A boundary element method for

Stokes ows with interfaces

Edoardo Alinovi and Alessandro Bottaro

DICCA, Scuola Politecnica University of Genova,

1 via Montallegro, 16145 Genova, Italy

Abstract

The boundary element method is a widely used and powerful technique to nu-merically describe multiphase ows with interfaces, satisfying Stokes' approxi-mation. However, low viscosity ratios between immiscible uids in contact atan interface and large surface tensions may lead to consistency issues as far asmass conservation is concerned. A simple and eective approach is describedto ensure mass conservation at all viscosity ratios and capillary numbers withina standard boundary element framework. Benchmark cases are initially con-sidered demonstrating the ecacy of the proposed technique in satisfying massconservation, comparing with approaches other solutions present in the litera-ture. The methodology developed is nally applied to the problem of slippageover superhydrophobic surfaces.

1. Introduction

Multiphase ows are ubiquitous in Nature and their prediction is of greatimportance in scientic and engineering applications. Over the last few decades,many computational methods has been developed to numerically investigate thisimportant class of complex ows. Among these, the boundary integral methods(BIM) assumes a considerable relevance from both theoretical and numericalpoint of view, particulary when the Stokes' approximation is applicable. Inthe Stokes ow regime, the Reynolds number is negligible and the governingequations become linear, allowing the reconstruction of the total ow eld byappropriate point sources and dipoles distribution at the boundaries of the uiddomain. The method has found noteworthy success in the simulation of emul-sions [1, 2] and droplet interactions [3, 4, 5] both in free space and boundeddomains. The main advantage of this method is that the ow equations aresolved only for the unknown stress and velocity elds at the domain boundariesand at uid interfaces, rather than in the bulk ow. The numerical counterpartof BIM is the boundary element method (BEM), which, in practice, recaststhe integral equations in discrete form for a set of elements of xed shape, ap-proximating the boundary of the domain/uid interfaces. The BEM is thus amesh-less method and this feature renders it a suitable method to study complex

Preprint submitted to Elsevier March 31, 2017

geometries and moving interfaces, without any additional complication relatedto the mesh quality or mesh deformation. In this paper attention is focusedon the problem of mass conservation inside closed volumes of uid creating aninterfaces with an outer uid. Calling λ the viscosity ratio between the innerand the outer uid, mass leakage may occurs when λ < 1, with increasing mag-nitude as the viscosity ratio decreases. This phenomenon was already pointedout by Pozrikidis [6], who also oered a correction, which involves a slight mod-ication of the governing integral equations. Here, we propose an alternativemethod based on the constraint of the interfacial velocity, forcing it to respectthe continuity equation. The procedure is applicable to any type of boundaryelement code and involves simple modications with respect to the standard al-gorithm. In the following we will rst present the mathematical formulation ofthe problem, describing the boundary integral equations arising from the anal-ysis of two-phase Stokes ows and their numerical treatment into a boundaryelement framework. We then present the results of the calculations of selectedbenchmarks cases. As nal example, we use the numerical method developed toapproach the problem of the slippage over superhydrophobic surfaces, calculat-ing protrusion heights (or Navier slip lengths), taking into account the viscosityratio and the deformation of the interface.

2. Mathematical formulation

We consider problems involving creeping ow in a region with dierent do-mains lled with immiscible, incompressible uids. The well-known governingequations for this type of problems read

µ∇2u = ∇p, ∇ · u = 0, (1)

where u is the uid velocity, p is the pressure and µ is the dynamic viscosity ineach given domain.

We use the boundary integral method to transform the dierential probleminto an integral one to be solved by the boundary element technique. Even if wederive the boundary integral formulation for a particular domain, sketched ingure 1, the same procedure applies to dierent shapes of the boundary, leadingto formally similar integral equations.

We start by considering a two-dimensional domain lled with two viscous

uids of viscosity ratio λ =µ1

µ2; uid 1 is found in domain Ω1, while uid

2 is contained within Ω2. The position of the interface, I, of unit normal nwhen seen from Ω1, must be found from the conservation equations. The basicidea underlying the boundary integral method is that a Stokes velocity eldin a generic domain Ω may be reconstructed using only values of the velocityand stress elds on the closed boundary C of the domain; this can be done byintroducing two integral operators. One is called the single-layer-potential andemploys a function Gij(x,x0) which represents the velocity eld generated by

2

y

x

2

1

h

ks

s

I

T

L R

W2W1

W3

n

y

x

Ω2

Ω1

Figure 1: Sketch of two dierent uid domains separated by the interface I. The letters T, L,R, I and W denote, respectively, the top, left, right boundaries, the interface and the wall. Inthe gure, L and R boundaries are periodic and W comprises all the walls of the cavity

a single point force at x; integrating with respect to the arclength around Cthe product of Gij times the surface force of strength fi(x) yields the velocityeld generated by the distribution of surface forces. The second integral, thedouble-layer-potential, is interpreted as a linear distribution in C of sources (orsinks) of strength u ·n plus a symmetric placement of point forces (i.e. dipoles).The single- and the double-layer potentials for Stokes ow in two dimensionsare:

FSLPj (x0,f ; C) =1

4πµ

∫Cfi(x)Gij(x,x0) dl(x), (2)

FDLPj (x0,u; C) =1

4π

∫Cui(x)Tijk(x,x0)nk(x) dl(x), (3)

where:

• Gij is the velocity Green's function for the two-dimensional Stokes ow,eventually satisfying specic boundary conditions for convenience;

• Tijk is the stress tensor associated to Gij ;

• x and x0 are, respectively, the so-called eld point and the singular pointin the interior of the domain Ω;

• fi = σijnj is the ith component of the stress acting on C, i.e. the boundary

traction;

• ui is the ith component of the velocity vector;

• nk is the kth component of the vector normal to C, conventionally pointinginwards, i.e. towards the uid;

3

We start with the boundary integral representation for the velocity u(1)j (x0)

in the lower uid, in the generic point x0 ∈ I [7]

1

2u

(1)j (x0) = − 1

λFSLPj (x0,f

(1);W3+I)+FDLPj (x0,u(1);W3)+FDLPj (x0,u

(1); I),

(4)where FDLP denotes the principal value of the double layer potential.

