Theory - Derivation of the Symmetric Pressure Gradient

Asalih3d

Hello!

I have been reading Monaghan's paper from 1982, where the approach is stated as such:

Equation 3.1, 3.2 and 3.3 are very clear to me and I derive the exact same result in these cases. The problem arises at 3.4, since when I plug the expressions in 3.2 and 3.3 into 3.1 I get (in my slightly different notation, but should still be clear);

So I am a bit confused with how 3.4 is actually constructed - does anyone happen to have a more thorough walk through?

The only one I have been able to find is done by Cossins (https://arxiv.org/abs/1007.1245), in which he uses the identity (page 8);

And is able to construct an operator very similar to 3.4;

Which is equivivalent to 3.4, since one divides by density and replaces f with rho.

So basically I would really like to understand how Monaghan derived 3.4 - maybe what Cossins did it the approach that one fact should use?

Kind regards

Asalih3d

Never mind, I figured it out. One has to remember that;

And I had forgotten to multiply with the factor in yellow. When I do this then I finally get

Which is the correct answer.

This paper here, also shows the exact calculation from eqn. 16 to 18;

https://www.researchgate.net/publication/326171915MeshlessMethods_for_Large_Deformation_Elastodynamics

Kind regards