Corrected trapezoidal rule for near-singular integrals in axi-symmetric Stokes flow

Abstract

The paper presents a method to evaluate near-singular boundary integrals in axi-symmetric interfacial Stokes flow for target points close to but not on the interface. The method consists of appoximating the integrand by a function with analytically available integral and numerically approximating the difference by a standard 4th-order trapezoidal rule with uniform mesh size h. The resulting correction is applied to all points at distance d sufficiently close to the boundary. The maximum errors are reduced from order O(h/d) to $O(h^m)$, where m depends on the number of terms in the approximating function. The paper presents all necessary details in the axi-symmetric Stokes case, implements specific $O(h^2)$, $O(h^3)$ and $O(h^4)$ versions, and shows that the analytically predicted convergence rates are attained for sample test cases. The method is applied to compute the evolution of double emulsions through a tapered nozzle and shown to resolve the unphysical tangling of interfaces that occurs with standard quadratures.

Appendices

A: The functions $M_{ij}, Q_{ijk}$  

$$\begin{aligned} M_{11}= & {} r(I_{10}+\xi ^2I_{30})/(8\pi ) \nonumber \\ M_{12}= & {} r\xi (rI_{30}-r_oI_{31})/(8\pi )\nonumber \\ M_{21}= & {} r\xi (rI_{31}-r_o I_{30})/(8\pi )\nonumber \\ M_{22}= & {} r[I_{11}+(r^2+r_o^2)I_{31}-rr_o(I_{30}+I_{32})]/(8\pi ),\nonumber \\ Q_{111}= & {} -6r\xi ^3I_{50}/(8\pi ),\nonumber \\ Q_{112}= & {} -6r\xi ^2(rI_{50}-r_oI_{51})/(8\pi ),\nonumber \\ Q_{122}= & {} -6r\xi (r_o^2I_{52}+r^2I_{50}-2rr_oI_{51})/(8\pi ),\nonumber \\ Q_{211}= & {} -6r\xi ^2(rI_{51}-r_oI_{50})/(8\pi ),\nonumber \\ Q_{212}= & {} -6r\xi [(r^2+r_o^2)I_{51}-rr_o(I_{50}+I_{52})]/(8\pi ), \nonumber \\ Q_{222}= & {} -6r[r^3I_{51}-r^2r_o(I_{50}+2I_{52}) +rr_o^2(I_{53}+2I_{51})-r_o^3I_{52}]/(8\pi ), \end{aligned}$$

(A.1)

where $\xi =z-z_0$ and $Q_{121}=Q_{112}, Q_{121}=Q_{212}$ with

$$\begin{aligned} I_{10}= & {} {4\over c}F(k),\nonumber \\ I_{11}= & {} {4\over c} a[bF(k)-E(k)],\nonumber \\ I_{30}= & {} {4\over c^3}{E_{3/2}(k)},\nonumber \\ I_{31}= & {} {4\over c^3}a[bE_{3/2}(k)-F(k)],\nonumber \\ I_{32}= & {} {4\over c^3}a^2 [b^2{E_{3/2}(k)}-2bF(k)+E(k)],\nonumber \\ I_{50}= & {} {4\over c^5}E_{5/2}(k),\nonumber \\ I_{51}= & {} {4\over c^5}a[bE_{5/2}(k)-E_{3/2}(k)],\nonumber \\ I_{52}= & {} {4\over c^5}a^2[b^2E_{5/2}(k) -2bE_{3/2}(k)+F(k)],\nonumber \\ I_{53}= & {} {4\over c^5}a^3[b^3E_{5/2}(k)-3b^2E_{3/2}(k)+3bF(k)-E(k)], \end{aligned}$$

(A.2)

where $\xi =z-z_o$, $k^2={4rr_o/(\xi ^2 + (r+r_o)^2)}$,

$$\begin{aligned} E_{3/2}=\frac{E(k)}{1-k^2}~,\quad E_{5/2}(k)= \frac{1}{3(1-k^2)} \left[ \frac{2(2-k^2)}{1-k^2}E(k)- F(k)\right] ~, \end{aligned}$$

