Viscosity

\(F_{viscosity}= m_i\,\nu \; \triangle {v}_i\)

1. Use symetric laplacian b/w (i,j)
- \(F_{viscosity}= m_i\,\nu \, \triangle (\mathrm{v}_j-\mathrm{v}_i)\) \(\;\;\;\)-\(\;\) viscosity depends on velocity differences
- \(F_{viscosity}= m_i\,\nu \, \sum_j m_j \frac{(\mathrm{v}_j-\mathrm{v}_i)}{\rho_j}\;\triangle W_h(\|p_i-p_j\|)\)

2. Weight function
- Second derivative should remain positive.
- Can use the spiky kernel

- \(W_h^{spiky}(d)=\frac{15}{2 \pi h^6}\left(h-d\right)^3\;\;\;0\leq d \leq h\)
- \(\triangle W_h^{spiky}(d)=\frac{45}{\pi h^6}\left(h-d\right)\;\;\;0\leq d \leq h\)

Increasing viscosity \(\nu\)