Consider the non lifting flow over a circular cylinder of a given radius, where free stream velocity is \({V_\infty }\). Explain if there is any change in the shape of streamlines.
Consider the non lifting flow over a circular cylinder of a given radius, where free stream velocity is \({V_\infty }\).If \({V_\infty }\) is doubled,explain if there is any change in the shape of the stream lines.
\[{V_r} = \frac{1}{r}\frac{{\partial \psi }}{{\partial \theta }} = \left( {{V_\infty }\cos \theta } \right)\left( {1 – \frac{{{R^2}}}{{{r^2}}}} \right)\] and \[{V_\theta } = – \frac{{\partial \psi }}{{\partial r}} = – \left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right){V_\infty }\sin \theta \] therefore, \[\frac{{{V_r}}}{{{V_\infty }}} = \left( {1 – \frac{{{R^2}}}{{{r^2}}}} \right)\cos \theta \]\[\frac{{{V_\theta }}}{{{V_\infty }}} = – \left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right)\sin \theta \]So, at any given point \(\left( {r,\theta } \right)\), \({V_r}\) and \({{V_\theta }}\) are both directly proportional to\({{V_\infty }}\).Therefore, the direction of the resultant,\(\mathop V\limits^ \to \) is the same, for any value of \({{V_\infty }}\).This infers that the shape of the streamlines remains the same.
