We study theoretically the deformation of polyelectrolyte brushes in strong flows of good solvent using a scaling theory which calculates the deformation of grafted chains and the solvent flow profile within the brush in a mutually consistent fashion. We consider the cases of permeation flows normal to the grafting surface and shear flows parallel to the grafting surface. For the case of permeation flows, we find that strongly charged brushes are more uniformly extended than weakly charged ones and that the crossover region separating the weak deformation and strong deformation regimes shifts to higher solvent flow rates and becomes somewhat broader with increasing charge fraction. For the case of shear flows, we find that chains are more uniformly stretched and less strongly tilted with increasing charge fraction. Furthermore, we demonstrate that with increasing charge fraction, there is a reversal of the dependence of brush thickness on shear rate : weakly charged brushes expand in strong shear flows, while strongly charged brushes collapse somewhat in the same flows, in qualitative agreement with previous theoretical predictions.