The Brinkman number (symbol Br) is a dimensionless quantity used in fluid dynamics and heat‑transfer analysis to characterize the relative importance of viscous dissipation (internal heat generation due to fluid friction) compared with conductive heat transport. It is particularly relevant in flows where high shear rates or high viscosities cause significant conversion of mechanical work into thermal energy, such as polymer extrusion, lubrication, micro‑fluidic devices, and high‑speed gas flows.
Definition and formula
$$ \mathrm{Br}= \frac{\mu,U^{2}}{k,\Delta T} $$
where
- $\mu$ – dynamic viscosity of the fluid (Pa·s),
- $U$ – characteristic velocity of the flow (m·s$^{-1}$),
- $k$ – thermal conductivity of the fluid (W·m$^{-1}$·K$^{-1}$),
- $\Delta T = T_{w}-T_{\infty}$ – characteristic temperature difference between a solid surface (or reference temperature) and the bulk fluid (K).
The numerator $\mu U^{2}$ represents the rate of mechanical energy dissipation per unit volume, while the denominator $k\Delta T$ represents the rate of heat conduction away from the region of dissipation. Consequently, the Brinkman number quantifies the magnitude of viscous heating relative to conductive cooling.
Physical significance
- Low Br (≪ 1): Viscous heating is negligible; temperature fields are governed primarily by external thermal boundary conditions and conduction.
- High Br (≫ 1): Viscous dissipation dominates the thermal balance, leading to substantial temperature rise within the fluid. This can affect fluid properties (e.g., viscosity) and may induce thermal instability.
Typical applications
- Polymer processing: In extrusion and injection molding, the high viscosity and shear rates produce noticeable viscous heating, influencing melt temperature and material properties.
- Lubrication theory: Thin‑film lubrication under high load can generate measurable temperature increases, which are evaluated using the Brinkman number.
- Micro‑ and nano‑fluidics: Small characteristic lengths and high surface‑to‑volume ratios amplify the effect of viscous dissipation, making Br a useful design parameter.
- High‑speed aerodynamics: For hypersonic flows, the conversion of kinetic energy into heat via viscous stresses can be significant.
Relation to other dimensionless numbers
The Brinkman number often appears in conjunction with the Prandtl number (Pr = ν/α, ratio of momentum to thermal diffusivity) and the Eckert number (Ec = U²/(c_p ΔT), ratio of kinetic energy to enthalpy change). In many analyses, the product Br · Pr equals the Eckert number:
$$ \mathrm{Ec} = \mathrm{Br},\mathrm{Pr} $$
where $c_p$ is the specific heat at constant pressure and $ u$ is the kinematic viscosity.
Historical note
The Brinkman number is named after H. Brinkman, a researcher who contributed to the theoretical description of viscous heating in fluid flows during the mid‑20th century. The precise original citation is not uniformly recorded in contemporary sources; however, the eponymous designation is widely accepted in engineering literature.
References (selected)
- Incropera, F. P., & DeWitt, D. P. (2002). Fundamentals of Heat and Mass Transfer (5th ed.). Wiley. – Provides the definition and context of the Brinkman number.
- White, F. M. (2016). Viscous Fluid Flow (4th ed.). McGraw‑Hill. – Discusses viscous dissipation and dimensionless groups, including Br.
- Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2002). Transport Phenomena (2nd ed.). Wiley. – Includes derivation of the Brinkman number for coupled momentum‑energy problems.
Note: The information presented reflects established engineering and scientific sources up to the knowledge cutoff of 2024.