Symmetrization of a tensor in spherical coordinate

[oliveriandrea]

Hello, i don't know if my question is well posed,

if i have a symmetric tensor S_ij = (∂_ix_j + ∂_jx_i) / 2
with x_i cartesian coordinates, how can i transform it in a spherical coordinates system (ρ,θ,\varphi)?
(I need it for the calculus of shear stress tensor in spherical coordinate in fluid dynamics)

 

[Chestermiller]

Hello, i don't know if my question is well posed,

if i have a symmetric tensor S_ij = (∂_ix_j + ∂_jx_i) / 2
with x_i cartesian coordinates, how can i transform it in a spherical coordinates system (ρ,θ,\varphi)?
(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