Question 4.8: Investigate the stream function in polar coordinates ψ = U s...

The streamlines are lines of constant ψ, which has units of square meters per second. Note that ψ/(UR) is dimensionless. Rewrite Eq. (1) in dimensionless form
\frac{\psi}{UR}=\sin \theta \left(\eta -\frac{1}{\eta}\right)                               \eta=\frac{r}{R}                                   (2)
Of particular interest is the special line ψ = 0. From Eq. (1) or (2) this occurs when (a) θ = 0 or 180° and (b) r = R. Case (a) is the x axis, and case (b) is a circle of radius R, both of which are plotted in Fig. E4.8.
For any other nonzero value of ψ it is easiest to pick a value of r and solve for θ:
\sin \theta=\frac{\psi /(UR)}{r/R-R/r}                                 (3)
In general, there will be two solutions for θ because of the symmetry about the y axis. For example, take ψ/(UR) = +1.0:

This line is plotted in Fig. E4.8 and passes over the circle r = R. Be careful, though, because there is a second curve for ψ/(UR) = +1.0 for small r < R below the x axis:

This second curve plots as a closed curve inside the circle r = R. There is a singularity of infinite velocity and indeterminate flow direction at the origin. Figure E4.8 shows the full pattern.
The given stream function, Eq. (1), is an exact and classic solution to the momentum equation (4.38) for frictionless flow. Outside the circle r = R it represents two-dimensional inviscid flow of a uniform stream past a circular cylinder (Sec. 8.4). Inside the circle it represents a rather unrealistic trapped circulating motion of what is called a line doublet.

\rho g_z-\frac{\partial p}{\partial z}+\mu \left(\frac{\partial^2 w}{\partial x^2}+\frac{\partial^2 w}{\partial y^2}+\frac{\partial^2 w}{\partial z^2}\right)=\rho \frac{dw}{dt}                              (4.38)