (14) are functions of the distances from x to the vortices and sides of the triangle (x1 , x2 , x3 ) and can be easily defined to be continuous across the element sides. The approximation x p constructed in the above mentioned way is of second order and is exact for the spherical parts of the interfaces. Important properties of this approximation are that h(x p ) is smooth and k(x p ) is continuous. By using h(x p ) simulations of foam drops at high volume fraction (up to 98%) are possible. For comparison,using the initial surface discretization with triangles, h(x), at about 60% volume fraction the numerical scheme become unstable due to the insuficient accuracy of the interface-to-interface distance calculation.

Time Integration At given interface position and velocity the convection-diffusion equation (11) can be written in the form (after using approximations of the spatial terms, see [6]):

∂Γj = ∑ A ji Γi , j = 1, 2, . . . , N, ∂t iin(N U j)

(15)

j

where the coefficients A ji depend on the coordinates of the nodal points and the interface velocity. Here N j contains the numbers of the nodal points that are directly connected with x j . For time discretization we use the theta method and thus (15) reads: (16) Γ j (t + ∆t) = Γ j (t) + ∆t ∑ A ji˙[θ Γi (t + ∆t) + (1 − θ )Γi (t)]. i

In the present study Euler explicit, θ = 0, and Crank-Nicolson, θ = 1/2, methods are considered. The Euler explicit method is very simple to realize, however it has only first-order approximation of the time derivative. The CrankNicolson method has second-order accuracy, however it is implicit, i.e. (16) is a system of N algebraic equations. An advantage of the system (16) is that the corresponding matrix is sparse. For the meshes considered here only 6 or 7 elements per row are non-zero (#(N j ) = 5 or 6). This makes the use of an iterative solver very efficient. In the present work (16) is solved for Γ j (t + ∆t) by means of the Jacobi or Gauss-Seidel method. Both methods give identical results and converge for about 3 - 6 iterations to an absolute tolerance of 10−9 . In order to verify the order of approximation of the time integration, for θ = 0 and θ = 1/2, the following test is performed. Consider convection-diffusion on unit

FIGURE 4. The relative error as a function of the time step ∆t.

sphere of surfactant with uniform initial concentration Γ(t = 0) = 1. The surfactant is convected on the sphere by the interface velocity us = (x2 − 1, xy, xz)/4, while the diffusion is moderated, Pe = 10. For each method the results for the concentration on a mesh of 2000 triangles at time t = 5, obtained by using different ∆t, are compared with the one for ∆t = 10−5 , and the relative error is shown in figure 4.

Dynamic Mesh Regularization Optimal mesh properties are important for the accuracy of every numerical method and not easy to obtain in cases like the one considered here, where regions of the interfaces with completely different characteristics exist: plateau borders and junctions with high curvature and curvature gradient, film regions with thickness of order 10−3 . The regions of higher gradients of the surfactant concentration, Γ, and respectively the interfacial tension σ also require denser mash. The foam dynamics also involves topological transition of the foam structure. For instance during the deformation in shear flow the drops change their neighbors, which is related to different kinds of transitions between films, plateau borders and junctions, see [2]. Thus the goal is not only to generate a proper mesh, but it is also essential to maintain desired mesh properties during the process. A suitable approach for this purpose is a dynamic mesh regularization, see for instance [3, 5]. For fixed mesh topology the mesh nodes are moved with the extra tangential velocity w(x,t), see (4). This velocity is determined based on the local characteristics of the mesh and interfaces (such as element size, curvature, film thickness and surfactant concentration gradients) by: (17) w(xi ) = (I − nn) ∑ {a + b.h(x j ) + c.|k(x j )| + d.|∇s Γ|}(x j − xi ) − u(xi ) + us , j∈Ni

where the summation involves only the nodes x j that are directly connected to xi ; us is an average velocity of the interface to which xi belongs. The term −u(xi ) + us in (17) eliminates possible mesh distortion due to the tangential component of the hydrodynamic velocity u. By a proper choice of the parameters a, b, c and d in (17) the mesh is maintained finer in the regions of higher gradients of k and h and Γ. This allows to achieve sufficiently good accuracy using reasonable number of elements. The mesh used for the present simulations consists of 8820 elements for the outer interface and 3380 elements per inner drop, in total 35860 elements. The steps for time integration of (4) used in the present study are of order 10−5 .

