Changes between Version 10 and Version 11 of AstroBearProjects/multiphysics


Ignore:
Timestamp:
10/28/19 13:47:32 (5 years ago)
Author:
Atma
Comment:

Density was missing from equations.

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  • AstroBearProjects/multiphysics

    v10 v11  
    22[[BR]]
    33The 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}} } $)]][[BR]]
    55For the applications we are considering, the fluid is isotropic. Under this assumption, the tensor can be expressed as the following combination of two scalar dynamic viscosities [[latex($\mu$)]] and [[latex($\mu'$)]]:[[BR]][[BR]]
    66[[latex($T = 2 \mu \nabla v + \mu' \nabla \cdot v I$)]][[BR]][[BR]]
     
    99[[latex($\mu' = -2/3\mu$)]][[BR]][[BR]]
    1010Combining 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]]
    1212Notice that this equation already counts the effect of all the elements in a viscosity tensor, assuming the fluid itself is isotropic.[[BR]]
    1313[[BR]][[BR]]