Repeating the same derivation for the velocity u(2)j (x0) in the upper uid,

we obtain an analogous representation

1

2u

(2)j (x0) = −FSLPj (x0,f

(2);T+W1 +W2 + I+ L+ R) +

FDLPj (x0,u(2);T+ L+ R) + FDLPj (x0,u

(2); I). (5)

We assume no-slip along W1,W2 and W3, while the left and right boundaries,L and R, are considered periodic. With these choices, equation (4) and (5)simplify in

1

2λu

(1)j (x0) = −FSLPj (x0,f

(1);W3 + I) + λFDLPj (x0,u(1); I), (6)

1

2u

(2)j (x0) = −FSLPj (x0,f

(2);W1 +W2 + I+ T) + FDLPj (x0,u(2);T+ I). (7)

It is worth noting that the contribution of the periodic boundaries cancels outfrom equation (5) only if the Green's function is chosen to be periodic. Next, weadd equations (6) and (7) and, recalling that the velocity is continuous acrossthe interface, we achieve the following nal form:

1 + λ

2uj(x0) = −FSLPj (x0,f ;W+ T) + FDLPj (x0,u;T)

−FSLPj (x0,∆f ; I) + (λ− 1)FDLPj (x0,u; I), (8)

with W = W1 + W2 + W3. The non-dimensional jump in traction through the

interface is ∆f = f1 − f2 =Kn

Ca, with Ca the capillary number, Ca =

µ1urefσs

;

σs is the surface tension present at the interface between uids 1 and 2, urefis some characteristic velocity of the problem and K is the local curvature ofthe interface. In the following, Ca will be understood to be a control parameterwhich tunes the rigidity of the uid interface.

Proceeding further, we reconsider an arbitrary point x0 ∈W3 but, this time,

we derive an alternative integral relation for the velocity u(1)j (x0) integrating

over the contour of domain Ω2 and taking advantage of the reciprocal theoremfor Stokes ow, leading to:

−FSLPj (x0,f(1);T+W1 +W2 + I) + FDLPj (x0,u

(1);T+ I) = 0. (9)

Recalling the orientation of the normal vector and the continuity of the velocityon the interface, summing with equation (6) we obtain:

4

1

2λuj(x0) = −FSLPj (x0,f ;T+W) + FDLPj (x0,u;T)

−FSLPj (x0,∆f ; I) + (λ− 1)FDLPj (x0,u(1); I) = 0. (10)

A third integral equation can be obtained proceeding in the same way as before:we take an arbitrary point x0 ∈W1,2, we integrate along the contour of domainΩ1 and we apply the reciprocal theorem, i.e

−FSLPj (x0,f(2);W3 + I) + λFDLPj (x0,u

(2); I) = 0. (11)

Again, we add equation (11) to equation (7) and end up with:

uj(x0)

2= −FSLPj (x0,f ;W+ T)−FSLPj (x0,f ;T) + FDLPj (x0,u;T)

−FSLPj (x0,∆f ; I) + (λ− 1)FDLPj (x0,u; I) = 0. (12)

If x0 ∈ T we obtain an equation formally similar similar to (12)

uj(x0)

2= −FSLPj (x0,f ;W+ T)−FSLPj (x0,f ;T) + FDLPj (x0,u;T)

−FSLPj (x0,∆f ; I) + (λ− 1)FDLPj (x0,u; I). (13)

Equations (8), (10), (12) and (13) are a system of integral equations for theunknown stresses along the solid walls, the interface velocity and the velocityor the stress on the top wall T, as function of the applied boundary conditions.

3. Numerical Method

The boundary integral problem described in the previous section is solvedusing the boundary element method (BEM). We reproduce the bottom bound-ary of the domain with NW discrete elements, the interface with NI elements,the top wall with NT elements and, nally, we apply the integral equations (8),(10), (12) and (13) at the elements' collocation points (at least one for each el-ement for low order BEM). Carrying out these operations, we produce a linearsystem, whose unknowns are the velocity or the stress at each collocation point.For the points laying over the top boundary we have

−DTT · uT +1

2uT + STW · fW

− (λ− 1)DTI · uI = −STT · fT − STI ·∆f I . (14)

Similarly, for the point on the bottom wall we have

−DWT · uT − (λ− 1)DWI · uI + SWW · fW = −SWT · fT − SWI ·∆f I . (15)

5

Finally, considering the collocation points on the interface

−DIT · uT + SIW · fW − (λ− 1)DII · uI +1 + λ

2uI = −SIT · fT − SII ·∆f I . (16)

The matrices S and D are called inuence matrices and are the discretizedcounterpart of the single-layer and the double-layer potential operators denedin (2) and (3). The rst letter in the superscript denotes the position of thecollocation point, while the second letter identies the piece of boundary overwhich the integral operator is being evaluated. Since the matrices D∗∗ havethe same size of the corresponding matrices S∗∗, only the size of the matricesS∗∗ is reported in table 1. Regarding the expression of the coecients of S andD, they strictly depends on the shape and the order of the interpolation of theboundary quantities along the elements.

STT 2NT × 2NT SWI 2NW × 2NI

SWT 2NW × 2NT SWW 2NW × 2NW

STW 2NT × 2NW SIT 2NI × 2NT

STI 2NT × 2NI SIW 2NI × 2NW

SWT 2NW × 2NT SII 2NI × 2NI

Table 1: Size of the discretized single-layer operator.

Equations (14), (15) and (16) can be condensed in a more suitable form as−DTT +

1

2I STW −(λ− 1)DTI

−DWT SWW −(λ− 1)DWI

−DIT SIW −(λ− 1)DII +1 + λ

2I

uT

fW

uI

=

−STT DTW −STI

−SWT DWW −SWI

−SIT DIW −SII

fT

0

∆f I

. (17)

where I denotes the identity matrix. Once the boundary quantities are known,

the internal eld can be reconstructed in both domains using the integral rep-resentation

λuj(x0) = −FSLPj (x0,f ;W+ T) + FDLPj (x0,u;T)

−FSLPj (x0,∆f ; I) + (λ− 1)FDLPj (x0,u; I), (18)

6

for x0 ∈ Ω1.

uj(x0) = −FSLPj (x0,f ;W+ T) + FDLPj (x0,u;T)

−FSLPj (x0,∆f ; I) + (λ− 1)FDLPj (x0,u; I), (19)

for x0 ∈ Ω2. The integral relations (18)-(19) for the internal eld reconstructioncan be treated as the boundary integrals equations without any added dicul-ties, since they do not contains singular integrals.

In our implementation, all the boundaries are approximated by cubic splinesegments, while we assume a linear variation of the boundary quantities alongeach element. The advantages of the cubic splines are the high delity in t-ting complex boundaries and the possibility of computing the curvature at thecollocation points directly from the polynomial representation of the target el-ement. We employ discontinuous elements at boundary's sharp corners to dealwith the singularity of the stress f , which often arises [7]. In this particularcase, the collocation points are placed in specied locations along the segments,obtained from the roots of the rst order Radau's polynomial. For all the otherelements, the collocation points are located at the extremities of the segment;in this case, since a collocation point is shared between two adjacent elements,the boundary quantities are approximated in a continuous way. The six pointsGauss-Legendre quadrature formula is employed for the numerical evaluation ofthe non-singular integrals arising from the boundary element formulation. Theweakly singular integral case, due to the structure of the Green's function for theStokes ow, which usually involves natural logarithms in the two dimensionalcase, is treated subtracting o the singularity from the integrand function andusing basic analytical techniques for the successive integration. The collocationpoints along the interface are advanced in time using the following rule

dx(i)

dt= (ui · ni)ni, i = 1, . . . , NI , (20)

where x(i) are the ith collocation point and ni is normal to the interface at x(i).Using the normal velocity to advanced the interface, instead of the velocityu, is found to be very eective in limiting the spreading of the collocationpoints, with the consequent advantage that the interface does not need to befrequently remeshed. Equation (20) can be discretized with any explicit schemefor ordinary dierential equations. We have implemented both the rst orderEuler and the second order Runge-Kutta (RK2) integration, nding very fewdierences between the two schemes. However, since the RK2 scheme requiresthe evaluation of the interfacial velocity at two dierent time steps, with theconsequent solution of the boundary element system, we prefer a simpler andfaster one-step integration.

A detailed review of the boundary element method for Stokes ows is givenin [8]. Details of the current implementation can be found in Appendix A.

7

3.1. Enforcement of mass conservation

One hidden issue in solving ows in the presence of interfaces, is that aunique solution of the integral equations cannot be found for arbitrary values ofthe viscosity ratio λ. This was shown in particular by Pozrikidis [7, 8], and thedrawback encountered in solving such equations is that a leak or an increase ofthe mass of the uid inside a closed domain may occur in time; this phenomenonbecomes more important as the viscosity ratio λ decreases [9]. One way to dealwith this problem and remove the non-uniqueness of the solution is proposed in[6] and requires adding the following term

zj(x0)

∫Cui(x)ni(x)dl (21)

to the double-layer potential along the interface into the integral equations.

Here zj(x0) is an arbitrary function such that

∫Czini 6= 0, with nj the normal

vector to the interface. The simplest choice is zj = nj and, since this terms shiftthe eigenvalues of the double-layer potential operator, the procedure is knownas deation.

Alternative method to ensure mass conservation is proposed here. We startby noting that each boundary integral problem can be reduced to a solution ofa linear system of the type:

Ax = b, (22)

where A and b are the boundary element matrix and the right-hand-side, de-pendent on the original boundary integral formulation of the problem, while xis the vector containing the unknowns (cf. equation (17)). For incompressibleows, the mass conservation inside a domain Ωi can be readily written as

∇ · u = 0, (23)

with u the velocity vector inside the domain Ωi. Integrating (23) over thevolume Ωi and taking advantage of Green's theorem we obtain∫

∂Ωi

u · n dS = 0. (24)

The integral relation (24) can be discretized in the same fashion as the single-layer and the double-layer potentials, leading to a simple linear equation of theform

c · u = 0, (25)

where c is a vector containing the coecients of the unknown velocity at thecollocation points. The form of the coecient depends, again, on the type ofcollocation method chosen to discretize the boundary integral equation. If weare in the presence of multiple uid volumes, we can easily extend expression(25) as

Cx = 0. (26)

8

The ith row of the matrix C contains the coecients arising from the dis-cretization of equation (24) for the ith uid domain. Clearly, the matrix C willpresent zero entries for those unknowns which are not the interfacial velocitiesto be constrained. We now add the set of constraints encapsulated in (26) tothe boundary element system (22). This is not an easy task since, usually, thesystem is already closed and simply adding an additional constraint equationwill lead to an over-determined system. Discharging as many equations as thenumber of constraints would be an available option, but it is not clear whichequations are to be substituted and a loss of accuracy might result. To solvethis issue, the idea is to introduce in the system each additional equation withassociated an unknown Lagrange multiplier Λ, which will render the boundaryelement system well balanced and force the solution to respect mass conservationfor any value of λ. We consider the following Lagrangian functional

L =1

2xTAx− xT b + ΛT (Cx), (27)

where the rst two terms in L function represent the potential energy of theunconstrained system, while the last term represents the energy needed to main-tain the constraints. Λ is a vector containing the Lagrange multipliers, one foreach interface within the domain Ω. Now, we proceed to minimize L, requiringthat its total variation, δL, is zero for every possible value of δx and δΛ, thus

δL =∂L∂x· δx +

∂L∂Λ· δΛ = 0, (28)

which leads to the following conditions over the gradient of the Lagrangianfunctional:

∂L∂x

= 0,∂L∂Λ

= 0. (29)

By imposing the conditions above, we produce a new linear system, which in-corporates the desired constraints:[

A CT

C 0

] [xΛ

]=

[b0

]. (30)

This method is of easy implementation and, since usually the boundary elementmatrix is dense, it does not destroy an eventually banded form of the nal ma-trix. However, the size of the matrix increases and this can become undesirablewhen a large number of interfaces is present.

