Contour-Integral Representation of Single and Double Layer Potentials for Axisymmetric Problems Emilia G. Bazhlekova and Ivan B. Bazhlekov Institute of Mathematics, BAS, acad. G. Bonchev str., bl. 8, 1113 Sofia, Bulgaria
Abstract. Based on recently proposed non-singular contour-integral representations of single and double layer potentials for 3D surfaces, formulas in the axisymmetric case are derived. They express explicitly the singular layer potentials in terms of elliptic integrals. The presented expressions are non-singular, satisfy exactly very important conservation principles and directly take into account the multivaluedness of the double layer potential. The results are compared with another method for calculating the single and double layer potentials. The comparison demonstrates higher accuracy and better performance of the presented formulas.
1
Introduction
Significant advance has been made recently in the numerical simulation of complex multiphase Stokes flows using boundary integral methods (BIM), see e.g. [1,2,3]. An essential part of BIM is the calculation of the single and double layer potentials in the case of constant density. The main difficulties are due to the singularity of the kernels: x /| x /| x| + x x|3 ; T(x, x0 ) = −6 xx x|5 ; P(x, x0 ) = 2 x/| x|3 , G(x, x0 ) = II/| )ij = x = x − x0 , ( xx i x j and II is the unit tensor. Consider also the where x following auxiliary kernel: x /| x/| x|3 − 6 xx x|5 . Q(x, x0 ) = T(x, x0 ) + IIP(x, x0 ) = 2II
(1)
Note that G has first order singularity at x = x0 and T, P and Q - second order. The following identities satisfied for arbitrary closed surface Sc with outward unit normal n (see e.g. [4]) express important conservation properties: G(x, x0 ) · n(x) ds(x) = Q(x, x0 ) · n(x) ds(x) = 0; (2) Sc
Sc
Corresponding author. Present address: Section Materials Technology, Faculty of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands;
[email protected] (This work was supported by the Dutch Polymer Institute)
I. Dimov et al. (Eds.): NMA 2002, LNCS 2542, pp. 387–394, 2003. c Springer-Verlag Berlin Heidelberg 2003
388
Emilia G. Bazhlekova and Ivan B. Bazhlekov
T(x, x0 ) · n(x) ds(x) = −cII;
Sc
The constant c in (3) represents the 8π, c = 4π, 0,
Sc
P(x, x0 ) · n(x) ds(x) = c.
(3)
discontinuity of the potentials at x = x0 : if x0 inside Sc ; if x0 on Sc ; if x0 outside Sc .
Different approaches to perform the integration of the singular kernels exist in the literature, for example: higher order integration rules combined with a mesh refinement around the singular point (see e.g. [1] and [3]), in this case the identities (2-3) are not satisfied exactly; ”near-singularity” subtraction (see e.g. [2]), where the identities (2-3) are satisfied exactly, however, the accuracy could worsen locally around the singular point. This problem is tackled also in our recent study [6], where layer potentials are expressed by contour integrals, (4-6). These representations serve as a starting point for the present work. Consider a surface, for example drop interface, and let D be a part of it. It is assumed that D and its contour Γ are bounded and piecewise smooth. Let nD be the unit normal vector to D, outward to the surface. Let t be the unit tangential to Γ vector, defined by t = b × nD , where b is the unit normal to Γ vector, lying in the tangential plane to D, see figure 1.
Fig. 1. Schematic sketch of the surface D and its contour Γ in 3D. The following formulas, proposed in [6], express the single and double layer potentials in the case of density f ≡ 1 by means of contour integrals: t(y) × y G(x, x0 ) · nD (x) ds(x) = dl(y); (4) | y | D Γ
D
D
Q(x, x0 ) · nD (x) ds(x) = 2
P(x, x0 ) · nD (x) ds(x) = 2
Γ
( y y × t(y)) dl(y) | y|3
(5)
a · ( y × t(y)) dl(y) + c, ) | y|(| y| + a · y
(6)
Γ
Contour-Integral Representation of Single and Double Layer Potentials
389
= y − x0 and a is an arbitrary unit vector. The constant c in (6) is where y 0, −4π or −8π depending of the orientation of a and Γ , see [6], and guarantees the satisfaction of (3). Compared to the existing methods for calculation of the single and double layer potentials, the above contour integral representations have some important advantages: they are non-singular, satisfy exactly the identities (2-3) and offer higher accuracy for the numerical integration. In the following section formulas analogous to (4-6) are derived in the case when D is an axisymmertic surface.
