# Theory - Tait equation of state?

Hello!

I read through the documentation of DualSPHysics and I see a line which says:

**Weakly compressible approach using Tait’s equation of state.**

And can someone explain why this is attributed to Tait? Because by reading Wikipedia on SPH method, I found that actually we are using Cole's equation of state:

https://en.wikipedia.org/wiki/Cole_equation_of_state

Could someone explain why a lot of SPH theory is attributing it to Tait?

Kind regards

## Comments

This can be a historical error, perhaps...

Yes, this was also my initial thought, but just wanted to make sure. When it is clarified I think it should be changed to Cole for two reasons; for correct attribution and to make it easier to find the correct theory.

Kind regards

@rvacondio made me aware of this paper;

Smoothed Particle Hydrodynamics and Its Diverse Applications, J.J. Monaghan, Annual Review of Fluid Mechanics 2012 44:1, 323-346

Which states:

Kind regards

Hi,

An interesting thread.

Tait around 1880 or similar proposed an equation to relate pressure P to volume V through a log function that has the same ratio of V/Vo (or ρ/ρ_0 for constant mass) and very similar form to what we use currently.

Then (and I might be wrong on this) this went through many iterations with very small changes for

solidsandfluids. Yes Cole1948"used" this formula so did MacDonald (who states the original eq. came from Murnaghan1944and before Cole), based again on volumes. MacDonald also claims some well deserved glory on this!In other words, and from a historical point of view, we used Tait as the first to link pressures to density (for solids) and in my (personal) opinion the father of those barotropic relations. Of course anyone who finds the original paper with the original derivation is welcomed to correct me.

Cheers,

George

PS and a bit of humour: Do not hold me hostage on this post and name dropping, reading these old papers (when available is not... easy! So, when you find that original paper please point me to the eq. directly and do not make me read the whole thing.

@gfourtakas

Monaghan 2012 cites

Harvard 1967 cites

and Tait 1900 is ready open for you on page 334 at https://archive.org/details/scientificpapers02taituoft/page/334/mode/2up

I let you browse the remaining few pages though :-)

My two cents:

Macdonald, J. Ross. "Some simple isothermal equations of state."

Reviews of Modern Physics38.4 (1966): 669.## Some Simple Isothermal Equations of State

https://doi.org/10.1103/RevModPhys.38.669Thank you for the link.

George

Interesting discussion all of you! I don't know exactly what the correct term is still, but atleast I understand the history now thanks to @gfourtakas . My reason for pointing it out was that I was not able to find anything relevant when searching for "Tait's equation of state", but "Cole's" showed the exact equation in use by DualSPHysics :-)

Kind regards

Hi,

I was looking some time ago to it and I found also only reference to Cole. However, I am with @gfourtakas to use Tait as we can consider it the father of it. Besides, my research was rather on the meaning of the polytropic index, which is actually an adiabatic index. For water the most proper value is 1-1.33. Nevertheless we use 7 to enforce incompressibility to a compressible fluid.

Something to think about...

If you happen to know more about this discrepancy, i.e. using 7 instead of the actual 1-1.33, would you care to explain more?

Kind regards