9

4. Results

4.1. Relaxation of a two-dimensional droplet

We start by studying a simple benchmark case: the relaxation of a twodimensional droplet from an ellipse of given aspect ratio, as show in gure 2,to a circle. We assume that the droplet lies, initially at rest, in an innite freespace, lled with a dierent uid, and we monitor its evolution until reachingthe steady nal shape.

b

a

Figure 2: An elliptic droplet deforming into a circle. The right gure shows the evolution ofthe interface in time.

In order to perform this study we take advantage of the free space Green'sfunction and its associated stress tensor, which read

Gij(x,x0) = −δij log(r) +xixjr2

, Tijk(x,x0) = −4xixj xkr4

, (31)

where r is the distance between the points x and x0, while xi = xi − x0i .Regarding the boundary integral formulation, we note that this case corre-

sponds to solving the system[−DII +

1

2I][u]

= SII∆f I , (32)

for the interfacial velocity u. We can force the system to respect the massconservation constraint following the formulation in equation (30). Since in thiscase we have only one interface, the matrix C degenerates to a single equation,and, in practice, only one line and one column must be added to the originallinear system.

For this problem we consider three dierent values of the viscosity ratio, λ =1

10,

1

20,

1

100, and three dierent values of the capillary number, Ca = 0.1, 1, 10,

which tunes the rigidity of the interface and the velocity of the relaxation. Since

the aspect ratio of the ellipse isa

b= 2, we aspect that, after a transient, the uid

interface assumes a circular shape with radius equal to√

2 (provided b is initiallyset to one). For this simulation we use 60 spline elements and we employed axed time step ∆t = 0.01 for the lower capillary number, while ∆t = 0.05 for theothers. For the simulations, we have developed a boundary element code withembedded the proposed method. We have validated our implementation (with-out the Lagrange multiplier approach) against the code written by Pozrikidis

10

and publicly available with the library BEMLIB [10], nding indistinguishabledierences and the the same issue with mass conservation.

The results, reported in gure 3 and 4, compare the evolution of semi-majoraxis, a, in time for both the Lagrange multiplier approach and the standardformulation. Even if the initial transient path is similar, we note (symbols) acontinuous decrease of the semi-axis a after the droplet has reached the circularshape. The eect of this mass loss is enhanced as the viscosity ratio and thecapillary number become smaller. Imposing the constraint (24), the radius ofthe droplet remains constant in time, when t is suciently large, and equal to√

2 for all the values of λ and Ca tested.

0 20 40 60 80 1001.35

1.4

1.45

1.5

t

a

λ = 0.1λ = 0.05λ = 0.01

a

Figure 3: Evolution of the major semi-axis of the droplet for dierent values of λ and Ca = 0.1.The solid lines display results obtained with the Lagrange multiplier approach, which reachsteady state on a =

√2 for every value of λ. The initial relaxation of the droplet is independent

of the viscosity ratio.

0 20 40 60 80 100

1.35

1.55

1.75

1.95

t

a

Ca = 10Ca = 1Ca = 0.1

a

Figure 4: Evolution of the major semi-axis of the droplet for λ = 0.01 at dierent Ca. Thecircles denotes the variation of a with time, without using the Lagrange multiplier approach.The initial relaxation of the droplet is slower the larger is Ca, i.e for small surface tension thedroplet reaches its nal shape in a longer time.

11

0 10 20 30 40 50 60 70 801.35

1.4

1.45

1.5

1.55

1.6

t

aa

Figure 5: Comparison between deation (solid line) approach, Lagrange multiplier approach(empty circles) and standard formulation (dashed line) at λ = 0.01 and Ca = 1.

0 20 40 60 8010

−10

10−8

10−6

10−4

10−2

100

t

max(|u n|)

Figure 6: Maximum normal velocity history. The dashed line corresponds to the standard (un-constrained) implementation, the line with empty circles correspond to the deation approach,while the solid line correspond to the Lagrange multiplier approach.

In the previous section, we have mentioned the possibility to modify the expres-sion of the double-layer potential to satisfy the mass conservation for all possiblevalues of λ. We have thus performed again the simulations implementing in ourcode the proposed deation correction and have compared the results with theLagrange multiplier approach, obtaining a very good agreement between the twomethods, as shown in gure 5. This agreement corroborates the validity of ourapproch. However, as will be shown also in the next cases treated, we have foundthat the method of Lagrange multipliers yields better performance in term ofmass conservation. Focusing on the normal velocity along the interface, we take

12

0 0.2 0.4 0.6 0.8 1−0.2

−0.15

−0.1

−0.05

0

x

y

y

x

Figure 7: Sketch of the cavity with a wavy interface (left), and successive positions assumedby the interface during its relaxation into a parabolic shape (right).

its maximum absolute value as a convergence indicator. If the problem admitsa steady state solution, the interface should assume a position such that themaximum normal velocity vanishes. The comparison between the methods isshown in gure 6. We observe that the maximum normal velocity decreases untilreaching a plateau, whose value is dependent on the number of element used todiscretize the droplet and goes down as the number of elements increases. TheLagrange multipliers approach oers a better performance in minimizing themaximum normal velocity along the droplet's interface at steady state, whichis several orders of magnitude lower with respect to the method proposed in [6]at the same spatial resolution and time step.

4.2. Relaxation of a pinned interface

For this numerical example, we consider a simple cavity bounded by threewalls of length L and a uid interface, pinned at the corners of the cavity,as sketched in gure 7. The initial shape of the interface is a cosine wave ofequation

y0(x) = b cos(2π

Lx)− b,

where b is a constant. One way to nd out the nal position of the interfaceis to consider the Young-Laplace equation, expressed for convenience in non-dimensional form

d2y

dx2

[1 +

(dydx

)2]− 32

= C1, (33)

with C1 = ∆P the non-dimensional pressure jump across the interface and ythe vertical displacement of the interface.

