In this work we investigate the behavior of a coupled reactor-separator system. The reactor considered is a continuous stirred tank reactor, and the coupling between the two units occurs via recycle of the reactant-rich stream (assumed to be the bottoms) from the downstream flash to the upstream reactor. We investigate two reactions: (i) an irreversible isothermal first-order reaction A --> B and (ii) an elementary cubic autocatalytic isothermal reaction of the form A + 2B --> 3B. The separator is modeled as a flash unit operating isothermally and isobarically. The feed to the flash is restricted to be a binary mixture. This uniquely fixes the composition of the two effluent streams, leaving the flash unit. The main focus in this work is to study the effect of delay arising from the transportation lag from the reactor to the separator. The behavior of the system is governed by delay differential equations. We want to study how the dynamic instability induced by delay crucially depends on the flow control strategy of the coupled reactor-separator system. It is shown that delay does not induce any instability if the fresh feed now rate F-0 is flow-controlled and the molar holdup is controlled using the reactor effluent flow rate F. In the second control strategy where the reactor effluent flow rate F is flow-controlled and the holdup is controlled using fresh feed flow rate F-0, delay above a threshold value can induce instability for the two reaction systems considered. In the third control strategy, both F and F-0 are flow-controlled and the holdup in the reactor is allowed to vary. Here again delay beyond a critical value can introduce a new region of instability for the isothermal first-order reaction, and it increases the region of dynamic instability for the cubic autocatalytic reaction. The main result of the paper is to demonstrate that some of the preferred control structures of Luyben that are stable can be rendered unstable by sufficiently large delay.