  This paper is concerned with a blood flow problem coupled with a slow plaque
growth at the artery wall. In the model, the micro (fast) system is the
Navier-Stokes equation with a periodically applied force and the macro (slow)
system is a fractional reaction equation, which is used to describe the plaque
growth with memory effect. We construct an auxiliary temporal periodic problem
and an effective time-average equation to approximate the original problem and
analyze the approximation error of the corresponding linearized PDE (Stokes)
system, where the simple front-tracking technique is used to update the slow
moving boundary. An effective multiscale method is then designed based on the
approximate problem and the front tracking framework. We also present a
temporal finite difference scheme with a spatial continuous finite element
method and analyze its temporal discrete error. Furthermore, a fast iterative
procedure is designed to find the initial value of the temporal periodic
problem and its convergence is analyzed as well. Our designed front-tracking
framework and the iterative procedure for solving the temporal periodic problem
make it easy to implement the multiscale method on existing PDE solving
software. The numerical method is implemented by a combination of the finite
element platform COMSOL Multiphysics and the mainstream software MATLAB, which
significantly reduce the programming effort and easily handle the
fluid-structure interaction, especially moving boundaries with more complex
geometries. We present some numerical examples of ODEs and 2-D Navier-Stokes
system to demonstrate the effectiveness of the multiscale method. Finally, we
have a numerical experiment on the plaque growth problem and discuss the
physical implication of the fractional order parameter.
