Viscous Stress Tensor

Vector Tensor Notation

\[ \tau=-\mu(\nabla \mathbf{v}+(\nabla \mathbf{v})^\intercal)+\left(\frac{2}{3}\mu-\kappa\right)\left(\nabla\cdot \vec{v}\right)\delta \](1)

\(\delta\) is the unit ten­sor with com­po­nents \(\delta_{ij}\).

\(\nabla \mathbf{v}\) is the ve­loc­ity gra­di­ent ten­sor with com­po­nents \(\dfrac{\partial}{\partial{x_j}}v_{ij}\).

\(\nabla\cdot\vec v\) is the di­ver­gence of the ve­loc­ity vec­tor.

\(\kappa\) is the di­la­tional vis­cos­ity which is \(0\) for monoatomic gases at low den­si­ties.

FIX SIGNS FROM BIRD!!!

Spherical Coordinates \((r,\theta,\phi)\)

\[ \tau_{rr}=\mu\left[2\frac{\partial v_r}{\partial r}-\frac{2}{3}(\nabla\cdot\vec v)\right] \](2)

\[ \tau_{\theta\theta}=\mu\left[2\left(\frac{1}{r}\frac{\partial v_{\theta}}{\partial \theta}+\frac{v_{r}}{r}-\frac{2}{3}(\nabla\cdot\vec v) \right)\right] \](3)

\[ \tau_{r\theta}=\tau_{\theta r}=\mu\left[ r\frac{\partial}{\partial r}\left( \frac{v_\theta}{r} \right) + \frac{1}{r}\frac{\partial v_r}{\partial \theta} \right] \](4)

\[ \tau_{\theta\phi}=\tau_{\phi\theta}=\mu\left[ \frac{\sin\theta}{r}\frac{\partial}{\partial\theta}\left( \frac{v_\phi}{\sin\theta}+ \frac{1}{r\sin\theta}\frac{\partial v_\theta}{\partial\phi} \right) \right] \](5)

\[ \tau_{\phi r}=\tau_{r\phi}=\mu\left[ \frac{1}{r\sin\theta}\frac{\partial v_r}{\partial\phi}+r\frac{\partial}{\partial r}\left(\frac{v_\phi}{r} \right) \right] \](6)

Cylindrical Coordinates \((r,\theta,z)\)

\[ \tau_{rr}=\mu\left[2\frac{\partial v_r}{\partial r} -\frac{2}{3}(\nabla\cdot v) \right] \](7)

\[ \tau_{\theta\theta}=\mu\left[ 2\left( \frac{1}{r}\frac{\partial v_\theta}{\partial\theta} + \frac{v_r}{r} \right) - \frac{2}{3}(\nabla\cdot v) \right] \](8)

\[ \tau_{zz}=\mu\left[ 2\frac{\partial v_z}{\partial z}-\frac{2}{3}(\nabla\cdot v) \right] \](9)

\[ \tau_{r\theta}=\tau_{\theta r}=\mu\left[ r\frac{\partial}{\partial r}\left( \frac{v_\theta}{r} \right) + \frac{1}{r}\frac{\partial v_r}{\partial\theta} \right] \](10)

\[ \tau_{\theta z}=\tau_{z\theta}=\mu\left[\frac{\partial v_\theta}{\partial z} + \frac{1}{r}\frac{\partial v_r}{\partial\theta} \right] \](11)

\[ \tau_{zr}=\tau_{rz}=\mu\left[ \frac{\partial v_z}{\partial r}+\frac{\partial v_r}{\partial z} \right] \](12)

Rectangular Coordinates \((x,y,z)\)

\[ \tau_{xx}=-\mu\left[ 2\frac{\partial v_x}{\partial x} - \frac{2}{3}(\nabla\cdot v) \right] \](13)

\[ \tau_{yy}=-\mu\left[ 2\frac{\partial v_y}{\partial y} - \frac{2}{3}(\nabla\cdot v) \right] \](14)

\[ \tau_{zz}=-\mu\left[ 2\frac{\partial v_z}{\partial z} - \frac{2}{3}(\nabla\cdot v) \right] \](15)

\[ \tau_{xy}=\tau_{yx}=-\mu\left( \frac{\partial v_x}{\partial y} + \frac{\partial v_y}{\partial x} \right) \](16)

\[ \tau_{yz}=\tau_{zy}=-\mu\left( \frac{\partial v_y}{\partial z} + \frac{\partial v_z}{\partial y} \right) \](17)

\[ \tau_{xz}=\tau_{zx}=-\mu\left( \frac{\partial v_x}{\partial z} + \frac{\partial v_z}{\partial x} \right) \](18)