In this study, linear stability analysis and Direct numerical simulations (DNS) are used to investigate the coupling between hydrodynamic instabilities, membrane transfer and osmotic pressure in a Taylor-Couette cell, as a model of rotating dynamic filtration devices. Performances of filtration techniques deteriorate as the retained materials accumulate near the semi-permeable membrane and the related osmotic pressure tends to cancel out the operating pressure driving the solvent through the membrane. Thus, rotating filtration devices make use of hydrodynamic instabilities to mix strongly the solution and abate the concentration boundary layer forming near the membrane. The emphasis here, is on characterizing the effect of the osmotic pressure related to the concentration boundary layer on the structure and dynamics of Taylor vortices. No quantitative results exist to assess the couplings between mixing, osmotic pressure and instabilities, the effectiveness of dynamic filtration and how to optimize it. So, this current work addresses those couplings in a controled fashion by considering a Taylor-Couette set-up, with a fixed outer cylinder and a rotating inner one. Moreover, the gap is filled with a solution and both cylinders are semi-permeable membranes totally rejecting the solute. We impose an operating pressure across the gap which drives a radial in- or outflow. As the rotation rate of the inner cylinder is increased, centrifugal instabilities emerge in the form of toroidal vortices carrying the solute. For fixed operating conditions, linear stability analysis shows that the osmotic pressure tends to alter centrifugal instabilities as a result of an original self-sustained mechanism coupling the advection of the concentration boundary layer by the vortices, molecular diffusion and osmotic pressure driving a transmembrane flow fostering the vortices. This mechanism can induce a substantial reduction of the critical rotation rate above which vortices are observed. However, depending on the thickness of the concentration boundary layer, i.e. on the Schmidt number, the mechanism can have a destabilizing effect on the Taylor vortices or no effect. Furthermore, stability analysis shows that critical conditions are also impacted by the radius ratio. These analytical results are compared to recent DNS based on a dedicated code using spectral methods that shows a good agreement with critical conditions.