If the curvature of the interface in small enough, the term in brackets inequation (33) tends to one, leading to the following approximate solution

y(x) =C1

2x(x− L), (34)

13

after imposing the boundary conditions

y(0) = 0, y(L) = 0. (35)

Particular attention should be payed to the constant C1: since the pressuredierence across the interface is not known a priori, its value can be calculatedimposing the conservation of mass inside the cavity through the relation:∫ L

0

y0dx =

∫ L

0

yfdx, (36)

which yields C1 =12b

L2. In order to obtain more precise results, equation (33)

can be solved without approximation using standard iterative techniques.For this simulation, we have employed 60 elements to discretize each edge of

the cavity and the interface. The uid in both the domains is initially at restand the interface moves as an eect of the surface tension. Periodic boundarycondition are applied to left and right boundaries by using the following Green'sfunction

A(x) =1

2log2[cosh(ωx2)− cos(αx1)], (37)

G11 = −A(x)− y ∂A(x)

∂y+ 1, (38)

G12 = y∂A(x)

∂x, (39)

G22 = A(x) + y∂A(x)

∂x. (40)

where x = x− x0 and α = 2πL . The components of the stress tensor Tijk are:

T111 = −4∂A(x)

∂x− 2x2

∂2A(x)

∂x∂y, T112 = −2

∂A(x)

∂y− 2x2

∂2A(x)

∂y∂y, (41)

T212 = 2x2∂2A(x)

∂x∂y, T222 = −2

∂A(x)

∂y+ 2x2

∂2A(x)

∂y∂y, (42)

with no need to specify the missing components of Gij and Tijk since they aresymmetric tensors.

In this case, we monitor the volume of the uid trapped between the cavitywalls and the interface, given, at each time, by the following integral relation

V =

∫Ω1

dS =1

2

∫Ω1

∇ · x dS =1

2

∫∂Ω1

x · n dl, (43)

which is integrated in the same fashion as other integral quantities. The results,reported in gures 8 and 9, shown a similar behavior to that observed in the

14

droplet relaxation benchmark. The total mass inside the cavity is not conservedin time and mass leakage becomes larger as the capillary number and the vis-cosity ratio become smaller. The usage of the deation approach (21) turnsout to be not as eective as in the previous case and the mass leakage (or cre-ation) persists, even if with a lower growth rate, as shown in gure 10. Instead,the Lagrange multiplier approach leads to very satisfactory results, maintainingconstant the mass inside the pocket and tting the theoretical steady-state po-sition of the interface prescribed by equation (33) for every test value of λ andCa, as shown in gure 11 for a representative case.Additional features arise from the analysis of the maximum absolute value ofthe normal velocity along the interface during the relaxation process, shownfor a representative set of parameters in gure 12. We note that the standardboundary element formulation is unstable: the initial decrease in the maximumvalue of the normal velocity is followed by an increase. This phenomenon brings,sooner or later, to the divergence of the simulation, with the interface breakingdown anomalously. The double layer-deation seems to counteract this unde-sirable eect, but it presents some diculties in bringing down the maximumnormal velocity below a reasonably low value. Again, the Lagrange multiplierapproach gives us the best result, yielding a much better convergence with re-spect to the other methods tested.

0 5 10 15 20 25 30 35 40−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

t

V−Vo

λ = 0.1λ = 0.05λ = 0.01

Figure 8: Time variation of the volume of uid contained inside the cavity (V0 is the initialvalue) for dierent values of λ and Ca = 0.1. The loss of uid within the cavity is enhancedas the viscosity ratio λ decreases. The solid line represents the mass variations in time for thesame cases when using the Lagrange multiplier approach.

15

0 5 10 15 20 25 30 35 40−0.04

−0.03

−0.02

−0.01

0

0.01

t

V−Vo

Ca = 10Ca = 1Ca = 0.1

Figure 9: Time variation of the volume of uid contained inside the cavity for dierent valuesof Ca and λ = 0.01. The loss of uid within the cavity is enhanced by a decreasing value ofCa. The solid line represents the mass variations in time for the same cases when using theLagrange multiplier approach.

0 5 10 15 20 25 30 35 40−10

−7.5

−5

−2.5

0

2.5x 10

−3

t

V−

Vo

Figure 10: Comparison between standard implementation (dashed line), double layer deation(empty circles) and Lagrange multiplier approach (solid line), for λ = 0.05 and Ca = 1.

16

0 0.25 0.5 0.75 1−0.2

−0.15

−0.1

−0.05

0

x

y

Figure 11: Position assumed by the interface starting from a co-sinusoidal shape (.− line)for λ = 0.05 and Ca = 0.1. The −N line represents the computed position for the standardboundary element implementation at t = 10.5, while the solid squares represent the nalsteady solution with the Lagrange multiplier correction, which at the same instant of time,agrees with the theoretical solution.

0 10 20 30 4010

−12

10−10

10−8

10−6

10−4

10−2

100

t

max

(|u n|)

Figure 12: Maximum normal velocity along the interface for λ = 0.01 and Ca = 1 forthe standard boundary element implementation (dashed line), double-layer deation (emptycircles) and Lagrange multiplier approach (solid line).

It is also interesting to compare the results obtained using the boundaryelement method with a standard Volume of Fluids method (VoF) implementedwith the nite volume framework provided by the openFoam software [11]. TheVoF method was introduced rstly by Hirt and Nichols [12] and it is based ondening an indicator function, called volume fraction, bounded between [0, 1].The extremities of the interval are associated to the two uids, while the inter-face is found in the cell with values between 0 and 1. This approach is widelyused to compute multiphase ows, but it is well know to suers of the undesired

17

phenomenon known as parasitic currents. This numerical issue consists in thegeneration of non-physical velocities near the uid interface and the phenomenonbecomes struggling in the presence of surface-tension-dominated ows. If themagnitude of these velocities is not very large, the method is able to capturethe interface with satisfactory accuracy, eventually generating small oscillationsof the volume fraction, but the ow elds will result unclean. To underline thisfact, we can look at the velocity magnitude inside the uid domain at steadystate, as shown in gure 13. For the nite volume computation, we have used a

ne Cartesian mesh, with a spacing between the grid points of1

300, over which

the stokes equation are solved. The viscosity ratio is set to λ = 0.018 and the

capillary number is Ca = 0.1, based on the velocity scale uref =ν2

L. We can

clearly see how the VoF produces signicant velocities in the proximity of theinterface, which persist in time, while the boundary element method does notsuers of this unwanted phenomenon.

Figure 13: Absolute velocity iso-surfaces in the proximity of the interface for the problemsketched in gure 7 using BEM with Lagrange multiplier correction (left) and VoF (right).

