Consider the stream function defined as[br][center][math]\psi=U\left(1-\frac{a^2}{x^2+y^2}\right)-\frac{\Gamma}{4\pi}\log\left(x^2+y^2\right)[/math][/center][justify]where [math]\Gamma[/math] is arbitrary, and represents the circulation about the cylinder.[br] [br]The resulting flow is shown in the following simulation, with a cylinder of radius [math]a=1[/math], circulation [math]-10\pi\le\Gamma\le10\pi[/math], and speed [math]-1.3\le U\le1.3[/math]. In this case, for convenience, we define the parameter[/justify][center][math]\gamma=\frac{\Gamma}{4\pi\cdot U\cdot a}[/math][/center]Observe what happens when you change the values of the parameter [math]\gamma[/math].
Describe the flow when [math]\gamma=0[/math].
When [math]\gamma=0[/math], there is no circulation and then we have a uniform flow past a circular cylinder, with stagnation points at [math]\left(a,0\right)[/math] and [math]\left(-a,0\right)[/math], with [math]a=1[/math].[br][br][br][br][br][br][br][br]
Describe the flow when [math]0<\gamma<1[/math].
As [math]\gamma[/math] increases, the circulation causes the stagnation points to move upwards around the cylinder.[br][br][br][br]
Describe the flow when [math]\gamma>1[/math].
If [math]\gamma>1[/math], then one stagnation point moves into the flow; the other one is inside the cylinder.[br][br][br][br]
Describe the flow when [math]\gamma=1[/math].
When [math]\gamma[/math] reaches the value 1, the two stagnation points coalesce at the top, or the bottom, of the cylinder [math]\left(0,a\right)[/math].[br][br][br][br]