Equation of motion for a fluid

• - Fundamental principle of dynamics on linear momentum

• • \(\displaystyle \rho\frac{\mathrm{d}u}{\mathrm{d}t}=F+\mathrm{div}(\sigma)\)
  • \(\Rightarrow \displaystyle \rho\,\frac{\partial u}{\partial t}=F+\mathrm{div}(\sigma)-\rho \, (u\cdot \nabla) u\). The term \((u\cdot \nabla) u\) is called advection.

• - External force: weight \(F=\rho\,g\)

• - Stress decomposed into
• • \(\sigma=\sigma_{viscous}+\sigma_{pressure}\)
  • \(\sigma_{pressure}=-p\,\mathrm{Id}\) (pressure acts along normal of surface elements)

• \(\displaystyle \rho\,\frac{\partial u}{\partial t}=\rho\,g-\rho\, u\cdot\nabla u+\mathrm{div}\left(\sigma_{viscous}-p\,\mathrm{Id}\right)\)

• \(\Rightarrow \displaystyle \rho\,\frac{\partial u}{\partial t}=\rho\,g-\rho\, u\cdot\nabla u-\nabla p+\mathrm{div}\left(\sigma_{viscous}\right)\)