F and E are the complete elliptic integrals of the first and second kind, respectively:

$$\begin{aligned} F(k)=\int _0^{\pi /2}\frac{d\theta }{\sqrt{1-k^2\sin ^2\theta }}~, \quad E(k)=\int _0^{\pi /2}\sqrt{1-k^2\sin ^2\theta }d\theta ~, \end{aligned}$$

and $a={2/k^2}$, $~b=(2-k^2)/2$, $~c^2={(r+r_o)^2+\xi ^2}$. These functions have singularities as $|{\mathbf{x}}-{\mathbf{x}}_0|^2=(r-r_o)^2+(z-z_0)^2\rightarrow 0.$ Using expansions for F, E as $k\rightarrow 1$ (see, eg, [1], we find that to leading order in this limit,

$$\begin{aligned} I_{1j}\approx & {} \frac{2}{r_0}\log |{\mathbf{x}}-{\mathbf{x}}_0|~,\nonumber \\ I_{3j}\approx & {} \frac{2}{r_0}\frac{1}{|{\mathbf{x}}-{\mathbf{x}}_0|^2}+\frac{a_{j}}{4r_0^3}\log |{\mathbf{x}}-{\mathbf{x}}_0|~,\nonumber \\ I_{5j}\approx & {} {4\over 3r_0}{1\over |{\mathbf{x}}-{\mathbf{x}}_o|^4} +\frac{b_{j}}{4r_0^3}{1\over |{\mathbf{x}}-{\mathbf{x}}_o|^2} +\frac{c_{j}}{64r_0^5}\log |{\mathbf{x}}-{\mathbf{x}}_o|~, \end{aligned}$$

(A.3)

where $a_{0}=1$, $a_{1}=-3$, $a_{2}=-7$, $b_{0}=1$, $b_{1}=-3$, $b_{2}=-7$, $b_{3}=-11$, $c_{0}=3$, $c_{1}=-5$, $c_{2}=19$, $c_{3}=75$ (obtained with Mathematica). These expressions yield equations (2.4) in §2.2.

B: The functions $F({\mathbf{x}},{\mathbf{x}_0})$

The functions $F_j({\mathbf{x}},{\mathbf{x}}_0)$ defined in Eq (2.4a) are listed here, in addition to their to leading order behaviour in $|{\mathbf{x}}-{\mathbf{x}}_0|$, as $|{\mathbf{x}}-{\mathbf{x}}_0|\rightarrow 0$:

