Hi everyone. I have a question on how to compute the fluid stress tensor in spherical polar coordinates. I actually just want a check on my method to see if it is correct. The stress tensor in cartesian coordinates (that I will indicate as i,j,k) is:
$$
\sigma_{ij}=\eta\left(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i}-\frac{2}{3}\delta_{ij}\frac{\partial v_k}{\partial x_k}\right).
$$
I will also indicate the spherical coordinates with indices a,b,c. The tensor in these new coordinates should be:
$$
\sigma_{ab}=\frac{\partial \bar x_a}{\partial x_i}\frac{\partial \bar x_b}{\partial x_j}\sigma_{ij}.
$$
Now, the point is that [itex]\sigma_{ij}[/itex] is still expressed in terms of [itex]v_{x,y,z}[/itex] as a function of x,y,z. Do I just need to express each term [itex]\partial v_i/\partial x_j[/itex] in terms of the spherical coordinates using the chain rule, metric tensor and so on?
Is that correct? Thank you very much.
$$
\sigma_{ij}=\eta\left(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i}-\frac{2}{3}\delta_{ij}\frac{\partial v_k}{\partial x_k}\right).
$$
I will also indicate the spherical coordinates with indices a,b,c. The tensor in these new coordinates should be:
$$
\sigma_{ab}=\frac{\partial \bar x_a}{\partial x_i}\frac{\partial \bar x_b}{\partial x_j}\sigma_{ij}.
$$
Now, the point is that [itex]\sigma_{ij}[/itex] is still expressed in terms of [itex]v_{x,y,z}[/itex] as a function of x,y,z. Do I just need to express each term [itex]\partial v_i/\partial x_j[/itex] in terms of the spherical coordinates using the chain rule, metric tensor and so on?
Is that correct? Thank you very much.