Motion of a solid particle in a bounded viscous flow using the Sparse Cardinal Sine Decomposition

Abstract

This work investigates the Sparse Cardinal Sine Decomposition (SCSD) method ability to efficiently deal with a Stokes flow about a solid particle immersed in a liquid. In contrat to Alouges and Aussal (Numer Algorithms 70:1–22, 2015), the liquid domain is bounded by a solid and motionless wall. The advocated procedure inverts on the particle and truncated wall boundaries the boundary-integral equation governing the stress there. This is numerically achieved by implementing a Galerkin method. The resulting linear system, with fully-populated and non-symmetric influence matrix, is both compressed and solved by the new SCSD method which allows to accurately deal with a large number of unknowns. Both analytical and numerical comparisons are reported for a spherical particle and several bounded liquid domains. Moreover, the rigid-body motion of spheroidal particles settling in a cylindrical tube is examined.

Appendix

The flow $(\mathbf{u},p)$ is obtained by superposing a few singularities placed at the sphere center $O'=O.$ Setting $r=|\mathbf{x}|,$$\mathbf{u}$ and p are sought in the following forms

$$\begin{aligned} \mathbf{u}= & {} {{\varvec{s}}\over {r}}+{{(\varvec{s}.\mathbf{x})\mathbf{x}}\over {r^3}} +3{{(\mathbf{d}.\mathbf{x})\mathbf{x}}\over {r^5}}-{{\mathbf{d}}\over {r^3}} +{{\varvec{\gamma }\wedge \mathbf{x}}\over {r^3}}+\mathbf{c}\nonumber \\&+\,\varvec{{\omega }}\wedge \mathbf{x}+2r^2\varvec{S}-(\varvec{S}.\mathbf{x})\mathbf{x} {\text{ for }} {a\le r\le R,} \end{aligned}$$

(38)

$$\begin{aligned} p= & {} 2\mu \left\{5\varvec{S}.\mathbf{x}+{{\varvec{s}.\mathbf{x}}\over {r^3}}\right\} {\text{ for }} {a\le r\le R.} \end{aligned}$$

(39)

Hence, $(\mathbf{u},p)$ consists of a rigid-body motion $\mathbf{c}+\varvec{{\omega }}\wedge \mathbf{x}$ and of four flows induced by singularties located at the sphere center: a Stokeslet with intensity $\mathbf{s},$ a potential dipole with intensity $\mathbf{d},$ a rotlet with intensity $\varvec{\gamma }$ and a Stokeson with intensity ${\varvec{S}}.$ The unknown vectors $\mathbf{c}, \varvec{{\omega }}, \varvec{s}, \mathbf{d}, \varvec{\gamma }$ and $\varvec{S}$ are obtained by enforcing the no-slip boundary conditions $\mathbf{u}=\mathbf{0}$ on the $r=R$ cavity and $\mathbf{u}=\mathbf{U}+\varvec{\varOmega }\wedge \mathbf{x}$ on the $r=a$ sphere surface. The following linear equations are obtained

$$\begin{aligned} \varvec{S}-{{\varvec{s}}\over {R^3}}-3{{\mathbf{d}}\over {R^5}}= & {} \mathbf{0}, 2R^2\varvec{S}+\mathbf{c}+{{\varvec{s}}\over {R}}-{{\mathbf{d}}\over {R^3}}=\mathbf{0}, \end{aligned}$$

(40)

$$\begin{aligned} \varvec{S}-{{\varvec{s}}\over {a^3}}-3{{\mathbf{d}}\over {a^5}}= & {} \mathbf{0}, {{\varvec{s}}\over {a}}-{{\mathbf{d}}\over {a^3}}+\mathbf{c}+2a^2\varvec{S}=\mathbf{U}, \end{aligned}$$

(41)

$$\begin{aligned} \varvec{{\omega }}+{{\varvec{\gamma }}\over {R^3}}= & {} \mathbf{0},{{\varvec{\gamma }}\over {a^3}}+\varvec{{\omega }}=\varvec{\varOmega }. \end{aligned}$$

(42)

Setting $\lambda =a/R,$ elementary algebra easily yields the analytical solution

$$\begin{aligned}&\varvec{s}={{3a(1-\lambda ^5)\mathbf{U}}\over {4[1-{{9}\over {4}}\lambda +{{5}\over {2}}\lambda ^3-{{9}\over {4}}\lambda ^5+\lambda ^6]}}, \end{aligned}$$

(43)

$$\begin{aligned}&\mathbf{d}={{(1-\lambda ^3)a ^2\varvec{s}}\over {3(\lambda ^5-1)}}, \mathbf{c}={{(4+5\lambda ^3-9\lambda ^5)\varvec{s}}\over {3a(\lambda ^5-1)}}+\mathbf{U}, \end{aligned}$$

(44)

$$\begin{aligned}&\varvec{\gamma }={{a^3\varvec{\varOmega }}\over {1-\beta ^3}}, \varvec{{\omega }}=-{{\lambda ^3\varvec{\varOmega }}\over {a^3}}, \varvec{S}={{\lambda (\lambda ^2-1)\varvec{s}}\over {aR^2(\lambda ^5-1)}} . \end{aligned}$$

(45)

From (38) the sphere experiences a force $\mathbf{F}$ and a torque ${\varvec{\Gamma }}$ (about its center $O')$ given by $\mathbf{F}=-8\pi \mu \varvec{s}$ and ${\varvec{\Gamma }}=-8\pi \mu \varvec{\gamma }.$ Accordingly, one gets the announced results (31)–(32).