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Boundary-integral formulation in 2d-axisymmetric configuration
Since we restrict our analysis to the axisymmetric√configuration depicted in Figure 1.1, we adopt cylindrical coordinates (r, φ, z) with r = x2 + y2, z = x3 and φ the azimuthal angle in the range [0, 2π]. The boundary-integral equation (1.52) is then reduced to a one-dimensional boundary-integral equation over the entire contour L = L0 ∪ L1 of the boundaries in the azimuthal plane (L0 and L1 being the contour of the free surface and the bubble, respectively). Therefore, the boundary-integral equation (1.52) for the unknown velocity u on the liquid boundary is expressed in our 2d-axisymmetric configuration as 4πuα(x0) − ZL1 1 Z Bαβ(x, x0)[−ρlgz + γ0rS · n]nβ(x)dl(x) L.
− Cαβ(x, x0)uβ(x)dl(x) = −µ 0.
− ZL1 Bαβ(x, x0)[−ρlgz + γ1rS · n]nβ(x)dl(x) for x0 on L (2.1) µ.
where α, β are either r or z (the radial or axial components) and dl denotes the diﬀerential arc length in the φ = 0 plane. By performing an exact integration of the velocity Green tensor G and the stress Green tensor T (see §1.4.2) over the azimuthal angle φ, one obtains the single-layer and double-layer 2 × 2 square matrices Bαβ(x , x0) and Cαβ(x, x0) given in Pozrikidis  and recalled in Appendix A. As outlined in this latter Appendix, all the components of those matrices are solely expressed in terms of the following complete elliptic integrals of the first E(k) and second kind F (k).
Sensitivity to the number of boundary elements on the bubble contour
Numerical simulations have been performed for a bubble and a free surface having identical surface tensions γ1 = γ0, therefore the considered Bond number in the present test is Bo1 = Bo0 = 1. The initial (normalized by the bubble diameter) distance between the spherical bubble and the plane undeformed free surface is h = 0.5.
We first investigate the sensitivity to the number of boundary elements on the bubble by running simulations for diﬀerent numbers Neb of elements on the bubble taking Neb = 5, 10, 15 and 20 with a given truncated free surface extending over 5 bubble diameters and meshed using 25 boundary elements. Moreover, 4 collocation points are uniformly spread on each boundary element and the smallest element on the free surface therefore presents a typical length of 1/5 in bubble diameter. In the same manner, at initial normalized time t = 6µ/(ρlga), the smallest boundary element length on the bubble surface are π/10, π/20, π/30 and π/40 for Neb = 5, 10, 15, 20, respectively.
In the case of a distant bubble shown in Figure 2.7 (a), the bubble shapes nicely gather on the same curve. We further call “error” the diﬀerence between two solutions obtained respectively using a low number of element and a large number of element. The biggest error appears between the shapes obtained for Neb = 10 and Neb = 20 and corresponds to an error of 1% in bubble diameter. In contrast, Figure 2.7 (b) shows a strong sensitivity of both the bubble and the free surface shapes to the number of boundary elements on a close bubble. This sensitivity decreases when the number of boundary element increases. Focusing on the upper part of the bubble shape along the z = 0 axis, a diﬀerence of Δhb = 3.8 10−2 is observed (hb is the height of the bubble) with shapes obtained for Neb = 5 and Neb = 20 corresponding then to an error of 3.8% in bubble diameter while the comparison between the case Neb = 10 and Neb = 20 exhibits a variation of 0.6% in bubble diameter.
Sensitivity to the number of boundary elements on the free surface
Similarly to the previous test, the bubble and free surface shapes sensitivity to the number of boundary elements, Nef , spread on the free surface for Bo1 = 1 and a spherical bubble initially distant of one radius from the free surface have been examined. Those shapes are computed for a bubble surface consisting of 20 boundary elements and a truncated free surface extending over 5 bubble diameters with diﬀerent numbers Nef = 5, 10, 20, 25 of boundary elements.
Sensitivity to the free surface truncature
As explained in §I.2.2.1, the numerical treatment of the prescribed bubble-free surface interaction problem is achieved using a truncated free surface.
The size of such a truncated free surface directly dictates the computational time cost since one requires to supplement or to remove boundary elements from the surface to keep the typical length of those elements constant at initial time. For instance, boundary elements with typical length of 1/5 correspond to either a free surface extending over 5 bubble diameters and composed of N ef = 25 boundary elements or a 10 bubble diameters free surface with Nef = 50 boundary elements. One aims then at reducing the computational time cost by investigating the sensitivity of the bubble and truncated free surface to the distance L of truncation for the free surface.
As for the previous benchmark tests, the bubble is initially distant of one bubble diameter, the surface tensions are identical on the bubble and the free surface and Bo1 = 1.The number of boundary elements spread on the bubble and free surfaces are equal to Neb = 20 and Nef = 25, respectively and the computations are performed for various truncation distances L = 2, 3, 4, 5, 10 (in bubble diameters) with Nef = 10, 15, 20, 25, 50 boundary elements spread on the free surface. The number of elements Nef has been selected to keep the typical length (1/5) of the smallest element on the free surface constant as the distance L increases.
