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For a 2D flow (z velocity component is zero, so are all derivatives with respect to the z axis), write down the components of the strain rate tensor 2) The following strain rate tensor is NOT valid...

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For a 2D flow (z velocity component is zero, so are all derivatives with respect to the z axis), write down the components of the strain rate tensor 2) The following strain rate tensor is NOT valid for incompressible flow. Why? 3) Could the following be a strain rate tensor? If yes, explain the two properties that this tensor satisfies that makes it valid. If no, explain why not. a) 4) Give an example of a 3D velocity gradient (i.e. the deformation tensor) that corresponds to a purely rotational flow 5) Given the following strain rate tensors, draw a square shaped fluid element and show the shape the fluid element would take after being deformed by the fluid flow
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Playing with the deformation and strain rate tensors 1) For a 2D flow (z velocity component is zero, so are all derivatives with respect to the z axis), write down the components of the strain rate tensor 2) The following strain rate tensor is NOT valid for incompressible flow. Why? 3) Could the following be a strain rate tensor? If yes, explain the two properties that this tensor satisfies that makes it valid. If no, explain why not. a) 4) Give an example of a 3D velocity gradient (i.e. the deformation tensor) that corresponds to a purely rotational flow 5) Given the following strain rate tensors, draw a square shaped fluid element and show the shape the fluid element would take after being deformed by the fluid flow a) b) c)

Answered Same Day Dec 20, 2021

Solution

Robert answered on Dec 20 2021
123 Votes
1) Given data : w = 0
Therefore we can write strain rate tensor matrix as
0
0
0
0
2
1
0
2
1
y
v
y
u
x
v
x
v
y
u
x
u
¶
¶
÷
÷
ø
ö
ç
ç
è
æ
¶
¶
+
¶
¶
÷
÷
ø
ö
ç
ç
è
æ
¶
¶
+
¶
¶
¶
¶
=
ÃŽ
&
Because,
EMBED Equation.3 0
=
¶
¶
=
¶
¶
=
z
v
z
u
w
(Ans)
2) Given strain-rate tensor
1
1
1
1
1
1
1
1
1
=
ÃŽ
&
As we know for incompressible flow condition
0
=
¶
¶
+
¶
¶
+
¶
¶
z
w
y
v
x
u
But the sum of the diagonal terms = 3
Therefore the given strain rate tensor is not valid for incompressible flow.
3) Given strain rate tensor
1
1
1
1
0
1
1
1
1
-
-
-
=
ÃŽ
&
It’s not a valid strain rate tensor. The reason behind this is the given matrix is not a symmetric matrix.
Transpose of the above matrix will be
1
1
1
1
0
1
1
1
1
-
-
-
=
ÃŽ
T
&
Now clearly...
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