Eckert Number

Everything moving carries kinetic energy, and bringing it to rest turns that energy into heat. Most of the time this is far too small to notice: stop a stream of air moving at 5 m/s and the temperature rises by about a hundredth of a degree. The Eckert number is how you decide whether "most of the time" applies to the problem in front of you. It compares the kinetic energy in the flow with the enthalpy change the heat transfer is already producing, $\mathrm{Ec} = v^{2}/(c_p\,\Delta T)$, and its practical job is to license dropping the viscous dissipation term out of the energy equation — which is exactly what almost every textbook derivation quietly does without saying so.

The threshold is soft and the reasoning is proportional. Below about 0.01 the dissipation contributes a percent or less and can be ignored with a clear conscience. Near and above 1 the kinetic energy is comparable with the enthalpy the heat transfer is moving, and bringing the flow to rest at a surface releases a temperature rise of the same size as the $\Delta T$ you started with. That is the regime of high-speed aerodynamics: the recovery temperature at a stagnation point rises above the free-stream temperature by roughly $v^{2}/(2c_p)$, which for air at 500 m/s is about 124 K. It is why the leading edge of an aircraft wing at cruise runs warm even in air at −55 °C, and why re-entry is a heat-transfer problem before it is anything else.

The velocity is squared, and that dominates everything. Doubling the speed quadruples the Eckert number, so the transition from "ignore it" to "it is the whole problem" happens over a narrow band of velocity. In air with a 100 K temperature difference, dissipation is negligible at 30 m/s, worth a footnote at 100 m/s, and dominant at 500 m/s. This is also why Eckert numbers matter in tiny passages: in a microchannel or a thin lubricating film the velocity gradients are enormous even when the bulk velocity is modest.

Two cautions. First, the $\Delta T$ in the denominator is a REFERENCE difference chosen by whoever wrote the correlation — usually wall minus free stream, but not always — so an Eckert number quoted without saying which difference it used cannot be compared with another one. And when $\Delta T$ approaches zero the group blows up, which is a signal that the framing has broken down rather than that dissipation has become infinite; with no imposed temperature difference the right question is simply how much the flow heats itself, and that is a Brinkman question, not an Eckert one. Second, Eckert has a close relative: $\mathrm{Br} = \mathrm{Ec}\cdot\mathrm{Pr}$ is the Brinkman number, which asks the same thing against conduction instead of enthalpy. For a fluid with a large Prandtl number — an oil, a polymer melt — Brinkman can be substantial while Eckert is small, so a small Eckert number is not by itself permission to ignore self-heating in a viscous fluid.