RESULTS In this section the numerical method described above is used for simulation of the dynamic behavior of a foam drop in simple shear flow in the presence of insoluble surfactant. The foam drop considered here consists of 8 inner drops (see figure 5), four of relative volume 13.5% and four of 10.25%. Thus the volume fraction of the foam drop is 95% and it has all structural elements of random polydisperse foams: plateau borders, junctions and films some of which with significant curvature. The initial shape is shown on the first row of figure 5, where the surfactant is distributed uniformly, at Γ = 1. The foam drop dynamics is shown on figure 5 at different time instances (the three rows respectively). The drop undergoes significant topological changes: the inner drops move inside the whole drop, changing their neighbors. This is related with disappearance of the existing films, plateau borders and junctions and formation of new ones. And the whole process being related to significantly nonuniform surfactant distribution. The results presented in this section are in reasonable qualitative agreement with the existing ones obtained for dry-film foams. Unfortunately, to the best of our knowledge, there are no other results in the literature suitable for a quantitative comparison.

CONCLUSIONS A 3D semi-implicit boundary-integral/finite-volume method is presented for simulation of foam dynamics in the presence of surfactant. A new element in the present study is the mutual effect of disjoining pressure and Marangoni stress. Both of them are results of the presence of insoluble surfactant and play very important role for the stability of foam configurations by suppressing the film drainage. Some of the elements of the numerical method, such as: nonsingular contour integration of the singular single layer potential, higher order approximation of the interfaces and dynamic mesh regularization are very important for the accuracy and the numerical stability. They allow simulation of film regions with significant curvature and interface-to-interface distance of order of 10−3 using linear triangular elements with side of order 10−1 . An important element here is the finite-volume method, allowing consideration of the effect of insoluble surfactant. The results for foam drop evolution in simple shear flow demonstrate the applicability of the method for simulation of foam dynamics at very high volume fraction (up to 95%) where most of the significant interfacial forces (Marangoni stress and van der Waals forces) are taken into account. The method can be extended straightforward for simulation of random polydisperse foams by applying periodic boundary conditions as in [3].

REFERENCES 1. 2. 3. 4. 5. 6. 7.

Kropinski M., Lushi E.: Efficient numerical methods for multiple surfactant-coated bubbles in a two-dimensional stokes flow Journal of Computational Physics 230 (2011) 4466–4487 Kraynik, A., Reinelt, D.: Foam microrheology: from honeycombs to random foams. in 15th Annual Meeting Polymer Processing Sosiety, ’s-Hertogenbosh, Netherlands, on CD-Rom (1999) Loewenberg, M., Hinch E.: Numerical simulation of concentrated emulsion in shear flow. Journal of Fluid Mechanics 321 (1996) 358–419 Yon, S., Pozrikidis C.: A finite-volume/boundary-element method for flow past interfaces in the presence of surfactants, with application to shear flow past a viscous drop. Computers & Fluids 27 (1998) 879–902 Bazhlekov, I., van de Vosse, F., Meijer, H.: Boundary integral method for 3D simulation of foam dynamics. Lecture Notes in Computer Science 2179 (2001) 401–408 Bazhlekov, I., Anderson, P., Meijer H.: Boundary integral method for deformable interfaces in the presence of insoluble surfactants. Lecture Notes in Computer Science 2907 (2004) 355-362 Bazhlekov, I.: Non-singular boundary-integral method for deformable drops in viscous flows. Ph.D. thesis, Eindhoven University of Technology, The Netherlands (2003)

FIGURE 5. Evolution of the foam drop in simple shear flow at Ca = 0.2; Pe = 100; β = 0.5 and A = 2.5 × 10−6 . The first column represents the outer interface and the second - the inner drops.

INTRODUCTION Liquid foams are multiphase structures of fluid particles at high concentration in another immiscible liquid. Their highly structured geometry (liquid films bounded by plateau borders and junctions, see figure 1) and mechanics at the film level determine their complex rheological behavior and consequently their practical importance. This is mainly due to the presence of relatively large interfacial area and correspondingly liquid films of many orders of magnitude thinner than the particle size. However, this large difference in the scales also introduces most of the difficulties that one faces during experimental and theoretical investigation of foam dynamics. In the vast majority of practical situations surfactants are present to stabilize the foam, which is related with additional difficulties. Because of that most of the theoretical investigations consider 2D [1] foams or 3D dry-film foams, see [2]. The recent advances of 3D boundaryintegral methods for simulation of drops in close approach (see [3]) have a stimulating effect on their application to investigations of more complicated problems such as foam dynamics. The work of Loewenberg and Hinch (see [3]) is, to the best of our knowledge, the first work in this direction. They have investigated emulsion up to 30% volume fraction in simple shear flow.

FIGURE 1. Foam-drop structure (4 inner drops at 95% volume fraction). The film regions (made transparent) are connected via plateau borders, joined in junctions

In the present study the application of boundary-integral/finite-volume method for simulation of polydisperse foam dynamics in Stokes flow is discussed: following section is devoted to the definition of the problem. Section 3 discusses some of the most important elements of the numerical method. Results for 3D foam-drop dynamics in a simple shear flow are presented in Section 4.

MATHEMATICAL MODEL We consider a compound drop, one which contains several smaller drops of another immiscible liquid, subjected to a simple shear flow, see figure 2. The mathematical model is based on the Stokes equations, i.e. the inertial forces are

FIGURE 2.

Schematic sketch of the problem

neglected. In dimensionless terms the governing equations are: −∇p + λi−1 ∇2 u = 0; ∇.u = 0 x ∈ Ωi , i = 0, 1, 2, ...

(1)

where p is the pressure, u the velocity and λi = λ Tfor i = 0, and λi = 1 if i > 0. Boundary conditions at the interface Si = Ωi Ω0 are stress balance boundary condition and continuity of the velocity across the interface: (Π0 − Πi ).n = f(x); (u0 − ui )(x) = 0 x ∈ Si , i = 1, 2, ...

(2)

where Π is the stress tensor Π = −pI + λi−1 (∇u + (∇u)T ), I being the unit tensor, n(x) is the unit vector normal to S. In equation (2) f(x) is the interfacial force and will be specified later. Simple shear flow is considered as a boundary condition at infinity, which in dimensionless form is: u∞ = (x2 , 0, 0) The position of the interface

Si (x,t)

(3)

is given by the kinematic condition: dx = u(x,t) + w(x,t), x ∈ Si dt

(4)

where w(x,t) can be an arbitrary velocity tangential to the interface. In the presence of surfactants on the interface the interfacial tension σ is not constant, but depends on the local surfactant concentration Γ. Thus the interfacial force f(x) has three components: The capillary pressure, which is normal to the interface and proportional to the local curvature and interfacial tension; The Marangoni stress, the surface gradient of the interfacial tension; the disjoining pressure due to the van der Waals intermolecular forces. In dimensionless form the interfacial force reads: f(x) =

A σ (x)2k(x)n − ∇s σ (Γ) − 3 n, x ∈ S Ca h (x)

(5)

where where k(x) = 0.5∇s · n is the dimensionless mean curvature of the interface, σ is the dimensionless interfacial tension scaled with σeq and Ca = Rγ˙ µ /σeq is the capillary number, where σeq is the interfacial tension at uniform

surfactant distribution on an equivalent spherical drop. The Marangoni stress is proportional to the surface gradient of the interfacial tension, ∇s σ (Γ). The term P = −A/h3 (x) is the disjoining pressure, where A is a parameter proportional to the Hamaker constant and h(x) is the distance to the closest interface, see [2]. The surface gradient ∇s is defined as: ∇s = (I − nn) · ∇. Due to deformation of the drop, the interfacial velocity and surface diffusion, the concentration of the surfactant Γ on the interface will, in general, be nonuniform. The equation governing the concentration of insoluble surfactant expresses the mass balance of the surfactant on the interface, known as convection- diffusion equation. In dimensionless terms it is (see for instance [4]):

∂Γ 1 − (w + us ) · ∇s Γ + ∇s · (Γus ) + 2k(x)Γun − ∇2s Γ = 0, x ∈ S (6) ∂t Pe where Γ is the dimensionless surfactant concentration scaled with Γeq , the uniform concentration on an equivalent spherical drop. The surface Peclet number is Pe = γ˙R2 /Ds , where Ds is surface diffusivity. In order to close the mathematical model (1)-(6) we need a relation between the interfacial tension and the surfactant concentration, σ (Γ), which is given by the equation of state. In the present work we use a linear equation of state given by: σ=

1 (1 − β Γ) , 1−β

(7)

More realistic non-linear relation between the interfacial tension and the surfactant concentration can also be used in the present method. However, their disadvantage is the larger number of parameters.

NUMERICAL METHOD The method described in the present section is an extension of our previous works [5, 6], where the applicability of boundary integral method for simulation of foam dynamics in the absence of surfactant and drop dynamics in the presence of surfactant is demonstrated, respectively. A new element in the present study is the mutual effect of disjoining pressure and insoluble surfactant. Both, the disjoining pressure and the Marangoni stress act to stabilize the foams, by suppressing the film drainage. Some of the elements of the numerical methods are discussed in the following subsections. More information can be found in [5, 6, 7].