4.3. Superhydrophobic surfaces: the microscopic, transverse problem

Superhydrophobic (SH) surfaces are becoming popular because of their pos-sible use for skin-friction drag reduction, which makes this kind of coatingsattractive for dierent applications, spanning from micro-uidic devices for lab-on-a-chip operations [13, 14] to marine and underwater engineering [15]. Theworking mechanism of such surfaces is based on the presence of tiny roughnesselements at the wall, permeated by air or another lling gas, over which a liquidcan ow with low friction. In the present section, we consider the Stokes owpast the geometry sketched in gure 1, which represents the elementary cell ofthe microscopic problem. The wall pattern is assumed to be periodic in x andcomposed by a plane wall indented with a rectangular cavity, lled with a uid(uid 1) dierent from the upper uid (uid 2). The ow is driven by a constantshear stress of unit magnitude imposed on the boundary T . We denote by sthe periodicity of the wall texture and we tune with the parameter k the regionwhere the two uids can be in contact. Moreover we dene the quantity of gasinside the cavity, which is taken into account by introducing the volume fractionΦ, dened as:

Φ = 1 +

∫ ks

0

y0(x) dx

ksh, (44)

18

with y0 the initial position of the interface.

0 0.5 1 1.5 2 2.5 3 3.5 4−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1x 10−3

t

V−

Vo

λ = 0.1λ = 0.05λ = 0.01

Figure 14: Time variation of the mass inside the cavity (Vo is the initial value) for dierentvalues of λ and Ca = 0.1. The loss of uid within the cavity is enhanced by a decreasingvalue of the viscosity ratio λ. The solid line represents the mass variations in time for thesame cases, but using the Lagrange multiplier approach.

0 2 4 6 8 10

−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

t

V−

Vo

Ca = 10Ca = 1Ca = 0.1

Figure 15: Time variation of the mass inside the cavity for dierent values of Ca and λ = 0.01.The loss of uid within the cavity is enhanced by a decreasing value of the capillary numberCa. The solid line represents the mass variations in time for the same cases, but using theLagrange multiplier approach.

19

(a) (b)

0 1 2 3 4 5 6 7 8 9 10

10−8

10−6

10−4

10−2

100

t

max

(|u n|)

(c)

Figure 16: Vertical velocity eld generated in the domain at t = 10. a) Standard implemen-tation; b) the Lagrange multiplier approach; c) history of the maximum absolute value ofthe velocity normal to the interface for the standard implementation (dashed line) and theLagrange multiplier approach (solid line).

The main issue arising using the boundary element method as solving strat-egy, is the mass conservation inside the pocket. In order to underline thisbehavior, we consider a geometry with s = 1, k = 0.5, h = 0.5, φ = 0.9. Weuse 100 spline elements to discretize the interface, 50 for each segment of thewall dening the cavity and for the two remaining parts of the at wall, at theleft and the right of the cavity. The periodicity of the ow is applied by using aproper periodic Green's function, already dened in the previous section. Alsoin this case, we employ the Lagrange multiplier approach obtaining satisfactoryresults, as shown in gures 14 and 15. Again, the problem depends on theviscosity ratio between the two uids and on the rigidity of the interface in asimilar fashion as described in the previous sections.

20

Focusing on the ow eld, inside the domain, in gure 16(a)-(b), we comparethe same case, at λ = 0.1 and Ca = 0.1, with and without the use of Lagrangemultiplier, and let the simulation reach a suitable long time; we can clearlynote that the vertical component of the velocity degenerates in the standardcase. The interface is pushed downwards in time and a consistent loss of gasoccurs. Moreover, as shown in gure 16(c), the simulation without the use ofthe Lagrange multiplier is unstable: the maximum absolute value of the velocitynormal to the interface presents an initial rapid decrease followed by an increasewhich eventually leads to divergence of the procedure. The use of the proposedcorrection guarantees a good convergence

Once the issue with mass conservation inside the wall pocket is solved, wewish to use the BEM technique to quantify the skin friction drag reductionproduced by SH's. The problem of slippage along these kinds of coatings hasbeen studied by dierent authors since the seminal work by Philip [16], whoanalyzed the ow along a at wall patterned with alternating regions of no-slip and no-shear. What is known from analytical considerations [17, 18], isthat the velocity far from the rough walls, or with superhydrophobic patternedprotrusions reads

u(y) = y + b (45)

where b is a constant known as protrusion height or slip length which representsthe virtual distance below (or above) the surface where the u velocity componentwould extrapolate to zero. In the literature, there are several extensions of thework by Philip. In particular, Davis & Lauga [19] have proposed an analyticalmodel for the slip length b in the presence of a curved meniscus, over whicha perfect slip boundary condition is applied. In our numerical simulations, wemaintain the same set-up shown in gure 1 and we calculate the slip lengthwhen varying the viscosity ratio, the capillary number, the length of the cavityk, and the volume ratio Φ. According to our denition, Φ > 0 means that themeniscus is protruding outside the cavity.

In gure 17 we propose a comparison between the analytical model andour numerical simulations. The agreement is good, especially for the values ofk = 0.3 and k = 0.5, but this is not surprising, since the analytical model isvalid in the dilute limit, i.e. for small values of k. Indeed, if one compares thevalue of the slip length with k = 0.7 and at interface (Φ = 1), the model by

Davis and Lauga yieldsb

s= 0.0963, while both Philip's and our calculations

yieldb

s= 0.126. Increasing the viscosity ratio λ has the eect of decreasing

the slip length, since for small λ the approximation of perfect slip along theinterface is better. The present simulations also conrm the existence of acritical value of Φ for which the slip length becomes negative. This condition,already pointed out in refs. [20, 21], occurs when the interface has an excessiveprotrusion outside of the cavity. The largest value of the protrusion height isfound, almost independently of Ca and λ, when Φ is close to 1.05, i.e. whenthe interface is very mildly protruding out of the cavity, and this is probably

21

the most promising condition in terms of the skin friction drag reduction. Theinterface deforms under the action of the shear ow, as shown in gure 18, butfor a suciently low capillary number, it presents a very small deviation fromthe steady shape position that it would have in the absence of forcing ow. Inparticular, for Φ < 1, the interface is quasi-parabolic, well approximated byequation (34), and for Φ > 1 it assumes a position close to a circular arch,whose equation is reported in [19]. The ow eld generated inside the domainare reported in gure 19 for both Φ larger and smaller than one.

