Two-Dimensional Irrotational Flow in Cylindrical Coordinates

In a two-dimensional flow pattern, we can automatically satisfy the incompressibility constraint, $\nabla\cdot{\bf v}=0$ , by expressing the pattern in terms of a stream function. Suppose, however, that, in addition to being incompressible, the flow pattern is also irrotational. In this case, Equation (5.10) yields

$$\nabla^{\,2}\psi = 0.$$ (5.60)

In cylindrical coordinates, because $\psi=\psi(r,\theta,t)$ , this expression implies that (see Section C.3)

$$\frac{1}{r}\,\frac{\partial}{\partial r}\!\left(r\,\frac{\partial... ...right) + \frac{1}{r^{\,2}}\,\frac{\partial^{\,2}\psi}{\partial\theta^{\,2}} =0.$$ (5.61)

Let us search for a separable solution of Equation (5.61) of the form

$$\psi(r,\theta) = R(r)\,{\mit\Theta}(\theta).$$ (5.62)

It is easily seen that

$$\frac{r}{R}\,\frac{d}{dr}\!\left(r\,\frac{dR}{dr}\right)= - \frac{1}{{\mit\Theta}}\,\frac{d^{\,2}{\mit\Theta}}{d\theta^{\,2}},$$ (5.63)

which can only be satisfied if

$$r\,\frac{d}{dr}\!\left(r\,\frac{dR}{dr}\right) =m^{\,2}\,R,$$ (5.64)

$$\frac{d^{\,2}{\mit\Theta}}{d\theta^{\,2}} = -m^{\,2}\,{\mit\Theta},$$ (5.65)

where $m^{\,2}$ is an arbitrary (positive) constant. The general solution of Equation (5.65) is a linear combination of $\exp(\,{\rm i}\,m\,\theta)$ and $\exp(-{\rm i}\,m\,\theta)$ factors. However, assuming that the flow extends over all $\theta$ values, the function ${\mit\Theta}(\theta)$ must be single-valued in $\theta$ , otherwise $\nabla\psi$ --and, hence, ${\bf v}$ --would not be be single-valued (which is unphysical). It follows that $m$ can only take integer values (and that $m^{\,2}$ must be a positive, rather than a negative, constant). The general solution of Equation (5.64) is a linear combination of $r^{\,m}$ and $r^{-m}$ factors, except for the special case $m=0$ , when it is a linear combination of $r^{\,0}$ and $\ln r$ factors. Thus, the general stream function for steady two-dimensional irrotational flow (that extends over all values of $\theta$ ) takes the form

$$\psi(r,\theta) = \alpha_0+ \beta_0\,\ln r + \sum_{m>0} (\alpha_m\,r^{\,m}+\beta_m\,r^{-m})\, \sin[m\,(\theta-\theta_m)],$$ (5.66)

where $\alpha_m$ , $\beta_m$ , and $\theta_m$ are arbitrary constants. We can recognize the first few terms on the right-hand side of the previous expression. The constant term $\alpha _0$ has zero gradient, and, therefore, does not give rise to any flow. The term $\beta_0\,\ln r$ is the flow pattern generated by a vortex filament of intensity $2\pi\,\beta_0$ , coincident with the $z$ -axis. (See Section 5.6.) The term $\alpha_1\,r\,\sin(\theta-\theta_1)$ corresponds to uniform flow of speed $\alpha_1$ whose direction subtends a (counter-clockwise) angle $\theta _1$ with the minus $x$ -axis. (See Section 5.4.) Finally, the term $\beta_1\,\sin(\theta-\theta_1)/r$ corresponds to a dipole flow pattern. (See Section 5.5.)

The velocity potential associated with the irrotational stream function (5.66) satisfies [see Equations (4.89) and (5.7)]

$$\frac{\partial\phi}{\partial r} =\frac{1}{r}\,\frac{\partial\psi}{\partial\theta},$$ (5.67)

$$\frac{1}{r}\,\frac{\partial\phi}{\partial\theta} =-\frac{\partial\psi}{\partial r}.$$ (5.68)

It follows that

$$\phi(r,\theta) = \alpha_0-\beta_0\,\theta+\sum_{m>0}(\alpha_m\,r^{\,m}-\beta_m\,r^{-m})\,\cos[m\,(\theta-\theta_0)].$$ (5.69)

Figure: Streamlines of the flow generated by a cylindrical obstacle of radius $a$ , whose axis runs along the $z$ -axis, placed in the uniform flow field ${\bf v}= V_0\,{\rm e}_x$ . The normalized circulation is $\gamma =0$ .