Streamlines from strain rate tensor

In summary: Therefore, the strain rate tensor alone is not sufficient to determine the flow field and flow direction. To fully determine the flow field, both the strain rate tensor and the vorticity tensor would be needed.As for the relationship between streamlines and potential lines in 3D, the streamlines will still be perpendicular to the potential surfaces (which are now planes instead of lines in 2D), but the set of streamlines will not necessarily form a single plane with a line perpendicular to it. This is because the potential surfaces can vary in orientation and shape throughout the 3D flow field. The strain rate tensor can be used to solve an eigenvalue problem to determine the orientation and shape of the potential surfaces, but it would not be sufficient
  • #1
vktsn0303
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I was reading about strain rate tensors and other kinematic properties of fluids that can be obtained if we know the velocity field V = (u, v, w). It got me wondering if I can sketch streamlines if I have the strain rate tensor with me to start with. Let's say I have the strain rate tensor:

eq0073SP.gif


Would it now be possible to sketch the flow field and determine the flow direction from this? If yes, how?

Also, in 2D potential flows streamlines are perpendicular to potential lines. Does this mean in 3D the set of streamlines will be a plane with a line perpendicular to it? And can this also be explained with just the help of the strain rate tensor above by solving an eigenvalue problem?
 
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  • #2
No. The strain rate tensor can be zero while maintaining a steady flow. Think a body of fluid in uniform translation and rotation.

vktsn0303 said:
Also, in 2D potential flows streamlines are perpendicular to potential lines. Does this mean in 3D the set of streamlines will be a plane with a line perpendicular to it?

For a potential flow ##\vec v = \nabla \phi## for some ##\phi##. The gradient is always orthogonal to the level surfaces of a scalar function. The streamlines for an irrotational flow will therefore always be perpendicular to the potential surfaces.
 
  • #3
vktsn0303 said:
I was reading about strain rate tensors and other kinematic properties of fluids that can be obtained if we know the velocity field V = (u, v, w). It got me wondering if I can sketch streamlines if I have the strain rate tensor with me to start with. Let's say I have the strain rate tensor:

View attachment 204609

Would it now be possible to sketch the flow field and determine the flow direction from this? If yes, how?

Also, in 2D potential flows streamlines are perpendicular to potential lines. Does this mean in 3D the set of streamlines will be a plane with a line perpendicular to it? And can this also be explained with just the help of the strain rate tensor above by solving an eigenvalue problem?
No. Aside from not knowing the velocity vector at any given location (which would provide the constant of integration), the strain rate tensor represents only the symmetric part of the velocity gradient tensor, and does not include the antisymmetric part (i.e., the vorticity tensor) which describes rotation of the fluid parcels.
 

Related to Streamlines from strain rate tensor

1. What are streamlines from strain rate tensor?

The streamlines from strain rate tensor are visual representations of the direction and magnitude of fluid flow. They are created by using the strain rate tensor, which is a mathematical representation of the rate at which a fluid is deforming or stretching at a specific point.

2. How are streamlines from strain rate tensor useful?

Streamlines from strain rate tensor are useful in understanding the behavior of fluids in various situations, such as in aerodynamics, fluid mechanics, and weather forecasting. They can also be used to identify areas of high and low fluid velocity and to predict the movement of fluid particles.

3. How are streamlines from strain rate tensor calculated?

To calculate streamlines from strain rate tensor, the strain rate tensor is first determined at various points in the fluid. Then, using mathematical equations, the direction and magnitude of fluid flow can be calculated and visualized as streamlines.

4. Can streamlines from strain rate tensor change over time?

Yes, streamlines from strain rate tensor can change over time as the fluid flow changes. This can be seen in situations such as turbulent flow, where the streamlines may change rapidly and be difficult to predict.

5. How are streamlines from strain rate tensor different from other flow visualization techniques?

Streamlines from strain rate tensor provide a more detailed and accurate representation of fluid flow compared to other visualization techniques. They take into account both the direction and magnitude of fluid velocity, whereas other techniques may only show one aspect of fluid flow.

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