  For the Stokes system in the half space, Kang [Math.~Ann.~2005] showed that a
solution generated by a compactly supported, H\"older continuous boundary flux
may have unbounded normal derivatives near the boundary. In this paper we first
prove explicit global pointwise estimates of the above solution, showing in
particular that it has finite global energy and its derivatives blow up
everywhere on the boundary away from the flux. We then use the above solution
as a profile to construct solutions of the Navier-Stokes equations which also
have finite global energy and unbounded normal derivatives due to the flux. Our
main tool is the pointwise estimates of the Green tensor of the Stokes system
proved by us in \cite{Green} (arXiv:2011.00134). We also examine the Stokes
flows generated by dipole bumps boundary flux, and identify the regions where
the normal derivatives of the solutions tend to positive or negative infinity
near the boundary.
