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I want to know the rate at which air will ascend above a stove element.

Is there some relationship between ambient air temperature, element temperature, and speed of the resulting updraft?

If nothing definite, is there anything approximate that I could use?

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    $\begingroup$ I assume that shaking a bit of flour over the stovetop and tracking the movement of the tiny particles carried by the air currents with a camcorder is out of the question? $\endgroup$ – DumpsterDoofus Oct 19 '13 at 18:40
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    $\begingroup$ @DumpsterDoofus That's not out of the question :) $\endgroup$ – user31346 Oct 19 '13 at 18:42
  • $\begingroup$ However, I am interested in a way of predicting the result, even approximately. $\endgroup$ – user31346 Oct 21 '13 at 6:11
  • $\begingroup$ Hi Sancho, how much detail do you want? Unless you solve the NS equations, scaling is the best approximate solution which gives you a bulk estimate on the same order as the quantity you are seeking. $\endgroup$ – Isopycnal Oscillation Oct 25 '13 at 17:15
  • $\begingroup$ Oh ok, its all detailed in that link, I will go ahead and list them for you. $\endgroup$ – Isopycnal Oscillation Oct 25 '13 at 17:30
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In this book they first non-dimensionalize the NS equations and then, assuming terminal velocity, small temperature differences, and using scaling arguments they arrive at the following relationship for the terminal speed of a particle moving buoyantly in a stratified flow:

$V = \frac{g \alpha \Delta T r^2}{6 \pi \nu}$

This is roughly what happens, I assume a lot of things as it is necessary for a sane understanding of the subject. Initially, a (stationary) stably stratified homogeneous fluid is above the stove element. There are roughly three stages, generation, evolution, steady state.

In the first stage, the element is turned on, fluid particles nearest to the element experience a change in temperature (positive), kinetic energy increases and nearby fluid parcels propagate such information upwards (can't go down, forget sides) via diffusive heat transfer (in the initial stages). At this point in time particles have barely moved from the surface (considering the total timescale we are interested in). However, this process effectively gives rise to considerable upwards fluid velocity in fluid that is adjacent to the heat element.

In the next stage, some fluid has accelerated to the point that we can now physically discern fluid motion. Diffusion is no longer important (it never really was), at this point advection takes the lead. A fluid parcel along the bottom, the warmest one with respect to all the other fluid parcels along the bottom, will, at this stage, begin to feel a little different from its neighbors. Since the fluid parcel is warmer, it is also lighter than all the fluid around it, so, according to Archimedes it must go up. This is very similar to the way the sun heats up the earth every morning and generates thermals. There will be lots of different convection cells owing to imperfections in the heating element, which will in turn give rise to eddying motion and turbulence.

At steady state, the fluid will be moving away from the element at a steady pace in a way that you can average out all the nuances of fluid dynamics, so that you can arrive at that formula. Sometimes though, I just wish we could give someone a million dollars.

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  • $\begingroup$ The units don't seem to work out for me. I have $g$ in $m/{s^2}$, $\alpha$ in $(1/K)$, $\Delta T$ in $K$, $r$ in $m$, and $\mu$ in $\frac{kg}{m \cdot s}$. That gives me a result in units of $\frac{m^4}{kg \cdot s}$. $\endgroup$ – user31346 Oct 26 '13 at 6:24
  • $\begingroup$ Hi Sancho, my apologies. Typically, the dynamic viscosity is represented by that greek letter, unfortunately, the book uses unconventional notation and has switched the symbols. Please check updated answer. I should have double checked. $\endgroup$ – Isopycnal Oscillation Oct 26 '13 at 6:48

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