# Polymeric Stress Regularity in a Slender Jet Axis-regularity constraints on $\boldsymbol{\sigma}_p$ used in [[polymeric-stress-regularity-slender-jet|Slender jet momentum equations]]. Use axisymmetric cylindrical coordinates $(r,\theta,z)$ with no swirl ($\partial_\theta() = 0$). ## 1) Axis regularity and rotational invariance At $r=0$, the transverse plane is rotationally symmetric. A physical tensor field must have a unique, direction-independent limit at the axis. For the polymer stress components in the $(r,\theta)$ plane, this implies $ \sigma_{p,rr}(0,z,t)=\sigma_{p,\theta\theta}(0,z,t), \qquad \sigma_{p,r\theta}(0,z,t)=0. $ So there is no preferred transverse direction at the axis. ## 2) Why $\sigma_{p,rr}-\sigma_{p,\theta\theta}=O(r^2)$ The radial divergence contains $ (\nabla\cdot\boldsymbol{\sigma}_p)_r =\partial_r\sigma_{p,rr}+\frac{\sigma_{p,rr}-\sigma_{p,\theta\theta}}{r}+\partial_z\sigma_{p,rz}. $ To avoid a singular $1/r$ forcing at $r=0$, the difference $ \Delta_\perp\equiv\sigma_{p,rr}-\sigma_{p,\theta\theta} $ must vanish at least as $O(r)$, and axis smoothness with Taylor-expandable fields gives the stronger form ([[r-2-parity|why?]]) $ \Delta_\perp=O(r^2). $ Hence $ \sigma_{p,rr}(0,z,t)=\sigma_{p,\theta\theta}(0,z,t), \qquad \sigma_{p,rr}-\sigma_{p,\theta\theta}=O(r^2). $ Equivalent decomposition near the axis: $ \Sigma_\perp\equiv\frac{\sigma_{p,rr}+\sigma_{p,\theta\theta}}{2} =\Sigma_{\perp,0}(z,t)+\Sigma_{\perp,2}(z,t)r^2+\cdots, $ $ \Delta_\perp\equiv\frac{\sigma_{p,rr}-\sigma_{p,\theta\theta}}{2} =\Delta_{\perp,2}(z,t)r^2+\cdots. $ ## 3) Why $\sigma_{p,rz}=r\,S+O(r^3)$ The axial divergence contains $ (\nabla\cdot\boldsymbol{\sigma}_p)_z =\partial_z\sigma_{p,zz}+\frac{1}{r}\partial_r(r\sigma_{p,rz}). $ If $\sigma_{p,rz}\to c\neq0$ as $r\to0$, then $ \frac{1}{r}\partial_r(r\sigma_{p,rz})\sim\frac{c}{r}, $ which is singular. Therefore boundedness requires $ \sigma_{p,rz}=O(r). $ With smooth odd parity in $r$, $ \sigma_{p,rz}(r,z,t)=r\,S(z,t)+O(r^3). $ ## 4) Leading-order form used in slender-jet reduction A consistent regular expansion is $ \sigma_{p,zz}(r,z,t)=\Sigma_{zz}(z,t)+O(\varepsilon^2 r^2), $ $ \sigma_{p,rr}(r,z,t)=\Sigma_{rr}(z,t)+O(\varepsilon^2 r^2), $ $ \sigma_{p,\theta\theta}(r,z,t)=\Sigma_{\theta\theta}(z,t)+O(\varepsilon^2 r^2), $ $ \sigma_{p,rz}(r,z,t)=r\,S(z,t)+O(\varepsilon^2 r^3), $ with $ \Sigma_{rr}=\Sigma_{\theta\theta} \quad\text{at leading order.} $ So the independent leading-order polymer contribution in the axial 1D model is the axial-radial normal-stress difference $ \Delta\Sigma\equiv\Sigma_{zz}-\Sigma_{rr}, $ while hoop stress is not an additional independent leading-order degree of freedom.