Two Connected Straight Channels

Most of the microfluidic devices consist of microchannel of different dimensions interconnected with each other. In the case of two straight channels of different dimensions connected to form one long channel, the translation invariance, that is, fully developed flow assumption, is broken. Hence, the expressions for the ideal Poiseuille flow no longer apply. However, it is expected that the ideal description is approximately correct if the Reynolds number (Re) of the flow is sufficiently small. This is because a very small value of Re corresponds to a vanishing small contribution from the nonlinear term (V - V)F in the Navier-Stokes equation (N-S equation), a term that is strictly zero in ideal Poiseuille flows due to translation invariance. 

Note that in microfluidics, the Reynolds number Re = pVqLq/h tends to be low because of the small length scales Lq involved and low velocities. Thus, Hagen-Poiseuille law can be applied beyond the ideal situation on various channel geometries. 

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