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Microfluidic Hydraulic Circuits-Hydraulic Resistance and Capacitance

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. 

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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<w)

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.

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