Skip to main content
Log in

Analytical solutions for two-dimensional singly periodic Stokes flow singularity arrays near walls

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

New analytical representations of the Stokes flows due to periodic arrays of point singularities in a two-dimensional no-slip channel and in the half-plane near a no-slip wall are derived. The analysis makes use of a conformal mapping from a concentric annulus (or a disc) to a rectangle and a complex variable formulation of Stokes flow to derive the solutions. The form of the solutions is amenable to fast and accurate numerical computation without the need for Ewald summation or other fast summation techniques.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Crowdy DG, Or Y (2010) Two-dimensional point singularity model of a low-Reynolds-number swimmer near a wall. Phys Rev E 81:036313

    Article  Google Scholar 

  2. Jeong J-T, Moffatt HK (1992) Free-surface cusps associated with flow at low Reynolds number. J Fluid Mech 241:1–22

    Article  MathSciNet  Google Scholar 

  3. Hasimoto H (1959) On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres. J Fluid Mech 5:317

    Article  MathSciNet  Google Scholar 

  4. Pozrikidis C (1996) Computation of periodic Green’s functions of Stokes flow. J Eng Math 30:79–96

    Article  MathSciNet  Google Scholar 

  5. Pozrikidis C (1987) Creeping flow in two-dimensional channels. J Fluid Mech 180:495–514

    Article  Google Scholar 

  6. Pozrikidis C (1992) Boundary integral and singularity methods for linearized viscous flow. Cambridge University Press, New York

    Book  Google Scholar 

  7. Davis AMJ (1993) Periodic blocking in parallel shear or channel flow at low Reynolds number. Phys Fluids 5:800–809

    Article  MathSciNet  Google Scholar 

  8. Lindbo D, Tornberg A-K (2011) Spectral accuracy in fast Ewald-based methods for particle simulations. J Comput Phys 230:8744–8761

    Article  MathSciNet  Google Scholar 

  9. Tornberg A-K, Greengard L (2008) A fast multipole method for the three-dimensional Stokes equations. J Comput Phys 227:1613–1619

    Article  MathSciNet  Google Scholar 

  10. Hernandez JP, de Pablo JJ, Graham MD (2007) Fast computation of many-particle hydrodynamic and electrostatic interactions in a confined geometry. Phys Rev Lett 98:140602

    Article  Google Scholar 

  11. Crowdy DG, Luca E (2018) Fast evaluation of the fundamental singularities of two-dimensional doubly periodic Stokes flow. J Eng Math 111:95–110

    Article  MathSciNet  Google Scholar 

  12. Langlois WE (1964) Slow viscous flows. Macmillan, New York

    Google Scholar 

  13. Luca E, Crowdy DG (2018) A transform method for the biharmonic equation in multiply connected circular domains. IMA J Appl Math 83:942–976

    Google Scholar 

  14. Moffatt HK (1964) Viscous and resistive eddies near a sharp corner. J Fluid Mech 18:1–18

    Article  Google Scholar 

  15. Davis AMJ, Crowdy DG (2012) Matched asymptotics for a treadmilling low-Reynolds-number swimmer near a wall. Q J Mech Appl Math 66:53–73

    Article  MathSciNet  Google Scholar 

  16. Mannan FO, Cortez R (2018) An explicit formulae for two-dimensional singly-periodic regularized Stokeslets flow bounded by a plane wall. Commun Comput Phys 23:142–167

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

DGC was supported by an EPSRC Established Career Fellowship (EP/K019430/10) and by a Royal Society Wolfson Research Merit Award. Both authors acknowledge financial support from a Research Grant from the Leverhulme Trust.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Elena Luca.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Transform method

Appendix A: Transform method

We briefly discuss how to extend a novel transform approach to biharmonic boundary value problems for both polygonal and circular domains recently described by the authors in [13] to reappraise the problem of a periodic array of point singularities in a channel geometry thereby offering an alternative to the solutions given in Sect. 5.

