Analitically gradient of pressure is the same using P=c*c*rho or P=c*c*(rho-rho_0). However when studying the effect of gravity in a system with constant inflow of air, I found that the second formula is more stable than the first. The problem is that the geometry is more complex and the SPH particles are dispesed an atractive force can arrise when rho
In case of the evaluation of the pressure gradient we usually prefer to fulfill conservativity over consistency, allowing the pressure gradient not to vanish even if the pressure is constant in space. The gradient operators summing the neighbors’ pressure values will therefore fail to reproduce zero gradient, however Newton‘s third law will always be satisfied, which is desirable in most cases.
Thank you for your reply. Since I put this question I got to read more. First of all the standard SPH fails badly for some applications involving pure gases.
SPH and its variants are pretty weird in this regard. In the standard SPH the Newton law is obeyed, but on the other hand we are solving a partial differential equation and computing gradients on a moving "grid/mesh/discretization". When we apply a constrain such as imposing that gradients of constant fields is zero and the gradient of linear field is constant, we lose this.
Consistent variants derived from Lagrangian ( https://lhe.epfl.ch/articles/2009ANM.pdf ) which impose the constrains on the derivatives work well in some cases, but put other problems such as what are the corrected kernels for the virtual particles used to described the solid walls and also in the case where the solid particles move and collide creating voids making it impracticle compute the correction factors
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