ϕ

(a) (b)

b/s

0.15

0.10

0.05

0

-0.05

-0.10

-0.15

0.8 0.9 1.0 1.1 1.2 1.3 1.4ϕ

0.8 0.9 1.0 1.1 1.2 1.3 1.4

b/s

0.15

0.10

0.05

0

-0.05

-0.10

-0.15

Figure 17: Comparison between the analytical model by Davis & Lauga [19] and the presentnumerical simulations. The solid lines correspond to the analytical model by Davis & Laugafor k = 0.3 (lower line), k = 0.5 (intermediate line), k = 0.7 (upper line). Symbols are thesimulations, for the same values of k, with λ = 0.018 (4), λ = 0.05 (2), λ = 0.1 (). a) Ca=1;b) Ca=0.1

22

Figure 18: Shape of the interface for Ca = 1 (left) and Ca = 0.1 (right) and λ = 0.1. Thevalues of Φ are 0.85, 0.90, 0.95, 1.10, 1.15, 1.25 and the ow is from left to right.

(a) (b)

(c) (d)

Figure 19: Iso-contours of the streamwise and wall normal velocity for λ = 0.1 and Ca = 1.The value of the volume ratio is Φ = 0.85 (a)-(b) and Φ = 1.15 (c)-(d).

23

5. Conclusions

A novel method to enforce mass conservation in multiphase Stokes owsproblems using the boundary element method has been presented. We have un-derlined how a viscosity ratio λ < 1 and the presence of a large interfacial tensioncause issues in mass conservation, oering clear examples. The method proposedis based on an easy-to-implement modication of the linear system obtained bythe discretization of the governing boundary integral equation, which consistsin adding one constraint equations for the interfacial velocity for each interfaceinside the domain of interest, by the use of Lagrange multipliers. The techniqueis very eective in limiting the mass leakage/creation for all the benchmarkproblems considered. In comparison with the deation method, introduced byPozrikidis [6], it achieves a better convergence, reducing the maximum absolutevelocity normal to the interface by several orders of magnitude and ensuringmass conservation even in cases where the double-layer deation approach fails.We have used our boundary element code to solve the problem of slippage oversuperhydrophobic surfaces, taking into account the eect of viscosity ratio andcapillary number while computing the deformation of the interface. The resultsare in good agreement with the theoretical predictions by Davis & Lauga [19]and conrm the presence of a maximum value of the slip length for a slightlyprotruding bubble.

6. Acknowledgment

This activity has started thanks to a gratefully acknowledged FincantieriInnovation Challenge grant, monitored by Cetena S.p.A. The authors thankGiacomo Gallino for interesting discussions.

24

A. Numerical details on the boundary element method

In section 3, we have briey described the numerical method employed inthis paper, giving an extended explanation only of its major features. In this ap-pendix, we will describe more extensively some numerical details. In particular,we consider the computation of the single-layer and the double-layer integraloperators, which are the entries of the matrices S∗∗ and D∗∗ in equation (17).

Ek-1

Ek

Figure 20: Sketch of two adjacent elements approximated by cubic splines. The • representthe collocations points dened at the end of each element.

The starting point is to dene the shape of the boundary element, which, inour case, is a spline connecting two collocation points, as shown in gure 20. Wedene a curvilinear abscissa, s, over the element Ek and we recast the operators(2) and (3) as

FSLPj (x0,u;Ek) =

∫ s2

s1

fki [x(s)]Gij [x(s),x0]hks(s) ds (46)

FDLPj (x0,u;Ek) =

∫ s2

s1

uki [x(s)]Tijl[x(s),x0]nl[x(s)]hks(s) ds, (47)

where hks(s) is the metric associated with the element:

hs(s) =

[(dx

ds

)2

+

(dy

ds

)2] 1

2

. (48)

We apply another coordinate transformation which maps an element fromthe global coordinate system based on the curvilinear abscissa to a local coordi-nate system such that the kth element's boundary points are mapped onto theinterval [−1, 1]. This mapping will result useful for the numerical quadrature ofboundary integrals and can be simply carried out using the following relation:

s(ζ) =(s1 + s2)

2+

(s2 − s1)

2ζ = sm + sdζ, (49)

from which we can easily dene the associated metric hζ = sd. Introducing this

25

Figure 21: Schematic view of an element parametrized using local coordinates ζ: the red crossmarks the position of the collocation points, while the black dot marks the starting and endingpoints of the element.

new parametrization into the integrals (46) and (47) we obtain:

FSLPj (x0,f ;Ek) = hkζ

∫ 1

−1

fki (x(s(ζ)))Gij(x(s(ζ)),x0)hs dζ, (50)

FDLPj (x0,u;Ek) = hkζ

∫ 1

−1

uki (x(s(ζ)))Tijk(x(s),x0)nk(x(s(ζ))) dζ.

(51)

Until now, no assumption has been made on the interpolation method of theboundary quantities over the element. We use a piecewise linear variation, whichis a good compromise between accuracy and programming diculty; thus, letus consider an element parametrised using the local coordinate ζ, as show ingure 21, and require that:

u(ζ) = ψ1(ζ)u1 + ψ2(ζ)u2, (52)

f(ζ) = ψ1(ζ)f1 + ψ2(ζ)f2, (53)

where ψ1 =l2 + ζ

Land ψ2 =

l1 + ζ

Lare shape functions. Introducing relations

(52) and (53) into the expression of the single- and double-layer integrals, wecan recast (50) and (51) as:

FSLPj (x0,f ;Ek) = fki A1ij + fk+1

i A2ij , (54)

FDLPj (x0,u;Ek) = ukiB1ij + uk+1

i B2ij , (55)

where Anij and Bnij are know tensors of the form:

Anij = hkζ

∫ 1

−1

ψn(ζ)Gij(ζ)hs(ζ) dζ, (56)

Bnij = hkζ

∫ 1

−1

ψn(ζ)Tijl(ζ)nl(ζ)hs(ζ) dζ. (57)

The integrals (56) and (57) can be computed numerically by using the Guass-Legendre quadrature rule, if the integrand is non-singular. The singular integralcase is more tricky and special techniques must be employed, as extensively

26

P2

P2

P1

Figure 22: Sketch of two boundary patches, with several collocation point dened over them.

illustrated in the following section. We consider now two adjacent elementssharing the kth collocation point, as shown in gure 20, and we write down thefollowing quantities

SLlk = A2ij |k−1k +A1

ij |kk, (58)

DLlk = B2ij |k−1k +B1

ij |kk, (59)

where in the notation |∗∗ the superscript stands for the element over which theintegral is being evaluated, while the subscript represents the collocation pointconsidered.