$$\begin{array}{lll}F_2^{Q_{222}}(\mathbf x,{\mathbf x}_0)&=-\frac2{rr_0\rho_2}\left[r^2-r_0^2+\xi^2\right]\,\left[r^2-r_0^2-\xi^2\right]^2&\sim\;\vert\mathbf x-{\mathbf x}_0\vert^3\\F_1^{Q_{222}}(\mathbf x,{\mathbf x}_0)&=-\frac3{2rr_0\rho_2^3}\left[(r-r_0)^2(r+r_0)^3(3r+r_0)\right.&\\&\quad+(r-r_0)\xi^2(r+r_0)^2(5r+3r_0)&\\&\quad\left.-3\xi^4(r^2+4rr_0+3r_0^2)-5\xi^6\right]&\sim\;\vert\mathbf x-{\mathbf x}_0\vert^2\\F_0^{Q_{222}}(\mathbf x,{\mathbf x}_0)&=\frac3{16rr_0\rho_2^5}\left[-(r-r_0)(r+r_0)^3(r^2+22rr_0+9r_0^2)\right.&\\&\quad+\xi^2(r+r_0)^2(17r^2+46rr_0+33r_0^2)&\\&\quad\left.+3\xi^4(11r^2+24rr_0+13r_0^2)+15\xi^6\right]&\sim\;\vert\mathbf x-{\mathbf x}_0\vert^1\\F_2^{Q_{221}}(\mathbf x,{\mathbf x}_0)&=-\frac{4\xi}{r_0\rho_2}\left[(r^2-r_0^2)^2-\xi^4\right]&\sim\;\vert\mathbf x-{\mathbf x}_0\vert^3\\F_1^{Q_{221}}(\mathbf x,{\mathbf x}_0)&=-\frac{3\xi}{r_0\rho_2^3}\left[(r^2-r_0^2)^2+4(r+r_0)^2\xi^2+3\xi^4\right]&\sim\;\vert\mathbf x-{\mathbf x}_0\vert^3\\F_0^{Q_{221}}(\mathbf x,{\mathbf x}_0)&=-\frac{3\xi}{8r_0\rho_2^5}\left[(r+r_0)^2(5r^2+22rr_0+5r_0^2)\right.&\\&\quad\left.+8(r+r_0)^2\xi^2+3\xi^4\right]&\sim\;\vert\mathbf x-{\mathbf x}_0\vert^1\\F_2^{Q_{211}}(\mathbf x,{\mathbf x}_0)&=-\frac{8r\xi^2}{r_0\rho_2}\left[r^2-r_0^2+\xi^2\right]&\sim\;\vert\mathbf x-{\mathbf x}_0\vert^3\\F_1^{Q_{211}}(\mathbf x,{\mathbf x}_0)&=\frac{6r\xi^2}{r_0\rho_2^3}\left[r^2+4rr_0+3r_0^2+\xi^2\right]&\sim\;\vert\mathbf x-{\mathbf x}_0\vert^2\\F_0^{Q_{211}}(\mathbf x,{\mathbf x}_0)&=-\frac{3r\xi^2}{4r_0\rho_2^5}\left[r^2+8rr_0+7r_0^2+\xi^2\right]&\sim\;\vert\mathbf x-{\mathbf x}_0\vert^2\\F_2^{Q_{122}}(\mathbf x,{\mathbf x}_0)&=-\frac{4\xi}{r\rho_2}(r^2-r_0^2-\xi^2)^2&\sim\;\vert\mathbf x-{\mathbf x}_0\vert^3\\F_1^{Q_{122}}(\mathbf x,{\mathbf x}_0)&=-\frac{3\xi}{r\rho_2^3}\left[(r-r_0)(r+r_0)^2(5r+3r_0)\right.&\\&\quad\left.-2\xi^2(r^2+4rr_0+3r_0^2)-3\xi^4\right]&\sim\;\vert\mathbf x-{\mathbf x}_0\vert^2\\F_0^{Q_{122}}(\mathbf x,{\mathbf x}_0)&=\frac{3\xi}{8r\rho_2^5}\left[19r^4+48r^3r_0+16rr_0(r_0^2+\xi^2)\right.&\\&\quad\left.+3(r_0^2+\xi^2)^2+2r^2(21r_0^2+5\xi^2)\right]&\sim\;\vert\mathbf x-{\mathbf x}_0\vert^1\\F_2^{Q_{121}}(\mathbf x,{\mathbf x}_0)&=-\frac{8\xi^2}{\rho_2}(r^2-r_0^2-\xi^2)&\sim\;\vert\mathbf x-{\mathbf x}_0\vert^3\\F_1^{Q_{121}}(\mathbf x,{\mathbf x}_0)&=-\frac{6\xi^2}{\rho_2^3}\left[3r^2+4rr_0+r_0^2+\xi^2\right]&\sim\;\vert\mathbf x-{\mathbf x}_0\vert^2\\F_0^{Q_{121}}(\mathbf x,{\mathbf x}_0)&=\frac{3\xi^2}{4\rho_2^5}\left[7r^2+8rr_0+r_0^2+\xi^2\right]&\sim\;\vert\mathbf x-{\mathbf x}_0\vert^2\\F_2^{Q_{111}}(\mathbf x,{\mathbf x}_0)&=-\frac{16r\xi^3}{\rho_2}&\sim\;\vert\mathbf x-{\mathbf x}_0\vert^3\\F_1^{Q_{111}}(\mathbf x,{\mathbf x}_0)&=-\frac{12r\xi^3}{\rho_2^3}&\sim\;\vert\mathbf x-{\mathbf x}_0\vert^3\\F_0^{Q_{111}}(\mathbf x,{\mathbf x}_0)&=\frac{9r\xi^3}{2\rho_2^5}&\sim\;\vert\mathbf x-{\mathbf x}_0\vert^3\end{array}$$

