Blast Waves and Scaling Relations

[bombadil]

One of the outstanding questions I have in physics relates to scaling relations.

Say you're presented with a problem like: Find X--which has units of y--given that the relevant (dimensional) quantities of this problem are A, B, C, and D. Then you construct a solution using these quantities based not on physical principles, but rather on dimensional analysis!

An examplar of this sort of logic is the Sedov-Taylor solution of exploding bombs/stars in an atmosphere/interstellar medium.

You want to find the radius of the blast given the energy of the explosion and the density of the medium into which the blast wave propagates and you suddenly get:

$$R= \left(\frac{E}{\rho}\right)^{1/5}t^{2/5}$$

I often hear the phrase "self-similar" attached to this sort of conversation. I assume this has something to do with solutions to a certain class of partial differential equations. Can anyone enlighten me?

 

[AndyW]

Thank you for your question about scaling relations in physics. This is indeed a fascinating topic that has been studied extensively by scientists in various fields.

To begin, let's define what scaling relations are. In simple terms, scaling relations are mathematical relationships between different variables in a physical system that remain the same even when the system itself is scaled up or down in size or magnitude. This means that the relative proportions between the variables remain unchanged, regardless of the overall size of the system.

One of the most common examples of scaling relations is the one you mentioned in your post - the Sedov-Taylor solution for exploding bombs or stars in a medium. This solution is based on the principle of dimensional analysis, which is a powerful tool in physics that allows us to derive relationships between physical quantities without having to rely on specific physical laws or equations.

In the case of the Sedov-Taylor solution, we can see that the radius of the blast (R) is related to the energy of the explosion (E) and the density of the medium (ρ) through a power law relationship. This means that as the energy and density change, the radius of the blast will change in a predictable and proportional manner. This is what we mean by self-similar - the solution remains the same even when the variables are scaled up or down.

This self-similarity is closely related to the concept of similarity solutions in partial differential equations. These are solutions that remain unchanged under certain transformations, such as scaling. In the case of the Sedov-Taylor solution, the equations describing the blast wave are scale-invariant, meaning that the solution remains the same even when the physical quantities are scaled up or down.

Overall, scaling relations and self-similarity are powerful tools in physics that allow us to understand complex systems and phenomena without having to rely on specific physical laws or equations. They provide us with a deeper understanding of the underlying principles governing the behavior of physical systems and have applications in a wide range of fields, from astrophysics to fluid dynamics.

I hope this has helped to shed some light on this topic for you. If you have any further questions, please don't hesitate to ask.