Divergence of a dot product of tensor and vector
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Hei, I am trying to derive energy equation from Navier-Stokes equation and I come across this:
$$nabla.(sigma.v)=(nabla.sigma).v +sigma:nabla v$$
$sigma $ is the stress tensor
V :is the velocity vector
Could anyone thankfully explain this and if that is correct?
vector-analysis tensors classical-mechanics
edited Dec 22 '18 at 16:22
Harry49
8,80831346
asked Nov 23 '18 at 14:39
F.OF.O
788
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add a comment |
$begingroup$
Hei, I am trying to derive energy equation from Navier-Stokes equation and I come across this:
$$nabla.(sigma.v)=(nabla.sigma).v +sigma:nabla v$$
$sigma $ is the stress tensor
V :is the velocity vector
Could anyone thankfully explain this and if that is correct?
vector-analysis tensors classical-mechanics
edited Dec 22 '18 at 16:22
Harry49
8,80831346
asked Nov 23 '18 at 14:39
F.OF.O
788
$endgroup$
add a comment |
$begingroup$
Hei, I am trying to derive energy equation from Navier-Stokes equation and I come across this:
$$nabla.(sigma.v)=(nabla.sigma).v +sigma:nabla v$$
$sigma $ is the stress tensor
V :is the velocity vector
Could anyone thankfully explain this and if that is correct?
vector-analysis tensors classical-mechanics
edited Dec 22 '18 at 16:22
Harry49
8,80831346
asked Nov 23 '18 at 14:39
F.OF.O
788
$endgroup$
Hei, I am trying to derive energy equation from Navier-Stokes equation and I come across this:
$$nabla.(sigma.v)=(nabla.sigma).v +sigma:nabla v$$
$sigma $ is the stress tensor
V :is the velocity vector
Could anyone thankfully explain this and if that is correct?
vector-analysis tensors classical-mechanics
vector-analysis tensors classical-mechanics
edited Dec 22 '18 at 16:22
Harry49
8,80831346
asked Nov 23 '18 at 14:39
F.OF.O
788
edited Dec 22 '18 at 16:22
Harry49
8,80831346
asked Nov 23 '18 at 14:39
F.OF.O
788
edited Dec 22 '18 at 16:22
Harry49
8,80831346
edited Dec 22 '18 at 16:22
Harry49
8,80831346
edited Dec 22 '18 at 16:22
Harry49
8,80831346
8,80831346
asked Nov 23 '18 at 14:39
F.OF.O
788
asked Nov 23 '18 at 14:39
F.OF.O
788
asked Nov 23 '18 at 14:39
F.OF.O
788
788
add a comment |
add a comment |
1 Answer
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Let us consider an orthonormal basis of the euclidean space. The divergence reads
begin{aligned}
nablacdot (sigmacdot v) &= (sigma_{ij} v_j)_{,i} \ &= sigma_{ij,i} v_j + sigma_{ij} v_{j,i} \
&= sigma_{ji,i} v_j + sigma_{ji} v_{j,i} \
&= (nablacdotsigma)cdot v + sigma : nabla v
end{aligned}
using Einstein notation and the symmetry of the stress tensor.
answered Dec 22 '18 at 16:21
Harry49Harry49
8,80831346
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1 Answer
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1 Answer
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active
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active
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$begingroup$
Let us consider an orthonormal basis of the euclidean space. The divergence reads
begin{aligned}
nablacdot (sigmacdot v) &= (sigma_{ij} v_j)_{,i} \ &= sigma_{ij,i} v_j + sigma_{ij} v_{j,i} \
&= sigma_{ji,i} v_j + sigma_{ji} v_{j,i} \
&= (nablacdotsigma)cdot v + sigma : nabla v
end{aligned}
using Einstein notation and the symmetry of the stress tensor.
answered Dec 22 '18 at 16:21
Harry49Harry49
8,80831346
$endgroup$
add a comment |
$begingroup$
Let us consider an orthonormal basis of the euclidean space. The divergence reads
begin{aligned}
nablacdot (sigmacdot v) &= (sigma_{ij} v_j)_{,i} \ &= sigma_{ij,i} v_j + sigma_{ij} v_{j,i} \
&= sigma_{ji,i} v_j + sigma_{ji} v_{j,i} \
&= (nablacdotsigma)cdot v + sigma : nabla v
end{aligned}
using Einstein notation and the symmetry of the stress tensor.
answered Dec 22 '18 at 16:21
Harry49Harry49
8,80831346
$endgroup$
add a comment |
$begingroup$
Let us consider an orthonormal basis of the euclidean space. The divergence reads
begin{aligned}
nablacdot (sigmacdot v) &= (sigma_{ij} v_j)_{,i} \ &= sigma_{ij,i} v_j + sigma_{ij} v_{j,i} \
&= sigma_{ji,i} v_j + sigma_{ji} v_{j,i} \
&= (nablacdotsigma)cdot v + sigma : nabla v
end{aligned}
using Einstein notation and the symmetry of the stress tensor.
answered Dec 22 '18 at 16:21
Harry49Harry49
8,80831346
$endgroup$
Let us consider an orthonormal basis of the euclidean space. The divergence reads
begin{aligned}
nablacdot (sigmacdot v) &= (sigma_{ij} v_j)_{,i} \ &= sigma_{ij,i} v_j + sigma_{ij} v_{j,i} \
&= sigma_{ji,i} v_j + sigma_{ji} v_{j,i} \
&= (nablacdotsigma)cdot v + sigma : nabla v
end{aligned}
using Einstein notation and the symmetry of the stress tensor.
answered Dec 22 '18 at 16:21
Harry49Harry49
8,80831346
answered Dec 22 '18 at 16:21
Harry49Harry49
8,80831346
answered Dec 22 '18 at 16:21
Harry49Harry49
8,80831346
answered Dec 22 '18 at 16:21
Harry49Harry49
8,80831346
8,80831346
add a comment |
add a comment |
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