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Let $D$ be the exterior of a cone inside a ball, with its altitude angle atmost $\pi/6$ in $\mathbb{R}^3$, which touches the $x_3$ axis at the origin. Forany initial value $v_0 = v_{0,r}e_{r} + v_{0,\theta} e_{\theta} + v_{0,3}e_{3}$ in a $C^2(\overline{D})$ class, which has the usual even-odd-oddsymmetry in the $x_3$ variable and has the partial smallness only in the swirldirection: $ | r v_{0, \theta} | \leq \frac{1}{100}$, the axially symmetricNavier-Stokes equations (ASNS) with Navier-Hodge-Lions slip boundary conditionhas a finite-energy solution that stays bounded for all time. In particular, nofinite-time blowup of the fluid velocity occurs. Compared with standardsmallness assumptions on the initial velocity, no size restriction is made onthe components $v_{0,r}$ and $v_{0,3}$. In a broad sense, this result appearsto solve $2/3$ of the regularity problem of ASNS in such domains in the classof solutions with the above symmetry. Equivalently, this result is connected tothe general open question which asks that if an absolute smallness of onecomponent of the initial velocity implies the global smoothness, see e.g. page873 in \cite{CZZ17}. Our result seems to give a positive answer in a specialsetting.  As a byproduct, we also construct an unbounded solution of the forced NavierStokes equation in a special cusp domain that has finite energy. The forcingterm, with the scaling factor of $-1$, is in the standard regularity class.This result confirms the intuition that if the channel of a fluid is very thin,arbitrarily high speed in the classical sense can be attained under a mildlysingular force which is physically reasonable in view that Newtonian gravityand Coulomb force have scaling factor $-2$.

Finite speed axially symmetric Navier-Stokes flows passing a cone