As an example, referring to the gure 22, we consider the assembling ofthe inuence matrix DP1P2 relative to the double-layer potential operator fortwo arbitrary patches, called P1 and P2, with NP1 and NP2 collocation points,respectively.

The discretized double-layer operator DP1P2 reads

DP1P2 =

DL1

1 DL11 . . . DL

NP21

DL11 DL1

1 . . . DLNP21

......

......

DL1NP1

DL2NP1

. . . DLNP2

NP1

; (60)

here DLlk stands for the quantities (59) calculated at the kth point belongingto P2 considering the l

th collocation point belonging to P1. Particular attentionshould be paid when a collocation point is not shared by two adjacent segments.In this case DLk turns out to be

DLk = B1ij |kk, (61)

if the collocation point is located on the left of the element, while

DLlk = B2ij |kk, (62)

if the collocation point is located on the right of the element. The assemblingmethodology is the same for other cases, with no dierence in the procedure ifwe consider the single-layer potential.

27

Non-singular integrals

The integrals (56)-(57) are the building blocks for the numerical solutionof the boundary integral equations. Let us recall the periodic velocity Green'sfunction and its associated stress tensor in order to highlight the problems thatmay arise in their numerical evaluation.Starting from Gij :

A(x) =1

2log2[cosh(ωy)− cos(ωx)], (63)

G11 = −A(x)− y ∂A(x)

∂y+ 1, (64)

G12 = y∂A(x)

∂x, (65)

G22 = A(x) + y∂A(x)

∂x. (66)

where x = x− x0 and ω = 2πL , with L the period of the ow. The components

of the stress tensor Tijk are:

T111 = −4∂A(x)

∂x− 2y

∂2A(x)

∂x∂y, T112 = −2

∂A(x)

∂y− 2y

∂2A(x)

∂y∂y, (67)

T212 = 2y∂2A(x)

∂x∂y, T222 = −2

∂A(x)

∂y+ 2y

∂2A(x)

∂y∂y. (68)

If the point x0 does not lay over the same element over which we are performingthe integration, integrals (56) and (57) are not singular and can be approximatedusing the Gauss-Legendre formula, using Nq quadrature points, as:

hkζ

∫ 1

−1

ψn(ζ)Gij(ζ)hs(ζ) dζ = hkζ

Nq∑q=1

ψn(ζq)Gij(ζq)hs(ζq)wq, (69)

hkζ

∫ 1

−1

ψn(ζ)Gij(ζ)hs(ζ) dζ = hkζ

Nq∑q=1

ψn(ζq)Tijl(ζq)nl(ζ)hs(ζq), (70)

where ζq is the position of the qth quadrature point along the interval [−1, 1]and wq is the associated weight.

Singular integrals

In the case of two-dimensional ows, as considered here, since the integrandof the double-layer potential exhibits a discontinuity across the collocation pointx0 special accommodations are not necessary. In contrast, the single-layer po-tential exhibits a logarithmic singularity for the diagonal component of G. Thebasic idea to solve this problem is to subtract o the singularity. Thus, turning

28

attention only to the term which contains the logarithm we add and subtracthsψ(s) log(r), r = |x− x0|, to the integrand in (46), obtaining:

− 1

2

∫ s2

s1

hs(s)ψn(s) log2[cosh(ωx2)− cos(ωx1)] ds =

− 1

2

[ ∫ s2

s1

hs(s)ψn(s) log

2

r[cosh(ωx2)− cos(ωx1)]

+

hsψn(s) log(r) ds

]. (71)

The rst term of the integrand is non-singular and can be accurately computedby Gauss-Legendre quadrature, but the second term involving log(r) is stillsingular and further manipulations are necessary. Calling s0 the curvilinearabscissa of the singular point, we add and subtract hs(s)ψn(s)log(|s− s0|) andrecast the integral as:∫ s2

s1

hsψn(s) log(r) ds =

∫ s2

s1

hs(s)ψn(s) log

(r

|s− s0|

)ds+∫ s2

s1

hsψn(s)log(|s− s0|) ds. (72)

Again, the rst term in the integrand is non-singular, but we must proceed tode-singularize the second term:∫ s2

s1

hs(s)ψn(s)log(|s−s0|) ds =

∫ s2

s1

[hsψn(s)−hs(s0)ψn(s0)

]log(|s−s0|) ds+∫ s2

s1

hs(s0)ψn(s0) log(|s− s0|) ds. (73)

Finally we can conclude the de-singularization noting that hs(s0)ψn(s0) is con-stant thus:∫ s2

s1

hs(s0)ψn(s0) log(|s− s0|) ds = hs(s0)ψn(s0)[|s1 − s0| log(|s1 − s0|)+

|s2 − s0| log(|s− s0|)]. (74)

29

Summing up, we can compute numerically the singular integral on the left-hand-side of (71) as:

− 1

2

∫ s2

s1

hs(s)ψn(s) log2[cosh(ωx2)− cos(ωx1)] ds =

−Nq∑q=1

hζwq2

[hs(ζq)ψn(ζq) log

2

r[cosh(ωx2(ζq))− cos(ωx1(ζq))]

+

hs(ζq)ψn(ζq) log

(r

|s(ζq)− s0|

)+[hs(ζq)ψn(ζq)− hs(s0)ψn(s0)

]log(|s(ζq)− s0|)

]−

hs(s0)ψn(s0)

2

(|s1 − s0| log(|s1 − s0|) + |s2 − s0| log(|s− s0|)

).

30

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