Shape sensitivity and gradient-flow optimization for linear micropolar fluid flow

We study a shape optimization problem for a stationary linear micropolar fluid. The objective is the squared L 2 -gap between the microrotation and the local vorticity. This cost measures the deviation of the micropolar model from the limiting Stokes-type regime in which microrotation is aligned with vorticity. The analysis is carried out by transporting the state equation to a fixed reference domain by the Piola transformation. We state the dimensional convention explicitly, prove well-posedness of the state and adjoint systems, derive the weak material derivative in the Piola sense under sufficient data regularity, and obtain a Hadamard boundary representation of the shape derivative under trace regularity assumptions. The resulting boundary density is used in a continuous gradient-flow dynamical system and in a finite-dimensional gradient flow for a fixed set of shape parameters. The numerical section describes the supplied obstacle and pipe-design computations and reports only information visible in the provided source files and output histories.