Integral Formulation The solution of the mathematical model (1-3) at a given point x0 can be obtained by means of boundary integral formulation, see for instance [3] and [7]: (λ + 1)u(x0 ) = 2.u∞ −

1 4π

Z

S

f (x) · G(x0 , x).n(x)dx +

λ −1 4π

Z

S

u(x).T(x0 , x).n(x)dx

(8)

where S = ∪i Si ; G(x0 , x) = I/r + xˆ xˆ /r3 is the Stokeslet, T(x0 , x) = −6ˆxxˆ xˆ /r5 is the stresslet, xˆ = x − x0 and r = |ˆx| (see [3] and [7]). In order to pay more attention on the interfacial forces (capillary pressure, Marangoni stress and disjoining pressure) the present study considers the case in which all phases have equal viscosities, λ = 1. In this case the second integral in (8) disappears and it reduces to: u(x0 ) = u∞ −

1 8π

Z

S

f(x)G(x0 , x).n(x)dx.

(9)

The case λ 6= 1 can also be handled by the present method, see [6]. The accurate calculation of the integrals in (8) is very important. The main problem is related with the singularity at x = x0 . Different approaches exist in the literature to overcome this, see for instance [3] and [4]. In the present study

the non-singular contour integration of the single layer potential proposed in [7]1 is used: Z

Sj

I

G(x0 , x).n(x)dx =

Cj

(x − x0 ) × {n(x) × b(x)} dx, |x − x0 |

(10)

where × is vector product. More information about the application of the contour integration (10) for the velocity calculation (9) is given in [7], where the advantages of (10) are also discussed. A finite-volume form of equation (6) is obtained (see for instance [4]) by taking the surface integral over an element of the interface S: Z

Sj

∂Γ ds = ∂t

Z

Sj

(w + us ) · ∇s Γ ds −

Z

Cj

Γ us · b dl −

Z

Sj

Γk u · n ds +

1 Pe

Z

Cj

b · ∇s Γ dl

(11)

where the second and the last term in the r.h.s. are converted into contour integrals, using the divergence theorem. In the following sections, we discuss approximations of the spatial terms and the time derivative of formulation (9) and (11).

Spatial Discretization and Approximation First order triangular elements are used for a initial interface discretization, see figure 3. Based on this triangulation surface elements S j around the collocation points x j (•) are constructed. Thus the surface element S j is bounded by a polygon C j , connecting the centers of mass of the triangles () and element sides () to which x j belongs. The

FIGURE 3. Interface discretization by triangles with vortices ◦. The surface element S j is defined by the centers of mass of the triangles () and element sides ()

integrals in the boundary integral formulation (9) are approximated using the non-singular representation (10), while the values of the interfacial force f(x) in (9) is considered constant in every surface element S j . The curvature k(x j ) at the vortices x j as well as the normal vector, n(x j ), are calculated by the formula, see also [3]: k(x j )n(x j )

Z

Sj

ds = −

I

bdl

(12)

Cj

where S j is a part of the discretized interface around the collocation point x j (•) and b is the unit vector tangential to S j and perpendicular to C j , see figure 3. For the surfactant concentration Γ, and respectively the interfacial tension σ a linear approximation is used in each triangle, defined by the corresponding values in the three vertices of the triangle. Thus, ∇s (Γ) is a constant vector inside each triangle. The approximations of the spatial terms in the convection-diffusion equation (11) as well as the Marangoni stress in the interfacial force f(x) in (9) are based on this approximation of Γ and ∇s (Γ). More information

1

Similar contour integration for the double layer potential is also given.

