Spectral solver for the oscillatory Stokes frequency-based equation in doubly periodic confined domains

Oscillatory flows induced by a monochromatic forcing frequency $\omega$ close to a planar surface are present in many applications involving fluid–matter interaction such as ultrasound, vibrational spectra by microscopic pulsating cantilevers, nanoparticle oscillatory magnetometry, quartz crystal microbalance and more. Numerical solution of these flows using standard time-stepping solvers in finite domains present important drawbacks. First, hydrodynamic finite-size effects scale as $1/L_{\parallel }^2$ close to the surface and extend several times the penetration length $\delta \sim \omega ^{-1/2}$ in the normal $z$ direction and second, they demand rather long transient times $O(L_z^2)$ to allow vorticity to diffuse over the computational domain. We present a new frequency-based scheme for doubly periodic (DP) domains in free or confined spaces which uses spectral-accurate solvers based on fast Fourier transform in the periodic $(xy)$ plane and Chebyshev polynomials in the aperiodic $z$ direction. Following the ideas developed for the steady Stokes solver (Hashemi et al. J. Chem. Phys. vol. 158, 2023, p. 154101), the computational system is decomposed into an ‘inner’ domain (where forces are imposed) and an outer domain (where the flow is solved analytically using plane-wave expansions). Matching conditions leads to a solvable boundary value problem. Solving the equations in the frequency domain using complex phasor fields avoids time-stepping and permits a strong reduction in computational time. The spectral scheme is validated against analytical results for mutual and self-mobility tensors, including the in-plane Fourier transform of the Green function. Hydrodynamic couplings are investigated as a function of the periodic lattice length. Applications are finally discussed.