Consider a uniform flow with velocity \(V_{∞}\). Show that this flow is a physically possible incompressible flow and that it is irrotational.
Consider a uniform flow with velocity \(V_{∞}\). Show that this flow is a physically possible incompressible flow and that it is irrotational.
Let a uniform flow with velocity =\( V_{\infty}\).
\(\overrightarrow{V}=V_{\infty}\hat{i},V_{\infty}=u=\mathrm{constant}\)
\[\nabla \cdot \overrightarrow{V}=\frac{\partial u}{\partial x}+\frac{\partial v }{\partial y}+\frac{\partial w}{\partial z}=0+0+0=0\]
Therefore the flow is incompressible.
\[\nabla \times\overrightarrow{V} =\begin{vmatrix}
\overrightarrow{i} & \overrightarrow{j} & \overrightarrow{k}\\
\frac{\partial }{\partial x} &\frac{\partial }{\partial y} &\frac{\partial }{\partial z} \\
u& v& w
\end{vmatrix}\]
\[\Rightarrow\overrightarrow{i}\left ( 0-0 \right )-\overrightarrow{j}\left ( 0-0 \right )+\overrightarrow{k}\left ( 0-0 \right )
=0\]
Therefore
\[\nabla \times\overrightarrow{V}=0\]
The flow is irrotational.
