Chapter 10, Problem 108P

Calculate the nine components of the viscous stress tensor in cylindrical coordinates (see Chap. 9) for the velocity field of Prob 10-107. Discuss your results.

To determine

The nine component of the viscous stress tensor in cylindrical coordinates.

Answer to Problem 108P

The nine component of the viscous stress tensor in cylindrical coordinates is ${\tau }_{ij}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$.

Explanation of Solution

Given information:

The radial velocity component is $0$, angular velocity component is $\omega r$ and z-direction velocity component is $0$.

Write the expression for all the nine component of viscous stress tensor in cylindrical coordinates.

  ${\tau }_{ij}=\left[\begin{array}{ccc}{\tau }_{rr}& {\tau }_{r\theta }& {\tau }_{rz}\\ {\tau }_{\theta r}& {\tau }_{\theta \theta }& {\tau }_{\theta z}\\ {\tau }_{zr}& {\tau }_{z\theta }& {\tau }_{zz}\end{array}\right]$   ...... (I)

Here, viscous stress tensor in $i\text{-}j$ plane is ${\tau }_{ij}$.

Write the expression for viscous stress tensor in $r-r$ plane.

  ${\tau }_{rr}=2\mu \frac{\partial u_r}{\partial r}$   ...... (II)

Here, dynamic viscosity is $\mu$.velocity in radial direction is $u_r$.

Write the expression for viscous stress tensor in $r-\theta$ plane.

  ${\tau }_{r\theta }=\mu \left(r\frac{\partial }{\partial r}\left(\frac{u_{\theta }}{r}\right)+\frac{1}{r}\frac{\partial u_r}{\partial \theta }\right)$   ...... (III)

Here, velocity in angular direction is $u_{\theta }$.

Write the expression for viscous stress tensor in $r-\text{z}$ plane.

  ${\tau }_{rz}=\mu \left(\frac{\partial u_r}{\partial z}+\frac{\partial u_z}{\partial r}\right)$   ...... (IV)

Here, velocity in z-direction is $u_{\text{z}}$

Write the expression for viscous stress tensor in $\theta -r$ plane.

  ${\tau }_{\theta r}=\mu \left(r\frac{\partial }{\partial r}\left(\frac{u_{\theta }}{r}\right)+\frac{1}{r}\frac{\partial u_r}{\partial \theta }\right)$   ...... (V)

Write the expression for viscous stress tensor in $\theta -\theta$ plane.

  ${\tau }_{\theta \theta }=2\mu \left(\frac{1}{r}\frac{\partial u_r}{\partial \theta }+\frac{u_r}{r}\right)$   ...... (VI)

Write the expression for viscous stress tensor in $\theta -z$ plane.

  ${\tau }_{\theta z}=\mu \left(\frac{\partial u_{\theta }}{\partial z}+\frac{1}{r}\frac{\partial u_z}{\partial \theta }\right)$   ...... (VII)

Write the expression for viscous stress tensor in $z-r$ plane.

  ${\tau }_{zr}=\mu \left(\frac{\partial u_r}{\partial z}+\frac{\partial u_z}{\partial r}\right)$   ...... (VIII)

Write the expression for viscous stress tensor in $z-\theta$ plane.

  ${\tau }_{z\theta }=\mu \left(\frac{\partial u_{\theta }}{\partial z}+\frac{1}{r}\frac{\partial u_z}{\partial \theta }\right)$   ...... (IX)

Write the expression for viscous stress tensor in $z-z$ plane.

  ${\tau }_{zz}=2\mu \frac{\partial u_z}{\partial z}$   ...... (X)

Substitute $2\mu \frac{\partial u_r}{\partial r}$ for ${\tau }_{rr}$, $\mu \left(\frac{\partial u_r}{\partial z}+\frac{\partial u_z}{\partial r}\right)$ for ${\tau }_{rz}$, $\mu \left(r\frac{\partial }{\partial r}\left(\frac{u_{\theta }}{r}\right)+\frac{1}{r}\frac{\partial u_r}{\partial \theta }\right)$ for ${\tau }_{r\theta }$, $\mu \left(r\frac{\partial }{\partial r}\left(\frac{u_{\theta }}{r}\right)+\frac{1}{r}\frac{\partial u_r}{\partial \theta }\right)$ for ${\tau }_{\theta r}$, $2\mu \left(\frac{1}{r}\frac{\partial u_r}{\partial \theta }+\frac{u_r}{r}\right)$ for ${\tau }_{\theta \theta }$, $\mu \left(\frac{\partial u_r}{\partial z}+\frac{\partial u_z}{\partial r}\right)$ for ${\tau }_{zr}$, $\mu \left(\frac{\partial u_{\theta }}{\partial z}+\frac{1}{r}\frac{\partial u_z}{\partial \theta }\right)$ for ${\tau }_{\theta z}$, $\mu \left(\frac{\partial u_{\theta }}{\partial z}+\frac{1}{r}\frac{\partial u_z}{\partial \theta }\right)$ for ${\tau }_{z\theta }$, $2\mu \frac{\partial u_z}{\partial z}$ for ${\tau }_{zz}$ in Equation (I).

