Polymerization reactions are characterized by changes in physical properties, such as viscosity, as the reaction proceeds. The effect of a conversion-dependent viscosity on the nonlinear behavior of a polymerization reactor is investigated in this paper. This involves considering the coupling between the momentum balance, mass balance, and energy balance of the reactor. It is shown that, when the viscosity increases with the conversion steady-state multiplicity can occur when the pressure drop across the reactor is fixed. The different steady states have different inlet flow rates and exit conversions. Two models of ideal continuous reactors, i.e., CSTR and PFR, are investigated. We analyze these reactors considering operation under a constant pressure drop. Our results indicate that multiple steady states can occur in systems such as polymerization reactors where viscosity significantly increases with conversion in the reactor. We show even an isothermal reactor can exhibit multiple steady states. The physical source, i.e., the positive-feedback effect, that causes this multiplicity is identified. The applicability of the predicted results is illustrated for a case study of the manufacture of low-density polyethylene. We postulate that operation of a reactor with a fixed pressure drop leads to decoupling of the reactor from a downstream separator in a coupled reactor-separator network. This can hence form a desirable control strategy for operating coupled systems.