In this paper, we investigate two sufficient conditions, in terms of the horizontal components \(u^{h}=(u^{1}, u^{2})\) of the velocity field, for the breakdown of local smooth solutions to the 3D incompressible Navier-Stokes/Poisson-Nernst-Planck system arising from electrohydrodynamics. More precisely, we prove that if \(\begin{aligned} \int _{0}^{T}\frac{\Vert \nabla _{h}u^{h}(\cdot ,t)\Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }}^{\frac{2}{1-\alpha }}}{1+\ln (e+\Vert \nabla _{h} u^{h}(\cdot ,t)\Vert _{\dot{B}^{-\alpha }_{\infty ,\infty }})}dt<\infty \ \ \ \text {for}\ \ \ 0<\alpha <1 \end{aligned}\) or \(\begin{aligned} \int _{0}^{T}\frac{\Vert \nabla _{h}u^{h}(\cdot ,t)\Vert _{\dot{B}^{0}_{\infty ,\infty }}}{\sqrt{1+\ln (e+\Vert \nabla _{h} u^{h}(\cdot ,t)\Vert _{\dot{B}^{0}_{\infty ,\infty }})}}dt<\infty , \end{aligned}\) then the local solution can be smoothly extended past the time \(t=T\) .