The Goursat functions can be represented by

$$\begin{aligned} f(z)=f_{s}(z)+f_{R}(z), \qquad g'(z)=g'_{s}(z)+g'_{R}(z), \end{aligned}$$
(63)

where \(f_{s}(z)\), \(g'_{s}(z)\) are related to the point singularity at \(z_{0}\) (for example, for a point stresslet at \(z_{0}\), they are given by (7)) and \(f_{R}(z)\), \(g'_{R}(z)\) are the correction functions to be found. The functions \(f_{R}(z)\), \(g'_{R}(z)\) are analytic and single-valued in the fluid region and they have the following integral representations [13]:

$$\begin{aligned} f_{R}(z)=\frac{1}{2 \pi } \left[ \sum _{j=1}^{4} \int _{L_j} { \rho _{j}(k) \mathrm {e}^{\mathrm{i} k z} \mathrm {d}k} \right] , \qquad g'_{R}(z)=\frac{1}{2 \pi } \left[ \sum _{j=1}^{4} \int _{L_j} { \hat{\rho }_{j}(k) \mathrm {e}^{\mathrm{i} k z} \mathrm {d}k} \right] , \end{aligned}$$
(64)

where \(L_j\), \(j=1,2,3,4\) are oriented rays from 0 in the spectral k-plane [13] and \(\rho _{j}(k), \hat{\rho }_{j}(k)\), \(j=1,2,3,4\) are the spectral functions defined by

$$\begin{aligned} \begin{aligned} \rho _{1}(k)&=\int _{0}^{l} {f_{R}(z) \mathrm {e}^{-\mathrm{i} k z} \mathrm {d}z}, \qquad ~~~\rho _{2}(k)=\int _{l}^{l+\mathrm{i}h} {f_{R}(z) \mathrm {e}^{-\mathrm{i} k z} \mathrm {d}z}, \\ \rho _{3}(k)&=\int _{l+\mathrm{i}h}^{\mathrm{i}h} {f_{R}(z) \mathrm {e}^{-\mathrm{i} k z} \mathrm {d}z}, \qquad \rho _{4}(k)=\int _{\mathrm{i}h}^{0} {f_{R}(z) \mathrm {e}^{-\mathrm{i} k z} \mathrm {d}z}, \end{aligned} \end{aligned}$$
(65)

and

$$\begin{aligned} \begin{aligned} \hat{\rho }_{1}(k)&=\int _{0}^{l} {g'_{R}(z) \mathrm {e}^{-\mathrm{i} k z} \mathrm {d}z}, \qquad ~~~ \hat{\rho }_{2}(k)=\int _{l}^{l+\mathrm{i}h} {g'_{R}(z) \mathrm {e}^{-\mathrm{i} k z} \mathrm {d}z}, \\ \hat{\rho }_{3}(k)&=\int _{l+\mathrm{i}h}^{\mathrm{i}h} {g'_{R}(z) \mathrm {e}^{-\mathrm{i} k z} \mathrm {d}z}, \qquad \hat{\rho }_{4}(k)=\int _{\mathrm{i}h}^{0} {g'_{R}(z) \mathrm {e}^{-\mathrm{i} k z} \mathrm {d}z}. \end{aligned} \end{aligned}$$
(66)

The spectral functions satisfy the so-called global relations:

$$\begin{aligned} \sum _{j=1}^{4} {\rho _{j}(k)}=0, \qquad \sum _{j=1}^{4} {\hat{\rho }_{j}(k)}=0, \qquad \text {for}\; k \in {\mathbb {C}}. \end{aligned}$$
(67)

The analysis of the boundary and periodicity conditions allows us to deduce relations between the spectral functions. We omit the details and report the key expressions; these are:

$$\begin{aligned}&-\overline{\rho _{1}}(-k) -\frac{\partial [k \rho _{1}(k)]}{\partial k}+\hat{\rho }_{1}(k)+ l f_{R}(l) \mathrm {e}^{-\mathrm{i}kl}=R_{1}(k), \end{aligned}$$
(68)
$$\begin{aligned}&\rho _{4}(k)+\mathrm {e}^{\mathrm{i}kl} \rho _{2}(k)+d ~ q(k)=R_{2}(k), \end{aligned}$$
(69)
$$\begin{aligned}&-\mathrm {e}^{2kh} \overline{\rho _{3}}(-k)-\frac{\partial [k \rho _{3}(k)]}{\partial k}+2kh \rho _{3}(k)+\hat{\rho }_{3}(k) +r(k)=R_{3}(k), \end{aligned}$$
(70)
$$\begin{aligned}&-\mathrm{i}k l \rho _{4}(k)+\hat{\rho }_{4}(k)+\mathrm {e}^{\mathrm{i}k l} \hat{\rho }_{2}(k)-l f_{R}(0)+l f_{R}(\mathrm{i}h) \mathrm {e}^{kh} +\overline{d} ~ q(k)=R_{4}(k), \end{aligned}$$
(71)

