Changes between Version 10 and Version 11 of AstroBearProjects/multiphysics
- Timestamp:
- 10/28/19 13:47:32 (5 years ago)
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AstroBearProjects/multiphysics
v10 v11 2 2 [[BR]] 3 3 The viscosity term in the Navier-Stokes Equation can be separated as:[[BR]][[BR]] 4 [[latex($ dv/dt = \nabla \cdot T$)]][[BR]]4 [[latex($ {\displaystyle \rho \frac{d \mathbf{v}}{dt} = \nabla \cdot \overline{\overline{T}} } $)]][[BR]] 5 5 For the applications we are considering, the fluid is isotropic. Under this assumption, the tensor can be expressed as the following combination of two scalar dynamic viscosities [[latex($\mu$)]] and [[latex($\mu'$)]]:[[BR]][[BR]] 6 6 [[latex($T = 2 \mu \nabla v + \mu' \nabla \cdot v I$)]][[BR]][[BR]] … … 9 9 [[latex($\mu' = -2/3\mu$)]][[BR]][[BR]] 10 10 Combining these relations, we can write down the final form of the separated viscous equation:[[BR]][[BR]] 11 [[latex($ dv/dt=\nabla \cdot (2 \mu \nabla v) - 2/3 \nabla (\mu \nabla \cdot v)$)]][[BR]][[BR]]11 [[latex($ {\displaystyle \rho \frac{d \mathbf{v}}{dt} = \nabla \cdot (2 \mu \, \nabla \mathbf{v}) - \frac{2}{3} \nabla \cdot \{ (\mu \, \nabla \cdot \mathbf{v}) \overline{\overline{I}} \} } $)]][[BR]][[BR]] 12 12 Notice that this equation already counts the effect of all the elements in a viscosity tensor, assuming the fluid itself is isotropic.[[BR]] 13 13 [[BR]][[BR]]