Order of convergence estimates on the interaction term for a micropolar fluid

Abstract

The boundary-initial and boundary-final value problems are studied for the micropolar fluid theory of Eringen [1–5]. We investigate how the velocity field v to the micropolar equations converges to the velocity field u of the Navier-Stokes equations as the interaction parameter λ → 0. For the forward in time problem it is shown that the convergence in L 2-norm, ∥ u − v∥, will be at least O(λ). For the backward in time problem we show that for appropriately constrained solutions one can expect an order of convergence like O(λ δ) , on compact sub-intervals of a finite time interval [0, T); the power δ satisfies 0 < δ ≤ 1, with δ = 1 at t = 0 and δ → 0 as t → T.