Equations of Motion

Consider axisymmetric motion in two dimensions, so that the flow is confined to a plane. Use cylindrical coordinates \( (r, \phi, z) \), where \( z \) is the direction perpendicular to the plane, with velocity components \( (u^r, u^\phi, u^z) \). For axisymmetric flow \( u^z = u^r = 0 \) but \( u^\phi \neq 0 \).

Rigid body motion: For a body in rigid body rotation, the velocity distribution is given by: \[ u^\phi = \Omega r \] where \( \Omega \) is the angular velocity of the fluid and \( r \) is the distance from the axis of rotation. Associated with this rotation is a vorticity given by: \[ \mathbf{\omega} = \nabla \times \mathbf{v} = \omega^z \mathbf{k} \] where: \[ \omega^z = \frac{1}{r} \frac{\partial}{\partial r} (r u^\phi) - \frac{1}{r} \frac{\partial}{\partial r} (r^2 \Omega) = 2\Omega \] The vorticity of a fluid in solid body rotation is thus twice the angular velocity of the fluid about the axis of rotation, and is pointed in a direction orthogonal to the plane of rotation.

\begin{gather} {\small \text{curl } A = e_1 \!\left(\frac{\partial A^3}{\partial x^2} - \frac{\partial A^2}{\partial x^3} \right) + e_2 \!\left(\frac{\partial A^1}{\partial x^3} - \frac{\partial A^3}{\partial x^1} \right) + e_3 \!\left(\frac{\partial A^2}{\partial x^1} - \frac{\partial A^1}{\partial x^2} \right) \\[10pt] \text{curl } A = \begin{vmatrix} e_1 & e_2 & e_3 \\ \frac{\partial}{\partial x^1} & \frac{\partial}{\partial x^2} & \frac{\partial}{\partial x^3} \\ A^1 & A^2 & A^3 \end{vmatrix} \\[10pt] d\omega^1_A = \omega^2_{\text{curl} A} } \end{gather}

\(\because\) \( \boldsymbol{\omega} = \nabla \times \mathbf{v} \), \( \nabla \times \mathbf{v} = \begin{vmatrix} \hat{\mathbf{r}} & \hat{\boldsymbol{\phi}} & \hat{\mathbf{z}} \\ \frac{\partial}{\partial r} & \frac{1}{r} \frac{\partial}{\partial \phi} & \frac{\partial}{\partial z} \\ u^r & u^\phi & u^z \end{vmatrix} \)

\(\because\) \( u^r = 0, u^z = 0, u^\phi = \Omega r \) \( \therefore \mathbf{v} = (0, \Omega r, 0) \)

\(\because\) \( \omega^r = 0, \omega^\phi = 0 \), \( \omega^z = \frac{1}{r} \frac{\partial}{\partial r} (r u^\phi) \)

\(\because\) \(\frac{1}{r} \frac{\partial}{\partial r} (r u^\phi) = \frac{1}{r} \left( u^\phi + r \frac{\partial u^\phi}{\partial r} \right)\)

\(\because\) \( u^\phi = \Omega r \), \( \therefore r u^\phi = \Omega r^2 \)

\(\therefore\) \( \frac{\partial}{\partial r} (\Omega r^2) = 2 \Omega r \), \(\frac{1}{r} \frac{\partial}{\partial r} (r^2 \Omega) = \frac{1}{r} (2 r \Omega) = 2\Omega\)

\(\therefore \omega^z = \frac{1}{r} (2 \Omega r) = 2 \Omega \)