Brinkman Number

A polymer melt entering a die at 200 °C leaves it at 215 °C, and the tool is being actively cooled the whole time. Nothing is adding heat. The melt is heating itself, because forcing a very viscous fluid through a narrow passage means shearing it, and shearing dissipates work as heat throughout the fluid rather than at any surface. The Brinkman number is the group that says when this matters: $\mathrm{Br} = \mu v^{2}/(k\,\Delta T)$, heat generated by shear divided by heat conducted away.

The threshold at 1 is unusually meaningful for a dimensionless group, because it marks a change of SIGN rather than a change of degree. Below 1 the wall is cooling the fluid, as one would assume. Above 1 the fluid generates heat faster than conduction removes it, the interior runs hotter than the wall, and the temperature gradient points the other way — the "cooling" surface is now the only thing standing between the melt and thermal degradation. An isothermal analysis in that regime is not merely imprecise; it has the direction of heat flow wrong. Polymer processing lives above 1 routinely, with melt viscosities in the hundreds or thousands of Pa·s and conductivities near 0.25 W/(m·K), which is a hundred times worse than a metal at carrying heat away. Journal bearings live near 1. Water and air at ordinary speeds sit at $10^{-6}$ and below, which is why nobody in heat-exchanger work has ever needed to think about this.

What makes the polymer case genuinely dangerous is a feedback loop the group itself does not show. Viscosity falls steeply as the melt heats, so the hottest fluid shears most easily; the shear then concentrates into that thinner, hotter layer, which raises the local dissipation further. In extreme cases the result is a runaway that shows up as scorch, black specks, or a burnt smell at the die — degradation caused by the process rather than by the heaters. The engineering responses are all about geometry and residence time rather than about turning down a setpoint: shorter land lengths, larger gaps, lower throughput, or a melt with a flatter viscosity curve.

Brinkman is a composite: $\mathrm{Br} = \mathrm{Ec}\cdot\mathrm{Pr}$ exactly, since multiplying $v^{2}/(c_p\Delta T)$ by $\mu c_p/k$ cancels the specific heat. That identity explains a puzzle — how a slow-moving polymer melt can have a large Brinkman number while its Eckert number is minute. The answer is that Prandtl for a melt runs into the tens of thousands, and the product is what counts. It also means the two pages must agree, and reaching Brinkman either way is a real check rather than a restatement. One warning about the literature: definitions vary. Some texts insert a factor of 2, some use the wall-to-CENTRELINE difference rather than wall-to-bulk, and some write it on a shear rate and a gap instead of a velocity. All describe the same physics, but a threshold lifted from one convention into another is off by whatever factor separates them, so check the definition before comparing against a published number.