As seen in Figure 2.16(a), the bubble and the free surface shapes are unaﬀected by changing truncation. The diﬀerence with the shapes obtained for L = 2, Ne = 10 and L = 5, Ne = 25 is of order of Δh = 4.5 10−3. Denoting by = Δh/h the relative error on the free surface height at the z = 0 axis, on gets = 1.3 10−2 in the present case which therefore corresponds to a suitable error of 1.3% in bubble diameter.
Table of contents :
I One bubble interacting with a free surface
1 Governing Problem and advocated boundary approach
1.1 Addressed time-dependent problem
1.2 Formulation at each time step
1.3 Boundary formulation and resulting relevant boundary -integral equation .
1.3.1 Numerical method to track a time-dependent interface
1.3.2 Suitability of the boundary integral formulation
1.3.3 Free-space Green tensor and relevant velocity integral representation
1.3.4 Boundary integral equation
1.4 Summary step to track in time the entire liquid boundary
2 Numerical implementation and benchmark tests
2.1 Boundary-integral formulation in 2d-axisymmetric configuration
2.2 Numerical implementation
2.2.1 Mesh and collocation method
2.2.2 Isoparametric interpolation
2.2.3 Discretized boundary-integral equation
2.2.4 Regular and weakly singular integrals
2.2.5 Wielandt deflation
2.3 Benchmark tests
2.3.1 Sensitivity to the number of boundary elements on the bubble contour
2.3.2 Sensitivity to the number of boundary elements on the free surface
2.3.3 Sensitivity to the bubble initial location
2.3.4 Sensitivity to the free surface truncature
3 Numerical Results: surface tension effects
3.1 Bubble and free surface shapes evolution in time
3.1.1 Sensitivity to the surface tension
3.1.2 Princen shapes
3.2 Time evolution of the film thickness between the bubble and the free surface
3.2.1 Sensitivity to the Bond number
3.2.2 Sensitivity to the surface tension
4 Asymptotic analysis at small Bond number for a bubble or a solid particle near a free surface in 2D-axisymmetric configuration
4.1 Governing equations and resulting zeroth-order and first-order flows
4.1.1 Axisymmetric problem and assumptions. Flow expansion
4.1.2 Zeroth-order flow and first-order flow problems
4.2 Zeroth-order solution in bipolar coordinates
4.2.1 Bipolar coordinates
4.2.2 Stream function, pressure and resulting drag force
4.3 First-order free surface deformation
4.3.1 Governing problem for the free surface shape
4.3.2 Solution in cylindrical coordinates
4.3.3 Solution in bipolar coordinates
4.4 First-order bubble shape
4.4.1 Governing problem for the bubble shape
4.4.2 Solution in closed form
5 Asymptotic analysis: benchmark tests and results
5.1 Zeroth-order problem: results and validation
5.2 Numerical results for the disturbed free surface shape
5.2.1 Comparison of the two proposed methods: case of a solid sphere
5.2.2 Free surface shapes in the case of one bubble
5.2.3 Comparisons with the BEM
5.2.4 Free surface shapes at small Bond number
5.2.5 Bubble shape: preliminary results
5.3 Conclusions on the asymptotic analysis
II Clusters consisting in M 0 bubble(s) and N 1 solid particle(s) with N +M 2.
6 Gravity-driven migration of bubble(s) and/or solid particle(s) near a free surface
6.1 Governing general problem and advocated trick
6.1.1 Governing equation and key remarks
6.1.2 Determination of each solid body velocity
6.2 Advocated boundary formulation
6.2.1 Three dimensional velocity integral representation
6.2.2 Axisymmetric formulation
7 Numerical results for several bubble(s) and/or solid particles
7.1 Cluster made of two bubbles
7.1.1 Comparison with the case of one bubble
7.1.2 Two bubbles with identical size and surface tension
7.1.3 Two bubbles different in size or in surface tension
7.2 Cluster involving at least one solid body
7.2.1 Bubble-sphere cluster
A Simple and double-layer operators in axisymmetric formulation
B Static shapes problem
B.1 Bubble-liquid interface equation
B.2 Bubble-free surface (contact area) interface equation
B.3 Bulk interface equation
B.3.1 Equation for the fluid interface shape
B.3.2 Boundary condition to calculate the asymptotic value L
C Bipolar coordinates
C.2 Vectors and metric coefficients
C.2.1 Transformation of partial derivatives
C.2.2 Associated unit vectors and Jacobian
D Free surface shape functions using cylindrical coordinates
D.1 Solution to the homogenous differential equation
D.2 The free surface shape function
E Solution to the linear problem for the bubble shape
F Expression of the zeroth-order stress and velocity components in bipolar coordinates
F.1 Expression of the zeroth-order velocity in bipolar coordinates
F.2 Zeroth-order stress tensor at the undisturbed free surface in terms of the