$$\begin{array}{lll} F^{M_{22}}_1({\mathbf{x}},{\mathbf{x}}_0)= & {} \frac{1}{r_0\rho_2} \big[(r^2-r_0^2)^2-\xi^4\big] & \sim ~|{\mathbf{x}}-{\mathbf{x}_0}|^2\\ F^{M_{22}}_0({\mathbf{x}},{\mathbf{x}}_0)= & {} -\frac{1}{4r_0\rho _2^3}\big [(r+r_0)^2(5r^2-2rr_0+5r_0^2)& \\&+4\xi ^2(3r^2+4rr_0+3r_0^2) \big ]&\sim ~|{\mathbf{x}}-{\mathbf{x}_0}|^0\\&+\frac{|{\mathbf{x}}-{\mathbf{x}_0}|^2}{32r_0\rho _2^3}(13r^2+54 r r_0 + 13r_0^2) + O(|{\mathbf{x}}-{\mathbf{x}_0}|^4)&\\ F^{M_{21}}_1({\mathbf{x}},{\mathbf{x}}_0)= & {} \frac{2r\xi }{r_0\rho _2}(r^2-r_0^2+\xi ^2)&\sim ~|{\mathbf{x}}-{\mathbf{x}_0}|^2\\ F^{M_{21}}_0({\mathbf{x}},{\mathbf{x}}_0)= & {} \frac{r\xi }{2r_0\rho _2^3}(r^2+4rr_0+3r_0^2+\xi ^2)&\sim ~|{\mathbf{x}}-{\mathbf{x}_0}|^1\\&+\frac{|{\mathbf{x}}-{\mathbf{x}_0}|^2}{\rho _2^3}{\xi \over 4} + O(|{\mathbf{x}}-{\mathbf{x}_0}|^4)&\\ F^{M_{12}}_1({\mathbf{x}},{\mathbf{x}}_0)= & {} \frac{2\xi }{\rho _2}(r^2-r_0^2-\xi ^2)&\sim ~|{\mathbf{x}}-{\mathbf{x}_0}|^2\\ F^{M_{12}}_0({\mathbf{x}},{\mathbf{x}}_0)= & {} -\frac{\xi }{2\rho _2^3}(3r^2+4rr_0+r_0^2+\xi ^2)&\sim ~|{\mathbf{x}}-{\mathbf{x}_0}|^1\\& -\frac{|{\mathbf{x}}-{\mathbf{x}_0}|^2}{\rho _2^3}{\xi \over 4} + O(|{\mathbf{x}}-{\mathbf{x}_0}|^4)&\\ F^{M_{11}}_1({\mathbf{x}},{\mathbf{x}}_0)= & {} \frac{4r\xi ^2}{\rho _2}&\sim ~|{\mathbf{x}}-{\mathbf{x}_0}|^2\\ F^{M_{11}}_0({\mathbf{x}},{\mathbf{x}}_0)= & {} -\frac{r}{\rho _2^3}(2r^2+4rr_0+2r_0^2+3\xi ^2)&\sim ~|{\mathbf{x}}-{\mathbf{x}_0}|^0\\&-\frac{|{\mathbf{x}}-{\mathbf{x}_0}|^2}{2\rho _2^3}(r) + O(|{\mathbf{x}}-{\mathbf{x}_0}|^4)& \end{array}$$

where $\rho _2^2 = (r+r_0)^2+(z-z_0)^2$ and $\xi =z-z_0$. The Mathematica notebooks used to obtained these expansions are posted on Github [29], and a sample pdf posted in the supplementary materials.

C: Integrands for $r_0=0$. Evaluation for small $r_0=0$

If ${\mathbf{x}}_0$ lies on the axis, with radial component $r_0=0$, the expressions for $M_{ij}, Q_{ijk}$ given in Appendix A are not valid. Instead, the limiting expressions of the integrands in (2.3a) as $r_0\rightarrow 0$ are found by expanding M and Q about $r_0=0$ using known expansions of F(k) and E(k) about $k=0$. Here we list the resulting expressions, in addition to their 1st and 2nd derivatives in the radial direction at $r_0=0$ that are used for interpolation in an O(h) neighbourhood of $x_{ax}$. Note that the axial velocity components $u_s, u_d$ are even about $r_0=0$, while the radial components $v_s,v_d$ are odd. We only list the expressions for the nonzero functions that follow from this symmetry:

$$\begin{aligned} u_s(z_0,0)= & {} \frac{2\pi }{\mu }\int _a^b {r\over (r^2+\xi ^2)^{3/2}}[f_2r\xi +f_1(2\xi ^2+r^2)]\,d\alpha ~,\nonumber \\ \frac{\partial v_s}{\partial r_0}(z_0,0)= & {} \frac{\pi }{\mu }\int _a^b{r\over (r^2+\xi ^2)^{5/2}}(f_2r+f_1\xi )(r^2-2\xi ^2)\,d\alpha ~,\nonumber \\ \frac{\partial ^2 u_s }{\partial r_0^2}(z_0,0)= & {} \frac{\pi }{\mu }\int _a^b \frac{r}{(r^2+\xi ^2)^{7/2}} [3 f_2 r\xi (r^2-4\xi ^2)+ f_1(r^4+8r^2\xi ^2-8\xi ^4)]\,d\alpha ~,\nonumber \\ u_d(z_0,0)= & {} -12\pi \int _a^b {r\xi \over (r^2+\xi ^2)^{5/2}} (vr+u\xi )(r\dot{z}-\xi \dot{r})\, d\alpha ~,\nonumber \\ \frac{\partial v_d}{\partial r_0}(z_0,0)= & {} -{6\pi }\int _a^b \frac{r}{(r^2+\xi ^2)^{7/2}} \Big [ \dot{z}r[ vr(r^2-4\xi ^2) +u\xi (2r^2-3\xi ^2)]~,\nonumber \\&-\dot{r}\xi [ vr(2r^2-3\xi ^2) +u\xi (3r^2-2\xi ^2) ]\Big ]\, d\alpha ~,\nonumber \\ \frac{\partial ^2 u_d}{\partial r_0^2}(z_0,0)= & {} -{6\pi }\int _a^b \frac{r\xi }{(r^2+\xi ^2)^{9/2}} \Big [ \dot{z}(7r^4v+15r^3u\xi -26r^2v\xi ^2-20 r u\xi ^3+2v\xi ^4)~, \nonumber \\&-5\dot{r}\xi ( 3r^3v+5r^2u\xi -4rv\xi ^2-2u\xi ^3)\Big ] \, d\alpha ~. \end{aligned}$$

(C.1)

where throughout, $\xi =z(\alpha )-z_0$. The derivatives with respect to $\alpha$ of the integrands at the endpoints needed for the trapezoidal rule (3.3a) are computed using finite differences if ${\mathbf{x}}(a)$ and ${\mathbf{x}}(b)$ do not lie on the axis, or using exact expressions when they are on the axis, in which case $a=0$, $b=\pi$. These exact expressions are

$$\begin{aligned} {d\over d\alpha }[G^u_s]_e= & {} {4\pi f_{1,e}\dot{r}_e\over |\xi _e|},\quad {d\over d\alpha }\Big [\frac{\partial G^v_{s}}{\partial r_0}\Big ]_e =- {2\pi f_{1,e}\dot{r}_e\over |\xi _e|\xi _e},\quad {d\over d\alpha }\Big [\frac{\partial ^2 G^u_s}{\partial ^2 r_0}\Big ]_e =- {8\pi f_{1,e}\dot{r}_e\over |\xi _e|\xi _e^2},\nonumber \\ {d\over d\alpha }[G^u_d]_e= & {} {12\pi \dot{r}^2_e u_e\over \xi _e|\xi _e|},\quad {d\over d\alpha }\Big [\frac{\partial G^v_{d}}{\partial r_0}\Big ]_e = -{12\pi \dot{r}^2_e u_e\over \xi _e^2|\xi _e|},\quad {d\over d\alpha }\Big [\frac{\partial ^2 G^u_s}{\partial r_0^2}\Big ]_e = -{60\pi \dot{r}^2_e u_e \over \xi _e^3|\xi _e|}. \end{aligned}$$

(C.2)

where subscript e denotes evaluation at the endpoint (either $\alpha =0$ or $\alpha =\pi$). If $(z_0,0)$ is near the interface, the integrals (C.1) are near singular and are computed accurately using the corrections described in this paper for $p=3,5,7,9$.

The velocity and radial derivatives on the axis, given by (C.1), are used to evaluate the single and double layer Stokes potentials for target points in an $\epsilon$-neighbourhood of ${\mathbf{x}}_{ax}$, where the corrections $E^*[G]+E^*[B]$ lose accuracy, in order to obtain the uniform $O(h^3)$ convergence presented in Fig. 7. We obtain velocities in this region using a 6th order polynomial interpolant of the values on the axis and a value on the upper boundary of this region along an arc $r_a(\alpha ),z_a(\alpha )$ parallel to the interface.