Dynamic response of a compressible binary fluid mixture

Binary fluids are present in a wide variety of systems at microscales, such as microfluidic devices containing drops, fluids with air bubbles trapped in them, and devices designed to mix fluids or to make two fluid substances react. Microfluidics devices are often, intentionally or unintentionally, subject to pulsatile forces due to the passing of drops. We demonstrate that when a binary fluid is subject to a pulsatile forcing, the compressibility of the lower viscosity phase is so important that it is able to generate resonances in the dynamic permeability of the whole system. This implies that the flow amplitude of a binary-fluid system in a zero-mean flow could be optimized at certain frequencies. The dynamic permeability is a powerful concept to describe the dynamics of the system, in the regime where the flow and the pressure gradient are related linearly in the frequency domain. We find two regimes for the frequency at which the resonance occurs: one dominated by a characteristic frequency of the system, related to the compressibility of the lower viscosity phase; and another one dominated by a more complex frequency, involving both the characteristic viscous frequency of the lower viscosity phase and a characteristic frequency related to the compressibility. In order to guide potential experiments, we calculate relevant quantities for two sets of binary fluids in standard microfluidic setups. Our calculations imply that for systems of typical microfluidic dimensions, involving a compressible fluid, the existence of the compressibility-induced resonance has to be contemplated for a correct description of the dynamics.