If you’re new to microfluidics, you might be wondering how flow rate, pressure, and resistance are related. In this blog post, we’ll explain the theory behind these concepts so that you can better understand how they impact your microfluidic designs. Stay tuned for more details!
A fundamental formula behind (micro)fluidics: Navier-Stokes equation
Navier-Stokes equations were developed over several decades during the 19th century by Claude-Louis Navier and George Gabriel Stokes to describe the motion of fluids such as air, water, and gases. They are based on Newton’s Second Law of Motion, which states that an object’s acceleration is dependent on the force acting upon it. Navier-Stokes equations combine several forces including those acting due to gravity, acceleration, and viscosity of the fluid and can be used to calculate the flow rate, pressure drop and resistance across a channel.
Since the Navier-Stokes equation is highly complex and difficult to solve, it is often simplified for a given application. The simplified equations represent a trade-off between accuracy and complexity. They capture some general trends of fluid dynamics, yet are much easier to solve.
In microfluidics, we refer to the Navier-Stokes equation as its simplification for incompressible fluids:
\rho\frac{\partial \vec{u}}{\partial t} = -\rho\vec{u}\vec{\nabla}\vec{u} – \vec{\nabla} p + \eta\Delta\vec{u}
with \eta the kinematic viscosity, \mu the volumetric mass.
\rho\vec{u}\vec{\nabla}\vec{u} being the convective term, and \eta\Delta\vec{u} the viscous term.
Finding the link between flow rate, pressure drop and resistance in microfluidics
In order to simplify the Navier-Stokes equation even further, we assume the following:
- the flow is laminar -> the convective term can be neglected
- the flow is established and steady -> \rho\frac{\partial \vec{u}}{\partial t} = 0
- the flow is unidirectional
This leads to the Stokes equation: \vec{\nabla} p = \eta\Delta\vec{u}
This equation can be simply resolved using the geometry of the microfluidic channel. In a circular channel, the final Hagen-Poiseuille equation is \Delta P = \left(\frac{8\eta L}{\pi r^4}\right) Q = R_h Q
R_h is the hydraulic resistance of the microfluidic channel, which can be calculated from the channel shape and dimensions.
Hydraulic resistance in various channel shapes
The following table gives a summary of the hydraulic resistance in various channel shapes. These formulas result from simplifications, and are only valid in the case of:
- Steady, unidirectional and laminar flow
- Incompressible fluids
| Circular | Two plates | Square | Rectangular | |
|---|---|---|---|---|
| Parameters | a is the radius | h is the height w is the width h << w | h is the height w is the width h = w | h is the height w is the width 0.2 < h/w < 1 |
| Hydraulic resistance \mathbf{R_h} | \frac{8\eta L}{\pi a^4} | \frac{12\eta L}{h^3 w} | \frac{28.4\eta L}{h^4} | \frac{12\eta L}{1-0.63 h^4} |
Electrical/fluidic analogy
The electrical/fluidic analogy is a convenient tool to better understand the behavior of fluids in a microfluidic system. Indeed, you may have found a resemblance between the Hagen-Poiseuille equation and Ohm’s law:
Hagen-Poiseuille equation: \Delta P = R_h Q, with \Delta P the pressure drop, Q the flow rate, and R_h the hydraulic resistance.
Ohm’s law: U=R I, with U the voltage, I the current, and R the electrical resistance.
Therefore, we can consider R_h as the equivalent electrical resistance, pressure as the voltage, and flow rate as current.
But the analogy doesn’t stop here: the mesh and node analysis also applies to microfluidic systems!
| Electrical | Fluidic | |
|---|---|---|
| Parallel |
|
|
| Series |
|
|
How to measure the flow rate, pressure and resistance?
Now that you understand the concepts, let’s take a look at some common methods and devices used to measure flow rate, pressure, and resistance in microfluidic systems.
Flow sensors are commonly used to directly measure the volumetric flow rate of a fluid. Examples include Coriolis mass flow sensors, or thermal flow sensors. Pressure sensors are used to measure pressure differences between two points in a microfluidic system. Common examples include the piezoelectric transducer and variable capacitance diaphragm-based pressure sensors. The pressure sensors can either be relative (pressure given against the environmental pressure), or absolute (pressure given against 0 bar).
Finally, hydraulic resistance cannot be measured directly, but through its relationship with flow rate and pressure drop. This is usually done by fixing the flow rate and measuring the pressure at different points of your system. The data can then be used to calculate the hydraulic resistance according to Hagen-Poiseuille equation: Rh = (P2-P1)/Q.
This is also a good way to see if the theoretical hydraulic resistance is close to the resistance in your experiment to apply the electrical/fluidic analogy to your system.
Conclusion
In this article, we have discussed the basics of flow rate, pressure drop and hydraulic resistance in microfluidics. We have seen how to use the Navier-Stokes equation to calculate these parameters, as well as the electric/fluidic analogy to better understand the behavior of fluids in a microfluidic system. Finally, we have looked at common devices used to measure these parameters, and how to use them to calculate hydraulic resistance.
