Plug Motion in Capillary

The lumped elements between A and B, C and D, and E and F have pressure drop due to friction. The pressure drop between B and C, and D and E is attributed to the interface. The advancing and receding interface contributes either positively or negatively to the capillary pressure drop, which is a function of contact angle. For a spherical interface of curvature radius of the advancing front, the pressure difference can be written based on Young-Laplace equation as 

Note that R is signed and the sign of pressure difference depends on the orientation of the curvature. For the capillary radius a and advancing contact angle 6, we have 

We can use the expression for dynamic contact angle (equation (5.14)) for the advancing and receding front as 

There is pressure drop due to bulk fluid motion. According to Washburn law, the friction pressure drop is 

At low capillary number, that is, for microchannel flow, the pressure drop due to friction is much smaller than the capillary pressure. If we have N plugs, the total pressure drop due to capillary is 

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