One fluid forcing another out in a microchannel

[sir_manning]

Hi

I'm currently doing some work in microfluidics on a microchannel device. This microchannel has a known, constant pressure drop across it and is initially filled up with fluid 1. Fluid 2 then is allowed to flow into the channel, "pushing" fluid 1 out.

What I'd like to know is how to go about finding the pressure at the interface between fluid 1 and 2. I know the volumetric flow rate Q, one of the viscosities, and the total pressure drop across the channel.

The microchannel is rectangular and leads to a solution of:

$$Q=\alpha \frac{1}{\mu}\frac{dp}{dx}$$ with mu as viscosity and alpha as some geometric factor.

However, this is for a single fluid in a microchannel of constant length. In my situation, the effective length for each fluid gets larger or small as fluid 2 pushes out fluid 1. Do I need to treat the entire microchannel as one system whose density is changing (as fluid 2 replaces fluid 1), or can I cobble together two of the time-independent solutions for each fluid?

Could someone point out some resources for this type of problem? I am assuming the flow is laminar viscous and doesn't become stratified or form an annular core. Thanks!

 

[Andy Resnick]

Because your interface is constrained by the channel walls, the precise determination will be difficult. However, Laplace's equation $\Delta P = -2\sigma\kappa$ is still valid- the pressure jump across an interface is given by the curvature of the interface.