Vorticity.

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Vorticity.

The vorticity of the flow in Cartesian coordinates in term of dimensional variables is given by

$\begin{displaymath}\omega = \frac{\partial u}{\partial y}-\frac{\partial v}{\partial x}. \end{displaymath}$

(61)

If we non-dimensionalize the vorticity, according to

$\begin{displaymath}\omega = \frac{U}{L}\omega', \end{displaymath}$

(62)

the non-dimensional form of the equation (3.61) is

$\begin{displaymath}\frac{U}{L}\omega' = \frac{U}{\delta}\frac{\partial u'}{\partial y'}-\frac{U\delta}{L^{2}}\frac{\partial v'}{\partial x'}, \end{displaymath}$

(63)

and since $\delta \sim O(R^{-1/2}L)$ , we have that

$\begin{displaymath}\omega' = R^{1/2}\frac{\partial u'}{\partial y'}-R^{-1/2}\frac{\partial v'}{\partial x'}, \end{displaymath}$

(64)

and in the limit $R \rightarrow \infty$ we have

$\begin{displaymath}\omega' \sim R^{1/2}\frac{\partial u'}{\partial y'}. \end{displaymath}$

(65)

In the context of the boundary-layer approximation, the vorticity in terms of dimensional variables is given by

$\begin{displaymath}\omega = \mu\frac{\partial u}{\partial y}. \end{displaymath}$

(66)

We can write the vorticity for the Blasius boundary layer similarity solution by placing equation (3.47) for $\psi$ into the equation (3.66), which gives

$\begin{displaymath}\omega = \mu\frac{\partial^{2}\psi}{\partial y^{2}} = \mu\left(\frac{U^{3}}{2\nu x}\right)^{1/2}f''(\eta) \end{displaymath}$

(67)

Next: Stress. Up: Similarity Solution. Previous: Similarity Solution.

Karl P Burr
2003-03-12