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This tutorial video is from Cornell University Mechanical Engineering 6240 – Physics of Micro- and Nanoscale Fluid Mechanics – 29 Aug 2012. A discussion of the matrix formalism for solving P=QR in hydraulic circuit networks in microfluidic devices is presented.

Hydrodynamic resistance

Flow rate $Q$ in a channel is proportional to the applied pressure drop $\Delta P$. This can be summarized in

$\Delta P=R_h Q,$

with $R_h$  the hydrodynamic resistance. This expression is formally the analog of the electrokinetic law between voltage difference and current, $U=R I$.

The expression for the hydraulic resistance is:

• channel of circular cross-section (total length $L$, radius $R$):

$R_h=\frac{8\mu L}{\pi R^4}$

• rectangular cross-section (width $w$ and height $h$, expression valid when $h)

$R_h\approx \frac{12 \mu L}{w h^3(1-0.630 h/w)}$

In a network of channels, equivalent resistances can be computed (as in electrokinetics):

• two channels in series have a resistance $R_h=R_{h1}+R_{h2}$,
• two channels in parallel have a resistance $1/R_h=1/R_{h1}+1/R_{h2}.$

These laws provide useful tools for the design of complex networks. Actually Kirchhoff’s laws for electric circuits apply, being modified in:

• the sum of flow rates on a node of the circuit is zero
• the sum of pressure differences on a loop is zero

Hydrodynamic capacitance

The volume of fluid in a channel can change just because of a change in pressure: this is either due to fluid compressibility or channel elasticity. This behavior can be summarized with

$Q=C_h \frac{d\Delta P}{dt}$

with $C_h$ the hydrodynamic capacitance. It is the microfluidic analog of the electrokinetic law $I=C\,dU/dt$.