Navier-Stokes equation
- - Isotropic Newtonian fluid \(\Rightarrow \) Linear (scalar) relation between strain-rate \(\epsilon\) and stress-rate \(\sigma_{viscous}\)
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- - \(\sigma_{viscous}=2\mu\,\epsilon = \mu\left(\nabla u+\nabla u^{T}\right)\), \(\mu\) constant viscosity parameter
- - Incompressible fluid \(\Rightarrow \) \(\mathrm{div}(u)=0\)
- \(\Rightarrow \displaystyle \rho\,\frac{\partial u}{\partial t}=\rho\,g-\rho \,u\cdot\nabla u-\nabla p+\mathrm{div}\left(\mu\left(\nabla u+\nabla
u^{T}\right)\right)\)
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- - Noting that \(\mathrm{div}(\nabla u^{T})=\nabla \, \mathrm{div}(u) = 0\)
- - And \(\mathrm{div}(\nabla u)=\triangle u\)
- - Set \(\nu=\mu/\rho\)
- \(\Rightarrow \displaystyle \frac{\partial u}{\partial t}=g-(u\cdot\nabla) u-\frac{1}{\rho}\nabla p+\nu\triangle u\)
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- Navier-Stokes equation for incompressible Newtonian fluid.