2
Contour Integration
In this work we consider an axisymmetric surface D and apply formulas (4-6) to obtain explicit representations of the single and double layer potentials with constant density in terms of complete elliptic integrals of the first, second and third kind. Introduce a cylindrical coordinate system (z, r, ϕ). It is known that in the case of axisymmetric surface D none of the integrals (4-6) is a function of the azimuthal angle ϕ. Thus, a considerable simplification results from analytical integration along rings centered about the axis of symmetry.
Fig. 2. Schematic sketch of an axisymmetric surface D and its contour K 1 ∪ K 2 . In the axisymmetric case the contour Γ of D consists of two circles: Γ = K 1 ∪ K 2 , see figure 2. Denote the integrals in the r.h.s. of (4), (5) and (6), where Γ is a circle K, by IG (x0 , K), IQ (x0 , K) and I P (x0 , K), respectively: t(y) × y dl(y); IG (x0 , K) = | y| K ( y y × t(y)) Q I (x0 , K) = 2 dl(y); | y|3 K a · ( y × t(y)) dl(y) + c. I P (x0 , K) = 2 ) y|(| y| + a · y K |
390
Emilia G. Bazhlekova and Ivan B. Bazhlekov
Our goal is to find explicit representations of IG , IQ and I P as functions of x0 and x, where x is the intersection point of K and the half-plane ϕ = 0. Let x = (z, r, 0) and x0 = (z0 , r0 , 0) in cylindrical coordinates. Then y = = (z − z0 , r cos ϕ − r0 , r sin ϕ), t(y) = (0, − sin ϕ, cos ϕ), (z, r cos ϕ, r sin ϕ), y and dl(y) = r dϕ, see figure 2. Therefore, the above integrals depend only on x0 and x and read as follows ( note that in the axisymmetric case IG reduces to a vector with two non-zero components and IQ reduces to a 2 × 2 tensor ): 2π ( r0 cos ϕ − r , (z − z0 ) cos ϕ ) G dϕ, (7) I (x0 , x) = r | y| 0 IQ (x0 , x) = −2r
0
with A=
2π
A dϕ, | y|3
(8)
(z − z0 )(r0 cos ϕ − r) (z − z0 )2 cos ϕ 2 2 2 r0 r − (r0 + r ) cos ϕ + r0 r cos ϕ (z − z0 )(r cos ϕ − r0 ) cos ϕ
and I P (x0 , x) = 2r
2π
0
r − r0 cos ϕ dϕ + c, | y|(| y| + z0 − z)
,
(9)
where | y| = ((z − z0 )2 + r02 + r2 − 2rr0 cos ϕ)1/2 . It has been taken a = (−1, 0, 0). Formulas (7) and (8) imply that IG (x0 , x) = r( r0 E(1, 1) − rE(0, 1) , (z − z0 )E(1, 1) ) and Q
I (x0 , x) = −2r(z − z0 )
r0 E(1, 3) − rE(0, 3) (z − z0 )E(1, 3) (z − z0 )E(1, 3) rE(2, 3) − r0 E(1, 3)
(10) . (11)
Here E(i, m) are defined by E(i, m) = 0
2π
(A2
cosi φ dφ − B 2 cos φ)m/2
(12)
with A = ((z − z0 )2 + r2 + r02 )1/2 and B = (2rr0 )1/2 . Functions E(i, m) can be expressed in terms of elliptic integrals as follows 4 Em/2 (k) (13) Cm 4 E(1, m) = m 2 (A2 Em/2 (k) − C 2 E(m−2)/2 (k)) C B 4 E(2, m) = m 4 (A4 Em/2 (k) − 2A2 C 2 E(m−2)/2 (k) − C 4 E(m−4)/2 (k)), C B 4 E(3, m) = m 6 (A6 Em/2 (k) − 3A4 C 2 E(m−2)/2 (k) + 3A2 C 4 E(m−4)/2 (k) C B − C 6 E(m−6)/2 (k)) E(0, m) =
Contour-Integral Representation of Single and Double Layer Potentials
where
391
π/2
dφ , (14) 2 cos2 φ)m/2 (1 − k 0 √ C = (A2 + B 2 )1/2 = ((z − z0 )2 + (r + r0 )2 )1/2 and k = 2B/C = 2(rr0 )1/2 ((z − z0 )2 + (r + r0 )2 )−1/2 . Note that E1/2 (k) and E−1/2 (k) are the complete elliptic integrals of the first and second kind, respectively, and (see [5]): Em/2 (k) =
E3/2 (k) =
E−1/2 (k) 2(2 − k 2 ) 1 E1/2 (k). (15) , E (k) = E−1/2 (k) − 5/2 1 − k2 3(1 − k 2 )2 3(1 − k 2 )
In this way, substituting (13) and (15) in (10) and (11), we obtain explicit representations of IG (x0 , x) and IQ (x0 , x) in terms of complete elliptic integrals of the first and second kind. Concerning I P (x0 , x), (9) implies after somewhat longer calculations: 4(z − z0 ) r − r0 P Π(a, k) , (16) E1/2 (k) + I (x0 , x) = r + r0 ((z − z0 )2 + (r + r0 )2 )1/2 where k and E1/2 (k) are defined as above, a = 2(rr0 )1/2 /(r + r0 ) and Π(a, k) is the complete elliptic integral of third kind, defined by:
π/2
Π(a, k) = 0
(1 −
a2
cos2
dφ . φ)(1 − k 2 cos2 φ)1/2
(17)
It remains to recall that Γ = K 1 ∪ K 2 in the axisymmetric case and we obtain from (4-6) the final results: G(x, x0 ) · nD (x) ds(x) = IG (x0 , x2 ) − IG (x0 , x1 ); (18) D
D
D
Q(x, x0 ) · nD (x) ds(x) = IQ (x0 , x2 ) − IQ (x0 , x1 );
(19)
P(x, x0 ) · nD (x) ds(x) = I P (x0 , x2 ) − I P (x0 , x1 );
(20)
where x1 and x2 are the boundary points of the trace C of D in ϕ = 0, see figure 2, and IG (x0 , x), IQ (x0 , x) and I P (x0 , x) are given by (10), (11) and (16), respectively. The representation for the double layer potential follows applying (1) and the above results: T(x, x0 )·nD (x) ds(x) = IQ (x0 , x2 )− IQ (x0 , x1 )− II(I P (x0 , x2 )− I P (x0 , x1 )). D
(21) A numerical verification of the above-derived formulas (18-21) is presented in the following section.
392
3
Emilia G. Bazhlekova and Ivan B. Bazhlekov
Comparison with Other Methods
Different methods exist for calculation of the layer potentials in a vicinity of the singular point, x0 ∈ D. In general, they are based on the following representations of the single and double layer potentials for axisymmetric interfaces, see e.g. [4]: D
G(x, x0 ) · nD (x) ds(x) =
D
T(x, x0 ) · nD (x) ds(x) =
C
C
Mij nj (x) dl(x),
(22)
qijk nk (x) dl(x),
(23)
where C is the trace of the surface in the ϕ = 0 half-plane. The free index i refers to the r or z component, and the repeated indices j and k are summed over the r and z components. The components of the coefficient matrices M and q presented in [4], are given in the Appendix for completeness. For our comparison we take the following particular case. The surface D is considered here to be a segment of the unit sphere centered at (0, 0, 0) bounded π between the planes z = ± sin 72 . Thus, the trace C of D in ϕ = 0 half-plane is π π , cos 72 , 0) the arc of the unit circumference with boundary points x1 = (− sin 72 π π 2 and x = (sin 72 , cos 72 , 0). The pole x0 is chosen to move along the axis r: x0 = (0, r0 , 0). This choice of C and x0 is made to demonstrate the advantages of the proposed formulas (18-21) around the singular point r0 = 1. On figure 3 the present result, (18), for the second component I2G of the single layer potential IG (the first component is zero) is compared with the numerical ones based on
Fig. 3. Comparison for I2G between the two representations: (18) obtained using the contour integration (thicker line) and (22) - by integration on the line C (thinner lines).