where \(d \in {\mathbb {C}}\) is a constant and

$$\begin{aligned} R_{1}(k)&\equiv \int _{0}^{l} {[\overline{f_{s}(z)}-z f'_{s}(z)-g'_{s}(z)] \mathrm {e}^{-\mathrm{i}kz} \mathrm {d}z}, \end{aligned}$$
(72)
$$\begin{aligned} R_{2}(k)&\equiv \int _{\mathrm{i}h}^{0} {[-f_{s}(z)+f_{s}(z+l)] \mathrm {e}^{-\mathrm{i}kz} \mathrm {d}z}, \end{aligned}$$
(73)
$$\begin{aligned} R_{3}(k)&\equiv \int _{l+\mathrm{i}h}^{\mathrm{i}h} {[\overline{f_{s}(z)}-(z-2\mathrm{i}h) f'_{s}(z)-g'_{s}(z)] \mathrm {e}^{-\mathrm{i}kz} \mathrm {d}z}, \end{aligned}$$
(74)
$$\begin{aligned} R_{4}(k)&\equiv \int _{\mathrm{i}h}^{0} {[l f'_{s}(z) - g'_{s}(z)+g'_{s}(z+l)] \mathrm {e}^{-\mathrm{i}kz} \mathrm {d}z} \end{aligned}$$
(75)

and

$$\begin{aligned} q(k) \equiv \int _{\mathrm{i}h}^{0} {\mathrm {e}^{-\mathrm{i} kz} \mathrm {d}z}, \qquad r(k)=-\mathrm{i}h f_{R}(\mathrm{i}h) \mathrm {e}^{kh}-(l-\mathrm{i}h) f_{R}(l+\mathrm{i}h) \mathrm {e}^{-\mathrm{i}k(l+\mathrm{i}h)}. \end{aligned}$$
(76)

Addition of (68) and (70) and use of (67), (69), (71) gives, after some algebra,

$$\begin{aligned} \rho _{1}(k)=\frac{2kh W(k)-(\mathrm {e}^{2kh}-1) \overline{W}(-k)}{4 [\sinh ^2(kh)-k^2 h^2]}, \end{aligned}$$
(77)

where W(k) contains \(\rho _{4}(k), \hat{\rho }_{4}(k), f_R(0)\), \(f_R(\mathrm{i}h), d\) and known quantities. The spectral function \(\rho _{1}(k)\) is analytic everywhere in the complex k-plane which means that its numerator in (77) must vanish at zeros of its denominator in the k-plane, i.e. we must require

$$\begin{aligned} 2kh W(k)-(\mathrm {e}^{2kh}-1) \overline{W}(-k)=0, \qquad \text {for} \quad k \in \Sigma _{1} \equiv \{k \in {\mathbb {C}}| \sinh ^2(kh)-k^2 h^2]=0\}, \end{aligned}$$
(78)

together with conditions at \(k=0\) following from (77). Next, we use the series representations

$$\begin{aligned} f_{R}(z)= \sum _{m} {a_{m} T_m(z)}, \quad g_{R}(z)=\sum _{m} {b_{m} T_m(z)}, \quad \text {along} \quad z=\mathrm{i}y, y\in [0,h], \end{aligned}$$
(79)

where \(a_{m}, b_{m} \in {\mathbb {C}}\) are unknown coefficients and \(T_m(z)\) are basis functions (e.g. Fourier, Chebyshev), and truncate the sums in (79) to a finite number of terms. Then, we formulate a linear system for the unknown coefficients \(a_{m}\), \(b_{m}\), parameter d and their complex conjugates. The linear system comprises conditions (78) evaluated at as many points in the set \(\Sigma _{1}\) as needed, together with conditions at \(k=0\). Once the unknowns are computed from the solution of the truncated linear system, the spectral functions \(\rho _{4}(k)\) and \(\hat{\rho }_{4}(k)\) can be found. The remaining spectral functions can be found by back substitution into (68)–(71), and therefore the correction functions \(f_{R}(z)\) and \(g'_{R}(z)\) can be computed.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Crowdy, D., Luca, E. Analytical solutions for two-dimensional singly periodic Stokes flow singularity arrays near walls. J Eng Math 119, 199–215 (2019). https://doi.org/10.1007/s10665-019-10025-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-019-10025-7

Keywords

Navigation