Numerical implementation of the Stokes-Couette flow using the SPH method

Resumo

This article uses the Lagrangian numerical method SPH (Smoothed Particle Hydrodynamics) to solve
the Stokes-Couette flow. The laminar flow is created by horizontal oscillation of the lower plate while the upper
plate remains stationary. Stokes-Couette flow can be used as a benchmark to test codes using the meshfree SPH
method since this phenomenon is governed by the one-dimensional classical heat equation. The newtonian fluid
domain between the flat plates is discretized by particles in a vertical column, which would be free to move
according to the forces acting on the particles. Boundary conditions are treated with ghost particles, created outside
the physical domain to keep the number of neighbors of each particle constant. The initial condition considers the
fluid at rest. The heat equation is then solved for each particle inside the domain and the horizontal velocity profile
is obtained from the integration over time. The variation in time of the velocity profile from the oscillatory point of
view (non-zero oscillation frequency) and the evolution in time of the velocity profile (zero oscillation frequency)
are evaluated. The numerical solution of the SPH method reproduces the Stokes-Couette flow, which can be used
as a benchmark for more complex SPH-based codes.