Symmetrization of a tensor in spherical coordinate

Tensor Analysis and Continuum Mechanics, provides a more thorough and mathematical treatment of tensors in curvilinear coordinates. In particular, they give the transformation equations for tensors between cartesian, cylindrical, and spherical coordinates. They also cover the calculus of tensors in these coordinate systems, including the calculation of shear stresses.In summary, the question is asking for a transformation of a symmetric tensor from cartesian coordinates to spherical coordinates for the purpose of calculating shear stress in fluid dynamics. Useful resources for this topic include Bird, Stewart, and Lightfoot's Transport Phenomena and Chetan, et al.'s Tensor Analysis and Continuum Mechanics.
  • #1
oliveriandrea
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Hello, i don't know if my question is well posed,

if i have a symmetric tensor Sij = (∂ixj + ∂jxi) / 2
with xi cartesian coordinates, how can i transform it in a spherical coordinates system (ρ,θ,[itex]\varphi[/itex])?
(I need it for the calculus of shear stress tensor in spherical coordinate in fluid dynamics)
 
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  • #2
oliveriandrea said:
Hello, i don't know if my question is well posed,

if i have a symmetric tensor Sij = (∂ixj + ∂jxi) / 2
with xi cartesian coordinates, how can i transform it in a spherical coordinates system (ρ,θ,[itex]\varphi[/itex])?
(I need it for the calculus of shear stress tensor in spherical coordinate in fluid dynamics)
Bird, Stewart, and Lightfoot, Transport Phenomena, gives the components of the stress tensor for a Newtonian fluid in cartesian coordinates, cylindrical coordinates, and spherical coordinates. They also give the stress equilibrium equations and the Navier Stokes equations for these coordinate systems. One thing to be careful about is that they use an unconventional sign convention for the stress tensor: compressive stresses are considered positive and tensile stresses are considered negative. But once you get past this, they have a wealth of useful information tabulated.

Chet
 
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What is a tensor?

A tensor is a mathematical object that describes the linear relationship between different coordinate systems. It has components that transform in a specific way when the coordinate system is changed.

What is symmetrization of a tensor in spherical coordinates?

Symmetrization of a tensor in spherical coordinates is a process of rearranging the components of a tensor in a way that preserves its symmetry under rotations and reflections in a spherical coordinate system.

Why is symmetrization of a tensor important?

Symmetrization of a tensor is important because it simplifies the mathematical expressions involving tensors and makes them easier to work with. It also helps in identifying patterns and relationships between different components of the tensor.

How is symmetrization of a tensor done in spherical coordinates?

In spherical coordinates, symmetrization of a tensor is done by taking the average of its components that are symmetric with respect to rotations and reflections along different axes. This results in a new tensor with fewer components, but with the same symmetry properties.

What are some real-world applications of symmetrization of a tensor in spherical coordinates?

Symmetrization of a tensor in spherical coordinates has applications in many fields such as physics, engineering, and computer science. Some examples include analyzing stresses and strains in a spherical shell, computing magnetic fields in spherical magnets, and optimizing algorithms for image processing and computer vision.

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