Contour-Integral Representation of Single and Double Layer Potentials
393
(22). For the normal vector nD (x) in (22-23) the exact value for the sphere D is taken. The integration on C is performed dividing it into N − 1 even segments and applying the trapezoidal rule in each of them. T of the double The same procedure is used for the first diagonal element I11 T layer potential I (the non-diagonal elements are zero). On figure 4 the r.h.s.
T Fig. 4. Comparison for I11 between the results obtained using the contour integration (21) (thicker line) and the line integration (23) (thinner lines).
of our formula (21) is compared with the r.h.s. of (23). Note that the discontinuity of the diagonal elements of the double layer potential in the singular point x = x0 is automatically taken into account in our formula. Regarding the performance of the considered methods: the contour integration is equivalent to the line integration on C for N = 2 and the CPU time for the line integrals is proportional to N . The calculation in the present section for both methods: contour and line integration, are performed using the Mathematica software.
4
Conclusions
In the present study we offer contour-integral representations of layer potentials with constant density for axisymmetric problems. The proposed formulas (18-21) express explicitly the layer potentials over an axisymmertric surface D as functions of the boundary points, x1 and x2 , of its trace C in the ϕ = 0 half-plane. Thus, they offer an efficient and accurate method for numerical calculation of the layer potentials. They are non-singular when x0 ∈ D in contrast to the surface integrals. Another very important feature of the proposed contour-integral representations is that they satisfy exactly the identities (2-3) for a closed interface.
394
Emilia G. Bazhlekova and Ivan B. Bazhlekov
In addition, the normal vector is automatically taken into account in them. The presented comparisons in the previous section demonstrate the above-mentioned advantages.
5
Appendix
The components of the coefficient matrices M and q, presented in [4]: √ r (z − z0 )2 Mzz = 2k √ E (k) , E1/2 (k) + −1/2 r0 (r − r0 )2 z − z0 r2 − r2 + (z − z0 )2 Mzr = k √ E (k) , E1/2 (k) − 0 r0 r (z − z0 )2 + (r − r0 )2 −1/2 √ (z − z0 ) r r02 − r2 − (z − z0 )2 E1/2 (k) − Mrz = −k E−1/2 (k) , 3/2 (z − z0 )2 + (r − r0 )2 r0
2 k (r0 + r2 + 2(z − z0 )2 )E1/2 (k) Mrr = 3/2 r1/2 r0 2(z − z0 )4 + 3(z − z0 )2 (r02 + r2 ) + (r02 − r2 )2 E−1/2 (k) ; − (z − z0 )2 + (r − r0 )2 qzzz = −6(z − z0 )3 E(0, 5), qzzr = qzrz = −6r(z − z0 )2 (rE(0, 5) − r0 E(1, 5)), qzrr = −6r(z − z0 )(r02 E(2, 5) + r2 E(0, 5) − 2rr0 E(1, 5)), qrzz = −6r(z − z0 )2 (rE(1, 5) − r0 E(0, 5)), qrzr = qrrz = −6r(z − z0 )((r2 + r02 )E(1, 5) − rr0 (E(0, 5) + E(2, 5))), qrrr = −6r(r3 E(1, 5) − r2 r0 (E(0, 5) + 2E(2, 5)) + rr02 (E(3, 5) + 2E(1, 5)) − r03 E(2, 5)). Applying (13) and (15), all components can be written in terms of complete elliptic integrals of the first and second kind.
References 1. Stone H.A., Leal L.G.: Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid. J. Fluid Mech. 198 (1989) 399 2. Davis R.H.: Buoyancy-driven viscous interaction of a rising drop with a smaller trailing drop. Phys. of Fluids 11 no.5 (1999) 1016 3. Kwak S., Fyrillas M.M. and Pozrikidis C.: Effect of surfactants on the instability of a liquid thread. Part II: Extensional flow. Int. J. Multiphase Flow 27 (2001) 39 4. Pozrikidis C.: Boundary-Integral and Singularity Methods for Linearized Viscous Flow. Cambridge U.P., Cambridge (1992) 5. Byrd P.F., Friedman M.D.: Handbook of Elliptic Integrals for Engineers and Scientists. Springer-Verlag, New York (1971) 6. Bazhlekov I., Bazhlekova E.: Non-singular contour-integral representation of single and double layer potentials. (in preparation)
