An incompressible fluid flows steadily past a circular cylinder as shown in Fig. P3.8. The fluid velocity along the dividing streamline \((-\infty \leq x \leq-a)\) is found to be \(V=V_{0}\left(1-a^{2} / x^{2}\right)\), where \(a\) is the radius of the cylinder and \(V_{0}\) is the upstream velocity.

(a) Determine the pressure gradient along this streamline.

(b) If the upstream pressure is \(p_{0}\), integrate the pressure gradient to obtain the pressure \(p(x)\) for \(-\infty \leq x \leq-a\).

(c) Show from the result of part

(b) that the pressure at the stagnation point \((x=-a)\) is \(p_{0}+ho V_{0}^{2} / 2\), as expected from the Bernoulli equation.

Figure P3.8

Transcribed Image Text:

Bernoulli equation p+pv + z = constant along streamline