Two-Dimensional Sources and Sinks

Consider a uniform line source, coincident with the $z$-axis, that emits fluid isotropically at the steady rate of $Q$ unit volumes per unit length per unit time. By symmetry, we expect the associated steady flow pattern to be isotropic, and everywhere directed radially away from the source. See Figure 25. In other words, we expect

$${\bf v} = v_r(r)\,{\bf e}_r.$$ (487)

Consider a cylindrical surface $S$ of unit height (in the $z$-direction) and radius $r$ that is co-axial with the source. In a steady state, the rate at which fluid crosses this surface must be equal to the rate at which the section of the source enclosed by the surface emits fluid. Hence,

$$\int_S {\bf v}\cdot d{\bf S} = 2\pi\,r\,v_r(r)= Q,$$ (488)

which implies that

$$v_r(r) = \frac{Q}{2\pi\,r}.$$ (489)

Figure 25: Streamlines of the flow generated by a line source coincident with the $z$-axis.

According to Equations (480) and (481), the stream function associated with a line source of strength $Q$ that is coincident with the $z$-axis is

$$\psi(r,\theta) = -\frac{Q}{2\pi}\,\theta.$$ (490)

Note that the streamlines, $\psi=c$, are directed radially away from the $z$-axis, as illustrated in Figure 25. Note, also, that the stream function associated with a line source is multivalued. However, this does not cause any particular difficulty, since the stream function is continuous, and its gradient single-valued.

Note, from Equation (490), that $\partial\psi/\partial r=\partial^2\psi/\partial\theta^{\,2}=0$. Hence, according to (482), $\omega_z=-\nabla^2\psi=0$. In other words, the steady flow pattern associated with a uniform line source is irrotational, and can, thus, be derived from a velocity potential. In fact, it is easily demonstrated that this potential takes the form

$$\phi(r,\theta) = -\frac{Q}{2\pi}\,\ln r.$$ (491)

A uniform line sink, coincident with the $z$-axis, which absorbs fluid isotropically at the steady rate of $Q$ unit volumes per unit length per unit time has an associated steady flow pattern

$${\bf v} = - \frac{Q}{2\pi\,r}\,{\bf e}_r,$$ (492)

whose stream function is

$$\psi(r,\theta) = \frac{Q}{2\pi}\,\theta.$$ (493)

This flow pattern is also irrotational, and can be derived from the velocity potential

$$\phi(r,\theta) = \frac{Q}{2\pi}\,\ln r.$$ (494)

Consider a line source and a line sink of equal strength, which both run parallel to the $z$-axis, and are located a small distance apart in the $x$-$y$ plane. Such an arrangement is known as a dipole or doublet line source. Suppose that the line source, which is of strength $Q$, is located at ${\bf r}={\bf d}/2$ (where ${\bf r}$ is a position vector in the $x$-$y$ plane), and that the line sink, which is also of strength $Q$, is located at ${\bf r}=-{\bf d}/2$. Let the function

$$\psi_Q({\bf r})= -\frac{Q}{2\pi}\,\theta =- \frac{Q}{2\pi}\,\tan^{-1}(y/x)$$ (495)

be the stream function associated with a line source of strength $Q$ located at ${\bf r}={\bf0}$. Thus, $\psi_Q({\bf r}-{\bf r}_0)$ is the stream function associated with a line source of strength $Q$ located at ${\bf r}={\bf r}_0$. Furthermore, the stream function associated with a line sink of strength $Q$ located at ${\bf r}={\bf r}_0$ is $-\psi_Q({\bf r}-{\bf r}_0)$. Now, we expect the flow pattern associated with the combination of a source and a sink to be the vector sum of the flow patterns generated by the source and sink taken in isolation. It follows that the overall stream function is the sum of the stream functions generated by the source and the sink taken in isolation. In other words,

$$\psi({\bf r}) = \psi_Q({\bf r}-{\bf d}/2)-\psi_Q({\bf r}+{\bf d}/2)\simeq - {\bf d}\cdot\nabla\psi_Q({\bf r}),$$ (496)

to first order in $d/r$. Hence, if ${\bf d} = d\,(\cos\theta_0\,{\bf e}_x+ \sin\theta_0\,{\bf e}_y)=d\,[\cos(\theta-\theta_0)\,{\bf e}_r -\sin(\theta-\theta_0)\,{\bf e}_\theta]$, so that the line joining the sink to the source subtends a (counter-clockwise) angle $\theta_0$ with the $x$-axis, then

$$\psi(r,\theta) = -\frac{D}{2\pi}\,\frac{\sin(\theta-\theta_0)}{r},$$ (497)

where $D=Q\,d$ is termed the strength of the dipole source. Note that the above stream function is antisymmetric across the line $\theta=\theta_0$ joining the source to the sink. It follows that the associated dipole flow pattern,

$$v_r(r,\theta) = \frac{D}{2\pi}\,\frac{\cos(\theta-\theta_0)}{r^{\,2}},$$ (498)

$$v_\theta(r,\theta) = \frac{D}{2\pi}\,\frac{\sin(\theta-\theta_0)}{r^{\,2}},$$ (499)

is symmetric across this line. Figure 26 shows the streamlines associated with a dipole flow pattern characterized by $D>0$ and $\theta_0=0$. Note that the flow speed in a dipole pattern falls off like $1/r^{\,2}$.

Figure 26: Streamlines of the flow generated by a dipole line source coincident with the $z$-axis and aligned along the $x$-axis. The flow is outward along the positive $x$-axis and inward along the negative $x$-axis.

A dipole flow pattern is necessarily irrotational since it is a linear superposition of two irrotational flow patterns. The associated velocity potential is

$$\phi(r,\theta) = \frac{D}{2\pi}\,\frac{\cos(\theta-\theta_0)}{r}.$$ (500)