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In this paper we provide a sufficient condition, in terms of the horizontalgradient of two horizontal velocity components and the gradient of liquidcrystal molecular orientation field, for the breakdown of local in time strongsolutions to the three-dimensional incompressible nematic liquid crystal flows.More precisely, let $T_{*}$ be the maximal existence time of the local strongsolution $(u, d)$, then $T_{*}<+\infty$ if and only if \begin{align*}\int_{0}^{T_*} \big( \|\nabla_h u^h\|_{\dot{B}^0_{p,\frac{2p}{3}}}^q + \|\nablad\|_{\dot{B}^0_{\infty,\infty}}^2 \big)dt = \infty\ \ \text{with}\ \ \\frac{3}{p} +\frac{2}{q} = 2,\ \ \frac{3}{2} < p\leq\infty, \end{align*} where$u^{h}=(u^{1},u^{2})$, $\nabla_{h}=(\partial_{1}, \partial_{2})$. This resultcan be regarded as the generalization of the BKM's criterion in \cite{HW12},and is even new for the three-dimensional incompressible Navier-Stokesequations.

BKM's criterion for the 3D nematic liquid crystal flows via two velocity components and molecular orientations