Brinkman Equation
Theory: adding a viscous term to Darcy's law
Overview
I calculated a metal-foam cooling channel with Darcy's law, and it predicted more flow than the experiment. What's missing?
Probably the walls. Darcy's law says the pressure gradient is proportional to velocity, giving the same flat velocity everywhere — it cannot make velocity zero at a wall. For high-permeability materials like metal foam, the slow layer near the wall is no longer negligibly thin. The Brinkman equation adds the ordinary viscous term to Darcy's law so this wall layer can be represented.
Darcy's law and the Brinkman equation
$K$ is permeability (m²), $\mu$ fluid viscosity, $\mu_{eff}$ effective viscosity and $\mathbf{u}$ the superficial velocity (averaged over the whole cross-section). The added second term allows a no-slip condition at walls. $\sqrt{K}$ has units of length and indicates how far wall effects reach.
Analytical solution for a parallel-plate channel
The solution for a porous-filled channel of width $2h$ under pressure gradient $G$, with effective viscosity equal to fluid viscosity. Large $h/\sqrt{K}$ approaches Darcy's flat profile; small $h/\sqrt{K}$ approaches the parabolic Poiseuille flow of an empty channel.
From fountains to groundwater to particle swarms
Darcy's law was found in 1856 by Darcy, an engineer designing the fountain water supply of Dijon, France, by measuring water flow through sand beds. In 1947 Brinkman added a viscous term to Darcy's law to compute the drag on a fluid flowing through a dense swarm of particles, linking flow around individual particles with flow through the medium as a whole. Today it is widely used where walls and interfaces matter — metal-foam heat exchangers, fuel-cell electrodes, flow in biological tissue.
Worked examples
Example 1: permeability and flow-rate error
Flow rate relative to Darcy's law and to Poiseuille flow for a porous-filled channel 10 mm wide ($h$ = 5 mm):
| Permeability K | √K | Darcy number K/h² | Ratio to Darcy | Ratio to Poiseuille |
|---|---|---|---|---|
| 10⁻¹⁰ m² | 0.01 mm | 4×10⁻⁶ | 0.998 | 0.00003 |
| 10⁻⁸ m² | 0.10 mm | 4×10⁻⁴ | 0.980 | 0.0012 |
| 10⁻⁷ m² | 0.32 mm | 4×10⁻³ | 0.937 | 0.011 |
| 10⁻⁶ m² | 1.0 mm | 0.04 | 0.800 | 0.096 |
| 10⁻⁵ m² | 3.2 mm | 0.4 | 0.419 | 0.50 |
| 10⁻⁴ m² | 10 mm | 4 | 0.076 | 0.91 |
The larger the permeability, the more Darcy's law overestimates the flow.
Yes. For sand and soil (around 10⁻¹⁰ m²) the wall layer is 0.01 mm and Darcy's law is fine. For metal foams and coarse packed beds (10⁻⁶ m² and above) it overestimates flow by 20% or more. At 10⁻⁴ m² the porous resistance barely matters and the flow is close to Poiseuille flow in an empty channel — Darcy alone would overpredict by 13 times. The Brinkman equation smoothly bridges the two extremes.
Example 2: thickness of the wall layer
For K = 10⁻⁸ m² (√K = 0.1 mm), velocity reaches 90% of the Darcy value 0.23 mm from the wall and 99% at 0.46 mm. The wall layer is only a few √K thick, so CFD needs several cells across it to capture wall drag correctly.
Example 3: effect of effective viscosity
| μ_eff / μ | Wall-layer length scale | Flow ratio to Darcy |
|---|---|---|
| 1 | 1.00 mm | 0.800 |
| 2 | 1.41 mm | 0.718 |
| 5 | 2.24 mm | 0.563 |
For K = 10⁻⁶ m² and a 10 mm channel. Effective viscosity is hard to measure and often set to 1, but doubling it reduces flow by about 10%. Where wall effects matter, report that this assumption influences the result.
How to approach the analysis
Choosing the model
| Situation | Model |
|---|---|
| √K below 1/100 of channel width, slow flow | Darcy's law |
| √K not negligible, walls or free-fluid interfaces | Brinkman equation |
| Fast flow with inertial drag | Add the Forchheimer term |
CFD cautions
- Superficial versus interstitial velocity: $\mathbf{u}$ in the equations is superficial; actual pore velocity is that divided by porosity.
- Porous–fluid interfaces: continuous-velocity and slip (Beavers–Joseph) conditions give different results.
- Anisotropic permeability: fibrous and layered materials can differ several-fold by direction; use a tensor.
“Flow sneaks along the walls”
In beds packed with spheres, porosity rises near the wall, so flow tends to bypass along it: spheres line up against the wall, leaving larger voids about half a diameter out. When the tube diameter is less than roughly ten sphere diameters, this wall channelling noticeably affects overall flow and heat transfer. Combining the Brinkman equation with porosity that varies with distance from the wall captures the effect.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Darcy's law for high-K media | Flow overestimated | Compare √K with channel width |
| Coarse mesh at the wall layer | Wall drag wrong | Several cells within √K |
| Mixing superficial and pore velocity | Heat transfer or residence time wrong | Convert with porosity |
| Ignoring effective viscosity | Flow off by 10% or more | Check and report sensitivity |
| Ignoring inertia at high speed | Pressure drop underestimated | Add the Forchheimer term |
I'd like to try permeability and friction calculations myself.
Try the permeability and Darcy's law tool for permeability and Darcy's law, and the Darcy vs Fanning friction factor for pipe friction. Related equations and applications are covered in Darcy flow, the Forchheimer equation, packed beds and groundwater flow.
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