Mixing
deformation of a fluid
$begingroup$
Given a 2D flow $u=[Sx, Sy, 0]^T$, show that a fluid element which was initially circular will be deform into an ellipse with major axes aligned $frac{pi}{4}$ with the $x-$ axis.
I have started by finding the deformation/ strain rate tensor, and the corresponding eigenvectors and eigenvalues. This gives the eigenvectors
$begin{bmatrix}
1 & 0 & 0
end{bmatrix}$, $begin{bmatrix}
0 & 1 & 0
end{bmatrix}$, and $begin{bmatrix}
0 & 0 & 1
end{bmatrix} $.
I know that these eigenvectors form the rotation matrix, but I am not sure where to go from here. Any help would be great.
fluid-dynamics
asked Jan 29 at 23:07
lrs417lrs417
1
$endgroup$
add a comment |
$begingroup$
Given a 2D flow $u=[Sx, Sy, 0]^T$, show that a fluid element which was initially circular will be deform into an ellipse with major axes aligned $frac{pi}{4}$ with the $x-$ axis.
I have started by finding the deformation/ strain rate tensor, and the corresponding eigenvectors and eigenvalues. This gives the eigenvectors
$begin{bmatrix}
1 & 0 & 0
end{bmatrix}$, $begin{bmatrix}
0 & 1 & 0
end{bmatrix}$, and $begin{bmatrix}
0 & 0 & 1
end{bmatrix} $.
I know that these eigenvectors form the rotation matrix, but I am not sure where to go from here. Any help would be great.
fluid-dynamics
asked Jan 29 at 23:07
lrs417lrs417
1
$endgroup$
$begingroup$
Check the signs of the components. As it stands the flow is not incompressible since $nabla cdot mathbf{u} = 2S neq 0$. Are you sure you don't wan't $[Sx,-Sy,0]$?
$endgroup$
– RRL
Jan 30 at 17:34
add a comment |
$begingroup$
Given a 2D flow $u=[Sx, Sy, 0]^T$, show that a fluid element which was initially circular will be deform into an ellipse with major axes aligned $frac{pi}{4}$ with the $x-$ axis.
I have started by finding the deformation/ strain rate tensor, and the corresponding eigenvectors and eigenvalues. This gives the eigenvectors
$begin{bmatrix}
1 & 0 & 0
end{bmatrix}$, $begin{bmatrix}
0 & 1 & 0
end{bmatrix}$, and $begin{bmatrix}
0 & 0 & 1
end{bmatrix} $.
I know that these eigenvectors form the rotation matrix, but I am not sure where to go from here. Any help would be great.
fluid-dynamics
asked Jan 29 at 23:07
lrs417lrs417
1
$endgroup$
Given a 2D flow $u=[Sx, Sy, 0]^T$, show that a fluid element which was initially circular will be deform into an ellipse with major axes aligned $frac{pi}{4}$ with the $x-$ axis.
I have started by finding the deformation/ strain rate tensor, and the corresponding eigenvectors and eigenvalues. This gives the eigenvectors
$begin{bmatrix}
1 & 0 & 0
end{bmatrix}$, $begin{bmatrix}
0 & 1 & 0
end{bmatrix}$, and $begin{bmatrix}
0 & 0 & 1
end{bmatrix} $.
I know that these eigenvectors form the rotation matrix, but I am not sure where to go from here. Any help would be great.
fluid-dynamics
fluid-dynamics
asked Jan 29 at 23:07
lrs417lrs417
1
asked Jan 29 at 23:07
lrs417lrs417
1
asked Jan 29 at 23:07
lrs417lrs417
1
asked Jan 29 at 23:07
lrs417lrs417
1
asked Jan 29 at 23:07
lrs417lrs417
1
1
$begingroup$
Check the signs of the components. As it stands the flow is not incompressible since $nabla cdot mathbf{u} = 2S neq 0$. Are you sure you don't wan't $[Sx,-Sy,0]$?
$endgroup$
– RRL
Jan 30 at 17:34
add a comment |
$begingroup$
Check the signs of the components. As it stands the flow is not incompressible since $nabla cdot mathbf{u} = 2S neq 0$. Are you sure you don't wan't $[Sx,-Sy,0]$?
$endgroup$
– RRL
Jan 30 at 17:34
$begingroup$
Check the signs of the components. As it stands the flow is not incompressible since $nabla cdot mathbf{u} = 2S neq 0$. Are you sure you don't wan't $[Sx,-Sy,0]$?
$endgroup$
– RRL
Jan 30 at 17:34
$begingroup$
Check the signs of the components. As it stands the flow is not incompressible since $nabla cdot mathbf{u} = 2S neq 0$. Are you sure you don't wan't $[Sx,-Sy,0]$?
$endgroup$
– RRL
Jan 30 at 17:34
add a comment |
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$begingroup$
Check the signs of the components. As it stands the flow is not incompressible since $nabla cdot mathbf{u} = 2S neq 0$. Are you sure you don't wan't $[Sx,-Sy,0]$?
$endgroup$
– RRL
Jan 30 at 17:34