Adhesion through viscosity

(\(j\) are fluid neighbours, \(k\) are boundary neighbours)

• previously used viscosity:
  • \[a_{i,vis} = 2\nu \sum_{j} \frac{m_j}{\rho_{j}} \frac{\vec{v}_{ij} \cdot \vec{x}_{ij}} \nabla W_{ij} {\lVert \vec{x}_{ij}\rVert^2 + 0.01h^2}\]
• correct for dimensionality: factor \(2(\text{dimensions}+2)\) instead of \(2\) ?
• introduce \(\nu_2\)
  • projected on boundary surface \(\hat{\vec{n}}\Longrightarrow\) adhesion without tangential friction
  • \[\hat{\vec{n}} = \begin{cases} \frac{\sum_k \nabla W_{ik}}{\lVert\sum_k \nabla W_{ik}\rVert} & \lVert\sum_k \nabla W_{ik}\rVert>0\\ \vec{0} & \text{otherwise}\\ \end{cases}\]
  • \[a_{i,vis} = 2(\text{dimensions}+2)\left( \nu_1 \sum_{j} \frac{m_j}{\rho_{j}} \frac{\vec{v}_{ij} \cdot \vec{x}_{ij}} {\lVert \vec{x}_{ij}\rVert^2 + 0.01h^2} \nabla W_{ij} + \left(\nu_2 \sum_{k} \frac{m_k}{\rho_{0}} \frac{\vec{v}_i \cdot \vec{x}_{ik}} {\lVert \vec{x}_{ik}\rVert^2 + 0.01h^2} \nabla W_{ik}\right) \cdot \hat{\vec{n}} \odot \hat{\vec{n}} \right)\]
• no adhesion, \(\nu_1 = 0.002, \nu_2=0\):
• adhesion, \(\nu_1 = 0.001, \nu_2=0.020\) lower viscosity possible without instability!: