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4
questions
2
votes
1
answer
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views
Proving $σ_{ji,j} + f_i = ρ(Dv_i/Dt) \implies (σ_{ij} - ρv_iv_j) + f_i = \partial(ρv_i)/ \partial t$
Exercise :
Prove that
$$σ_{ji,j} + f_i = ρ(Dv_i/Dt) \implies (σ_{ij} - ρv_iv_j)_{,j} + f_i = \partial(ρv_i)/ \partial t$$
where $σ_{ij} = σ_{ji}$ is the stress tensor.
Attempt :
It is :
$$\...
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6
votes
1
answer
467
views
Net torque on a surface in Stokes flow
I'm having difficulty seeing how the net torque on a surface in Stokes flow is non-zero.
For Stokes flow, we have $\nabla\cdot\sigma = 0$, where $\sigma$ is a symmetric stress tensor. The net torque ...
1
vote
0
answers
358
views
Can you recover a stress tensor from a velocity field?
Consider the Cauchy momentum equation for a steady flow:
$$ \rho\mathbf{u}\cdot\nabla\mathbf{u} = \rho\mathbf{g} + \nabla\cdot\mathbf{\sigma} $$
Here, $\mathbf{\sigma}$ is the stress tensor and $\...
2
votes
2
answers
174
views
Why is this true: $\nabla \cdot (S\cdot \vec v )=S:(\nabla \otimes \vec v)+\vec v \cdot (\nabla \cdot S)$?
I am doing fluid mechanics and I don't understand a particular step that is being used.
It is the following step which I don't understand:
$\nabla \cdot (S\cdot \vec v )=S:(\nabla \otimes \vec v)+\...
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