SPH for Navier Stokes

Lagrangian representation on particle \(i\)
- \(\displaystyle m_i \frac{\mathrm{d}v_i}{\mathrm{d}t}=\underbrace{m_i\, g}_{F_{weight}} - \underbrace{\frac{m_i}{\rho_i}\,\nabla \mathrm{p}_i}_{F_{pressure}} + \underbrace{m_i\,\nu\triangle v_i}_{F_{viscosity}}\)

Objective:
- 1. Express \(\rho_i, \nabla \mathrm{p}_i, \triangle v_i\) using SPH formulation
- 2. Then integrate: ex. \(v_i^{k+1}=v_i^{k}+\Delta t\left(F_{weight}+F_{pressure}+F_{viscosity}\right)/m_i\)

Generic SPH representation:

Arbitrary field \(A\) at position \(p_i\): \(A(p_i)=\sum_j A(p_j)\,W_h(p_i-p_j)\,V_j\)
For a particle of total mass \(m_i\) in the volume \(V_i\): \(\rho_i V_i = m_i\) \(\;\;\;\Rightarrow\;\) \(A(p_i)=\,\sum_j A(p_j)\,m_j/\rho_j\,W_h(p_i-p_j)\)

Usually \(W_h\) are distance function :
\(\displaystyle A(p_i)=\sum_j A(p_j)\,m_j/\rho_j\,W_h(\|p_i-p_j\|)\)