about the approximations of the spatial terms in (11) can be found in [6], where also numerical test are performed. The tests show that the discussed approximations are of second order accuracy. An accurate calculation of the interface-to-interface distance h(x j ) in the film region, where h(x j ) is of order 10−3 , is not only important for the accuracy, but is also very essential for the numerical stability. This is due to the fact that the disjoining pressure in the interfacial force f(x) in (9) is proportional to h(x)−3 . The main difficulties are due to the fact that the element size in the film regions is about two orders of magnitude larger than h(x). In the case of monodisperse foams, where typically the film regions are flat, an approximation of the interfaces by linear elements could be sufficient for an accurate calculation of h(x). However, in the case of polydisperse foam the films may have significant curvature and then a first order approximation of the interfaces is insufficient for the calculation of the interface-to-interface distance h(x j ). In the present work a higher order approximation of the interfaces is used. It is based on the initial discretization by triangles and information about the curvature and the normal vector in the nodes of the mesh, see also [5]. To construct this approximation the steps below are followed: (i) - A sphere C j (O j , R j ) is associated with every nodal point x j . The sphere is determined using the mean curvature k(x j ) and the normal vector n(x j ): 2n(x j ) 2 ; Oj = xj − (13) Rj = |k(x j )| k(x j ) (ii) - Let x be an arbitrary point from the initial discretization of S and the triangle to which x belongs has vortices x1 , x2 , x3 . The radial projection of x on the three spheres, C j , j = 1, 2, 3 are denoted by x pj respectively. (iii) - The last step is to define the projection of x by a proper linear combination: 3

x p = ∑ wi .xip ; i=1

3

∑ wi = 1

(14)

i=1

The coefficients wi in (14) are functions of the distances from x to the vortices and sides of the triangle (x1 , x2 , x3 ) and can be easily defined to be continuous across the element sides. The approximation x p constructed in the above mentioned way is of second order and is exact for the spherical parts of the interfaces. Important properties of this approximation are that h(x p ) is smooth and k(x p ) is continuous. By using h(x p ) simulations of foam drops at high volume fraction (up to 98%) are possible. For comparison,using the initial surface discretization with triangles, h(x), at about 60% volume fraction the numerical scheme become unstable due to the insuficient accuracy of the interface-to-interface distance calculation.

Time Integration At given interface position and velocity the convection-diffusion equation (11) can be written in the form (after using approximations of the spatial terms, see [6]):

∂Γj = ∑ A ji Γi , j = 1, 2, . . . , N, ∂t iin(N U j)

(15)

j

where the coefficients A ji depend on the coordinates of the nodal points and the interface velocity. Here N j contains the numbers of the nodal points that are directly connected with x j . For time discretization we use the theta method and thus (15) reads: (16) Γ j (t + ∆t) = Γ j (t) + ∆t ∑ A ji˙[θ Γi (t + ∆t) + (1 − θ )Γi (t)]. i

In the present study Euler explicit, θ = 0, and Crank-Nicolson, θ = 1/2, methods are considered. The Euler explicit method is very simple to realize, however it has only first-order approximation of the time derivative. The CrankNicolson method has second-order accuracy, however it is implicit, i.e. (16) is a system of N algebraic equations. An advantage of the system (16) is that the corresponding matrix is sparse. For the meshes considered here only 6 or 7 elements per row are non-zero (#(N j ) = 5 or 6). This makes the use of an iterative solver very efficient. In the present work (16) is solved for Γ j (t + ∆t) by means of the Jacobi or Gauss-Seidel method. Both methods give identical results and converge for about 3 - 6 iterations to an absolute tolerance of 10−9 . In order to verify the order of approximation of the time integration, for θ = 0 and θ = 1/2, the following test is performed. Consider convection-diffusion on unit

FIGURE 4. The relative error as a function of the time step ∆t.

sphere of surfactant with uniform initial concentration Γ(t = 0) = 1. The surfactant is convected on the sphere by the interface velocity us = (x2 − 1, xy, xz)/4, while the diffusion is moderated, Pe = 10. For each method the results for the concentration on a mesh of 2000 triangles at time t = 5, obtained by using different ∆t, are compared with the one for ∆t = 10−5 , and the relative error is shown in figure 4.

Dynamic Mesh Regularization Optimal mesh properties are important for the accuracy of every numerical method and not easy to obtain in cases like the one considered here, where regions of the interfaces with completely different characteristics exist: plateau borders and junctions with high curvature and curvature gradient, film regions with thickness of order 10−3 . The regions of higher gradients of the surfactant concentration, Γ, and respectively the interfacial tension σ also require denser mash. The foam dynamics also involves topological transition of the foam structure. For instance during the deformation in shear flow the drops change their neighbors, which is related to different kinds of transitions between films, plateau borders and junctions, see [2]. Thus the goal is not only to generate a proper mesh, but it is also essential to maintain desired mesh properties during the process. A suitable approach for this purpose is a dynamic mesh regularization, see for instance [3, 5]. For fixed mesh topology the mesh nodes are moved with the extra tangential velocity w(x,t), see (4). This velocity is determined based on the local characteristics of the mesh and interfaces (such as element size, curvature, film thickness and surfactant concentration gradients) by: (17) w(xi ) = (I − nn) ∑ {a + b.h(x j ) + c.|k(x j )| + d.|∇s Γ|}(x j − xi ) − u(xi ) + us , j∈Ni

