Another example of the method uses flow past a cylinder of radius, \(R\), and its image to simulate the flow of fluid past a cylinder that lies close to a plane wall. The situation is shown in Figure P9.22.

The stream function for this flow is given by:

\[\psi=-2 v_{\infty} y+v_{\infty} R^{2}\left[\frac{y-a}{x^{2}+(y-a)^{2}}+\frac{y+a}{x^{2}+(y+a)^{2}}\right]\]

a. Determine the point where the maximum velocity is located.

b. The pressure within the irrotational flow field is:

\[P+\frac{1}{2} ho v^{2}+ho g z=\text { constant } \quad \text { Bernoulli's equation }\]

Here \(z\) is the height of the fluid above some reference (not important here, \(z=0\) ) and \(v^{2}=v_{x}^{2}+v_{y}^{2}\). Using this result (we derive it in Chapter 11) and the fact that \(P=P_{o}\) and \(v=v_{\infty}\) far away from the source, determine the pressure distribution along the wall.

Transcribed Image Text:

FIGURE P9.22 a Cylinder Cylinder Image X