  ${\tau }_{ij}=\left[\begin{array}{ccc}2\mu \frac{\partial u_r}{\partial r}& \mu \left(r\frac{\partial }{\partial r}\left(\frac{u_{\theta }}{r}\right)+\frac{1}{r}\frac{\partial u_r}{\partial \theta }\right)& \mu \left(\frac{\partial u_r}{\partial z}+\frac{\partial u_z}{\partial r}\right)\\ \mu \left(r\frac{\partial }{\partial r}\left(\frac{u_{\theta }}{r}\right)+\frac{1}{r}\frac{\partial u_r}{\partial \theta }\right)& 2\mu \left(\frac{1}{r}\frac{\partial u_r}{\partial \theta }+\frac{u_r}{r}\right)& \mu \left(\frac{\partial u_{\theta }}{\partial z}+\frac{1}{r}\frac{\partial u_z}{\partial \theta }\right)\\ \mu \left(\frac{\partial u_r}{\partial z}+\frac{\partial u_z}{\partial r}\right)& \mu \left(\frac{\partial u_{\theta }}{\partial z}+\frac{1}{r}\frac{\partial u_z}{\partial \theta }\right)& 2\mu \frac{\partial u_z}{\partial z}\end{array}\right]$   ...... (XI)

Substitute $0$ for $u_r$, $\omega r$ for $u_{\theta }$ and $0$ for $u_z$ in Equation (XI).

  ${\tau }_{ij}=\left[\begin{array}{ccc}2\mu \frac{\partial \left(0\right)}{\partial r}& \mu \left(r\frac{\partial }{\partial r}\left(\frac{\omega r}{r}\right)+\frac{1}{r}\frac{\partial \left(0\right)}{\partial \theta }\right)& \mu \left(\frac{\partial \left(0\right)}{\partial z}+\frac{\partial \left(0\right)}{\partial r}\right)\\ \mu \left(r\frac{\partial }{\partial r}\left(\frac{\omega r}{r}\right)+\frac{1}{r}\frac{\partial \left(0\right)}{\partial \theta }\right)& 2\mu \left(\frac{1}{r}\frac{\partial \left(0\right)}{\partial \theta }+\frac{\left(0\right)}{r}\right)& \mu \left(\frac{\partial \left(\omega r\right)}{\partial z}+\frac{1}{r}\frac{\partial \left(0\right)}{\partial \theta }\right)\\ \mu \left(\frac{\partial \left(0\right)}{\partial z}+\frac{\partial \left(0\right)}{\partial r}\right)& \mu \left(\frac{\partial \left(\omega r\right)}{\partial z}+\frac{1}{r}\frac{\partial \left(0\right)}{\partial \theta }\right)& 2\mu \frac{\partial \left(0\right)}{\partial z}\end{array}\right]$

  $=\left[\begin{array}{ccc}2\mu \left(0\right)& \mu \left(r\left(0\right)+\frac{1}{r}\left(0\right)\right)& \mu \left(0+0\right)\\ \mu \left(r\left(0\right)+\frac{1}{r}\left(0\right)\right)& 2\mu \left(\frac{1}{r}\left(0\right)+0\right)& \mu \left(0+\frac{1}{r}\left(0\right)\right)\\ \mu \left(0+0\right)& \mu \left(0+\frac{1}{r}\left(0\right)\right)& 2\mu \left(0\right)\end{array}\right]$

  $=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

Here, all component of viscous stress tensor is zero, so flow is inviscid and no viscous stress is present in flow field.

Conclusion:

The nine component of the viscous stress tensor in cylindrical coordinates is ${\tau }_{ij}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$.