  The first goal of our paper is to give a new type of regularity criterion for
solutions $u$ to Navier-Stokes equation in terms of some supercritical function
space condition $u \in L^{\infty}(L^{\alpha ,*})$ (with
$\frac{3}{4}(17^{\frac{1}{2}} -1)< \alpha < 3 $) and some exponential control
on the growth rate of $\dv (\frac{u}{|u|})$ along the streamlines of u. This
regularity criterion greatly improves the previous one in
\cite{smoothnesscriteria}. The proof leading to the regularity criterion of our
paper basically follows the one in \cite{smoothnesscriteria}. However, we also
point out that totally new idea which involves the use of the new supercritical
function space condition is necessary for the success of our new regularity
criterion in this paper. The second goal of our paper is to construct a
divergence free vector field $u$ within a flow-invariant tubular region with
increasing twisting of streamlines towards one end of a bundle of streamlines.
The increasing twisting of streamlines is controlled in such a way that the
associated quantities $\|u\|_{L^{\alpha}}$ and $\|\dv(\frac{u}{|u|})\|_{L^{6}}$
blow up while preserving the finite energy property $u\in L^{2}$ at the same
time. The purpose of such a construction is to demonstrate the necessity to go
beyond the scope covered by some previous regularity criteria such as
\cite{Vasseur2} or \cite{Esca}.
