# Viscous stress in axisymmetric cylindrical coordinates

I'd like to write the viscous stress for incompressible Newtonian fluids in axisymmetric cylindrical coordinates $(r,z)$ with velocity component $(u,w)$ (for axisymmetric $v_\theta=0$ and $\partial_\theta()=0$). I get confused with the component of $\tau_{\theta\theta}=2\mu\frac{u}{r}$ because I don't know whether or not it should be included in the tensor. If I included $\tau_{\theta\theta}$, I got a 3*3 matrix, if not a 2*2 matrix. Please see

$\tau_{2*2}= \begin{pmatrix} 2\mu u_r & \mu (w_r+u_z) \\ \mu (w_r+u_z) & 2\mu w_z \end{pmatrix}, \quad \mbox{or} \quad \tau_{3*3}= \begin{pmatrix} 2\mu u_r & 0 &\mu (w_r+u_z) \\ 0 & \color{red}{2\mu\frac{u}{r}} &0 \\ \mu (w_r+u_z) & 0 & 2\mu w_z \end{pmatrix}$

I need the stress tensor to formulate an axisymmetric problem in cylindrical coordinates. Any one can explain which one is correct and why? Thanks in advance.

• @ Chester Miller Thanks a lot. But I still don't understand why one mush include the hoop stress (direct in circumferential direction) in the radial force balance... Do you mean a stress balance in the circumferential direction instead, in which $\tau_{\theta\theta}$ should be included. Btw, yes, I do need to do a stress balance in the radial direction.