We consider the stability of rotating disk flow coupled, through the fluid viscosity, to the mass concentration field of a chemical species. This configuration refers to an electrochemical cell where the working electrode consists of an iron rotating rod, which is dissolved in the 1 M H2SO4 solution of the electrolyte. Polarization curves obtained in these cells present a current instability at the beginning of the region where the current is controlled by the mass transport. The instability appears at a certain value of the applied potential and is suppressed beyond another value. Dissolution of the electrode gives rise to a thin concentration boundary layer, due to a Schmidt number Sc = 2000 of the setup. This boundary layer, together with the potential applied to the electrode, leads to an increase in the fluid viscosity and to a decrease in the diffusion coefficient, both affecting the chemical species field. Since the current is proportional to the normal derivative of the species concentration at the interface, an instability of the coupled fields at sufficiently low Reynolds numbers would result in a current instability. This work deals with the question of whether the coupling reduces the critical Reynolds number to values comparable to those attained in experimental setups, and if a possible field instability would be large enough to drive a detectable current instability. A phenomenological law is assumed, relating the fluid viscosity to the concentration of the chemical species. Parameters appearing in this law are evaluated on the basis of experimental electrochemical data. The steady-state solution is obtained by solving the coupled hydrodynamic and mass concentration equations. A temporal stability analysis is made, showing that small variations in the fluid viscosity significantly affect the stability of the flow. The analysis reveals the existence of a new unstable region, not found in the case of constant viscosity fluids. We call modes in this new unstable region chemical modes, in contrast with the hydrodynamic modes, which are amplified in the case of constant viscosity fluids. The chemical modes are destabilized at much lower Reynolds numbers than the hydrodynamic ones and close to values attained in electrochemical setups, in the most relevant cases. Hydrodynamic modes are strongly affected by the coupling in three aspects: the critical Reynolds number of this region is of order of 50% smaller than in the case of constant viscosity fluids, the unstable region is enlarged to a wider range of wave numbers, and the rate of growth of unstable modes is 30% larger for comparable Reynolds numbers. Concentration eigenmodes in the new unstable region show a combination of properties including rate of growth, amplitude higher than the amplitude of the hydrodynamic variables, and high normal derivative at the interface, sufficiently strong to drive detectable current oscillations. The destabilizing effect of a higher interface viscosity attains a maximum when the ratio between the interface and the bulk viscosities, ν0/ν∞, takes a value close to 1.5. Below this value, as the viscosity stratification diminishes and the condition of uniform viscosity is restored, the unstable region of chemical modes moves to higher Reynolds and eventually disappears. Conversely, as the ratio ν0/ν∞ increases beyond the value 1.5, the new unstable region also collapses and the neutral curve of hydrodynamic modes tends to the one of constant viscosity fluids. A sustained increase of the interfacial viscosity and the high Schmidt number results in an in facto field discontinuity with a thin high viscosity layer at the interface and restored constant viscous hydrodynamic boundary layer. A link between the current instability and the stability of the coupled fields may be inferred from the present analysis.