# DEs with Dimensional Analysis.

## Main Question or Discussion Point

Hi everyone. Can someone recommend a book that covers dimensional analysis/scaling with Differential Equations? All the books I've looked at on DEs have very little on dimensional analysis/scaling.
For instance, I set up a DE for vertical projectile motion with air resistance, I want to use scaling to reject insignificant terms, find the time scale etc. and thus simplify the DE and make approximations. Can anyone recommend such a book? And what level will it be? I need it for a 2nd year course on DEs.

Thanks.

EDIT: Ok, I realise that was a simple, maybe trivial (I don't think so), example of using dimensional analysis. But I'd like something that starts from the beginning, i.e. assuming you have NO knowledge of dimensional analysis, up to examples like the one I gave, and then up to stuff like dimensional analysis for fluid flow, Buckingham Pi theorem of course, and other stuff etc.

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Andy Resnick
Most continuum mechanics or fluid mechanics books have a section on scaling and dimensional analysis. A cheap one (Dover) is:

Mathematics applied to continuum mechanics, Segel.

One trick to remember is that a term like d/dt has units of 1/[T], and d/dx has units of 1/[L]. If you post your simple example, we can step through it.

Thanks for offering to help me Resnik. Like I say, I am only doing a 2nd year course on Differential Equations, one of the modules is modelling. We haven't actually looked at fluid mechanics or continuum mechanics, but we look at them as problems of dimensional analysis.

Ok, take for instance this question we were given:

The one-dimensional heat equation is

$$\frac{\partial\theta}{\partial t}=\kappa \frac{\partial^2\theta}{\partial x^2}$$

where kappa is the thermal diffusivity and theta the temperature. If a substance has kappa = 10^-6 and is 50cm wide, estimate the time taken for a significant change in temperature.

So the are asking us to find the time scaling, i.e., we will scale time thusly: t = aT, where the new time variable is 'T', and 'a' the scaling, so find 'a'.

Note we haven't actually done the heat equation, we only want to non-dimensionalise it.

Another example:

A layer of honey flows down a surface and its motion is given by:

$$\mu \frac{\partial^2u}{\partial z^2} + \rho.g = 0$$

where u is the speed. Without solving the equation, estimate the flow velocity for a 3mm thick layer if rho = 1021, mu = 1, g = 9.8.

Which is a similar question as the first, asking for a velocity scale, simplifications etc.

Again, we haven't actually done fluid flow, we only want to scale it/ non-dimensionalise it.

I am completely lost with such questions. Any help is very much appreciated.

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Andy Resnick
I can give a couple of hints. For the second equation:

$$\mu \frac{\partial^2u}{\partial z^2} + \rho.g = 0$$

I can non-dimensionalize this to get:

$$\frac{\partial^2u}{\partial z^2} =-\frac{L^2}{U_0}* \frac{\rho.g}{\mu} = 0$$

Where $-\frac{L^2}{U_0}* \frac{\rho.g}{\mu}$ is dimensionless. To get a characteristic velocity set this equal to 1 (why not?), plug in what you have and see what U_0 is.

$$\frac{\partial\theta}{\partial t}=\kappa \frac{\partial^2\theta}{\partial x^2}$$

This tells me that the units of $\kappa$ are L^2/T. You have a number for the diffusivity (I think a hint is that kappa is not given in terms of units), and a number for the length, I get T = 250000 seconds. (kappa in MKS units, not cgs units). That seems long.....

Oh, sorry bout the late reply. Hmm, ok I get the first one, but what will we set L equal to?

And do you know a book that would go over this kinda stuff?

Thanks for the help.

Andy Resnick
Dimensional analysis is to get a *rough estimate*, nothing more. It's also good to get an idea of what is important. For example, the dimensionless quantity $\frac{L^2}{U_0}* \frac{\rho.g}{\mu}$ looks a lot like the ratio of gravitational force and viscous force (I haven't double checked, so YMMV). So if the quantity is large, gravitational effects are important, while if the number is small, viscous effects are important.