Brinkman number#
Named after: Henri Coenraad Brinkman (1908-1961).
$$\text{Br} \stackrel{\text{def}}{=} \frac{\mu U^{2}}{k (T_s - T_\infty)} \sim \frac{\text{viscous heat generation}}{\text{conductive heat removal}}$$
Description#
Measures viscous heat generation relative to conductive heat removal. It indicates whether dissipation heating must be included in an energy balance.
Quantities#
| Name | Symbol | SI units | Dimension |
|---|---|---|---|
| dynamic viscosity | \(\mu\) | \(\mathrm{Pa}\,\mathrm{s}\) | \(\text L^{-1}\,\text M\,\text T^{-1}\) |
| velocity | \(U\) | \(\mathrm{m}\,\mathrm{s}^{-1}\) | \(\text L\,\text T^{-1}\) |
| thermal conductivity | \(k\) | \(\mathrm{W}\,\mathrm{m}^{-1}\,\mathrm{K}^{-1}\) | \(\text L\,\text M\,\Theta^{-1}\,\text T^{-3}\) |
| surface-ambient temperature difference | \(T_s - T_\infty\) | \(\mathrm{K}\) | \(\Theta\) |
Regimes#
Viscous heating
| Range | Regime | Description |
|---|---|---|
| 0 – 0.01 | negligible | Viscous dissipation produces little heat compared with conductive transport. |
| 0.01 – 1 | important | Viscous heating can noticeably alter local temperatures and heat-transfer rates. |
| 1 – ∞ | dominant | Viscous dissipation is comparable to or larger than conductive heat removal. |