Fluid stress tensor in cylindrical coordinates?

In summary, the stress strain relationship for a fluid with viscosity \mu can be expressed as \sigma_{ij} = -p \delta_{ij} + 2 \mu u_{ij}. When considering cylindrical coordinates, the strain tensor can be calculated using eq 1.8 in [1], but the \delta_{ij} portion remains unchanged. This means that for a non-viscous fluid, the stress tensor in cylindrical coordinates would be the same as in rectangular coordinates, with the only difference being that i,j would index over \{r, \theta, z\}. Therefore, the stress tensor in cylindrical coordinates would be \sigma_{ij} = -p \delta_{ij} with components for \sigma_{rr
  • #1
Peeter
305
3
For fluid with viscosity [itex]\mu[/itex] our stress strain relationship takes the form

[tex]\sigma_{ij} = -p \delta_{ij} + 2 \mu u_{ij}.[/tex]

I was wondering how to express this in cylindrical coordinates. The strain tensor I can calculate in cylindrical coordinates (what I get matches eq 1.8 in [1]). But how would the [itex]\delta_{ij}[/itex] portion of the stress strain relationship be expressed in cylindrical coordinates?

For example, if we considered a non-viscous fluid, the very simplest stress tensor, we have in rectangular coordinates

[tex]\sigma_{ij} = -p \delta_{ij}.[/tex]

It's not obvious to me how this would be expressed in cylindrical form. I wanted to try some calculations with the traction vector [itex]T_i = \sigma_{ij} n_j[/itex] in a cylindrical coordinate system, but I'm not sure how to express it. I figured the place to start was with the stress tensor.

References:

[1] L.D. Landau, EM Lifgarbagez, JB Sykes, WH Reid, and E.H. Dill. Theory of elasticity: Vol. 7 of course of theoretical physics. 1960.
 
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  • #2
After some thought, I think the answer is that this would be exactly like the rectangular case, where for a non-viscious stress [itex]\sigma_{ij} = -p \delta_{ij}[/itex], the cylindrical representation would be exactly the same, with the only difference being that i,j have to index over [itex]\{r, \theta, z\}[/itex]. So, we'd have a -p component for [itex]\sigma_{rr}, \sigma_{\theta\theta}, \sigma_{zz}[/itex].
 
Last edited:
  • #3
Peeter said:
After some thought, I think the answer is that this would be exactly like the rectangular case, where for a non-viscious stress [itex]\sigma_{ij} = -p \delta_{ij}[/itex], the cylindrical representation would be exactly the same, with the only difference being that i,j have to index over [itex]\{r, \theta, z\}[/itex]. So, we'd have a -p component for [itex]\sigma_{rr}, \sigma_{\theta\theta}, \sigma_{zz}[/itex].

I know nothing about fluid mechanics but comparing this tensor with Maxwell stress tensor in electromagnetism, you seem to be right. However in Maxwell stress tensor, only two components of matter, the radial and the tangential. Hence I expect an expression like the following:

[itex]\sigma_{rt}[/itex]=-p[itex]\delta_{rt}[/itex]+2[itex]\mu[/itex][itex]\mu_{rt}[/itex]

r: radial
t: tangential

see http://en.wikipedia.org/wiki/Maxwell_stress_tensor
 

1. What is the fluid stress tensor in cylindrical coordinates?

The fluid stress tensor in cylindrical coordinates is a mathematical representation of the stresses that act on a fluid at a specific point in space. It takes into account both the normal stresses, which act perpendicular to a surface, and the shear stresses, which act parallel to a surface.

2. How is the fluid stress tensor calculated in cylindrical coordinates?

The fluid stress tensor in cylindrical coordinates is calculated using the Navier-Stokes equations, which describe the motion of a fluid and the forces acting on it. It involves taking the derivatives of the velocity components in the radial, axial, and circumferential directions and plugging them into the tensor equation.

3. What is the significance of the fluid stress tensor in cylindrical coordinates?

The fluid stress tensor in cylindrical coordinates is important in understanding the behavior of fluids, particularly in situations where there are changes in flow direction and velocity. It helps engineers and scientists analyze and predict the stresses on structures, such as pipes and turbines, that are subject to fluid flow.

4. How does the fluid stress tensor change in different coordinate systems?

The fluid stress tensor is a tensor, meaning it is a mathematical object that can be represented in different coordinate systems. In cylindrical coordinates, it is represented differently than in Cartesian coordinates, but the underlying principles and equations remain the same.

5. What factors can affect the fluid stress tensor in cylindrical coordinates?

The fluid stress tensor in cylindrical coordinates can be affected by factors such as the velocity and direction of fluid flow, the viscosity of the fluid, and the geometry of the system. Changes in these factors can lead to changes in the magnitude and direction of the stresses acting on the fluid.

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