where the summation involves only the nodes x j that are directly connected to xi ; us is an average velocity of the interface to which xi belongs. The term −u(xi ) + us in (17) eliminates possible mesh distortion due to the tangential component of the hydrodynamic velocity u. By a proper choice of the parameters a, b, c and d in (17) the mesh is maintained finer in the regions of higher gradients of k and h and Γ. This allows to achieve sufficiently good accuracy using reasonable number of elements. The mesh used for the present simulations consists of 8820 elements for the outer interface and 3380 elements per inner drop, in total 35860 elements. The steps for time integration of (4) used in the present study are of order 10−5 .

RESULTS In this section the numerical method described above is used for simulation of the dynamic behavior of a foam drop in simple shear flow in the presence of insoluble surfactant. The foam drop considered here consists of 8 inner drops (see figure 5), four of relative volume 13.5% and four of 10.25%. Thus the volume fraction of the foam drop is 95% and it has all structural elements of random polydisperse foams: plateau borders, junctions and films some of which with significant curvature. The initial shape is shown on the first row of figure 5, where the surfactant is distributed uniformly, at Γ = 1. The foam drop dynamics is shown on figure 5 at different time instances (the three rows respectively). The drop undergoes significant topological changes: the inner drops move inside the whole drop, changing their neighbors. This is related with disappearance of the existing films, plateau borders and junctions and formation of new ones. And the whole process being related to significantly nonuniform surfactant distribution. The results presented in this section are in reasonable qualitative agreement with the existing ones obtained for dry-film foams. Unfortunately, to the best of our knowledge, there are no other results in the literature suitable for a quantitative comparison.

CONCLUSIONS A 3D semi-implicit boundary-integral/finite-volume method is presented for simulation of foam dynamics in the presence of surfactant. A new element in the present study is the mutual effect of disjoining pressure and Marangoni stress. Both of them are results of the presence of insoluble surfactant and play very important role for the stability of foam configurations by suppressing the film drainage. Some of the elements of the numerical method, such as: nonsingular contour integration of the singular single layer potential, higher order approximation of the interfaces and dynamic mesh regularization are very important for the accuracy and the numerical stability. They allow simulation of film regions with significant curvature and interface-to-interface distance of order of 10−3 using linear triangular elements with side of order 10−1 . An important element here is the finite-volume method, allowing consideration of the effect of insoluble surfactant. The results for foam drop evolution in simple shear flow demonstrate the applicability of the method for simulation of foam dynamics at very high volume fraction (up to 95%) where most of the significant interfacial forces (Marangoni stress and van der Waals forces) are taken into account. The method can be extended straightforward for simulation of random polydisperse foams by applying periodic boundary conditions as in [3].

REFERENCES 1. 2. 3. 4. 5. 6. 7.

Kropinski M., Lushi E.: Efficient numerical methods for multiple surfactant-coated bubbles in a two-dimensional stokes flow Journal of Computational Physics 230 (2011) 4466–4487 Kraynik, A., Reinelt, D.: Foam microrheology: from honeycombs to random foams. in 15th Annual Meeting Polymer Processing Sosiety, ’s-Hertogenbosh, Netherlands, on CD-Rom (1999) Loewenberg, M., Hinch E.: Numerical simulation of concentrated emulsion in shear flow. Journal of Fluid Mechanics 321 (1996) 358–419 Yon, S., Pozrikidis C.: A finite-volume/boundary-element method for flow past interfaces in the presence of surfactants, with application to shear flow past a viscous drop. Computers & Fluids 27 (1998) 879–902 Bazhlekov, I., van de Vosse, F., Meijer, H.: Boundary integral method for 3D simulation of foam dynamics. Lecture Notes in Computer Science 2179 (2001) 401–408 Bazhlekov, I., Anderson, P., Meijer H.: Boundary integral method for deformable interfaces in the presence of insoluble surfactants. Lecture Notes in Computer Science 2907 (2004) 355-362 Bazhlekov, I.: Non-singular boundary-integral method for deformable drops in viscous flows. Ph.D. thesis, Eindhoven University of Technology, The Netherlands (2003)

FIGURE 5. Evolution of the foam drop in simple shear flow at Ca = 0.2; Pe = 100; β = 0.5 and A = 2.5 × 10−6 . The first column represents the outer interface and the second - the inner drops.