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.
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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.
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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
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]:
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
and
The spectral functions satisfy the so-called global relations:
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:
where \(d \in {\mathbb {C}}\) is a constant and
and
Addition of (68) and (70) and use of (67), (69), (71) gives, after some algebra,
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
together with conditions at \(k=0\) following from (77). Next, we use the series representations
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.
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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
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DOI: https://doi.org/10.1007/s10665-019-10025-7