Hydrodynamic control of the interface between two liquids flowing through a horizontal or vertical microchannel

We derive a Fourier expansion for estimating laminar hydrodynamic spreading when two immiscible liquids flow steadily, side by side, at low Reynolds number, in a horizontal microchannel of rectangular cross-section. When the aspect ratio of the height of the channel to its width is small this expansion can be truncated after a couple of terms and solved to yield the steady-state position of the boundary in terms of the constant volumetric flow rates, the aspect ratio and the viscosities of the two liquids. Our formula shows that we can control the location of the liquid boundary at the outlet of the microchannel by adjusting the relative volumetric flow rates of the two inlet streams. If the microchannel is oriented vertically and the volumetric flow rates are very low, the position of the boundary is also influenced by gravitational forces that depend on the relative densities of the two liquids. We have tested solutions to our analytical expansions by running computational fluid dynamics simulations and have found excellent agreement. When the aspect ratio is 0.01 the leading term in the Fourier expansion for the spreading has an accuracy of about 1%. This accuracy deteriorates to about 12% when the aspect ratio is 0.1. An understanding of these spreading mechanisms is fundamental to the design of T-sensors and related microfluidic devices.