Circulation
We saw at the beginning of the last section that if we have an incompressible fluid with no circulation, the flow satisfies the following two equations: \begin{equation} \label{Eq:II:40:19} \FLPdiv{\FLPv}=0,\quad \FLPcurl{\FLPv}=\FLPzero. \end{equation} They are the same as the equations of electrostatics or magnetostatics in empty space. The divergence of the electric field is zero when there are no charges, and the curl of the electrostatic field is always zero. The curl of the magnetostatic field is zero if there are no currents, and the divergence of the magnetic field is always zero. Therefore, Eqs. (40.19) have the same solutions as the equations for $\FLPE$ in electrostatics or for $\FLPB$ in magnetostatics. As a matter of fact, we have already solved the problem of the flow of a fluid past a sphere, as an electrostatic analogy, in Section 12-5. The electrostatic analog is a uniform electric field plus a dipole field. The dipole field is so adjusted that the flow velocity normal to the surface of the sphere is zero. The same problem for the flow past a cylinder can be worked out in a similar way by using a suitable line dipole with a uniform flow field. This solution holds for a situation in which the fluid velocity at large distances is constant—both in magnitude and direction. The solution is sketched in Fig. 40-11(a).  There is another solution for the flow around a cylinder when the conditions are such that the fluid at large distances moves in circles around the cylinder. The flow is, then, circular everywhere, as in Fig. 40-11(b). Such a flow has a circulation around the cylinder, although $\FLPcurl{\FLPv}$ is still zero in the fluid. How can there be circulation without a curl? We have a circulation around the cylinder because the line integral of $\FLPv$ around any loop enclosing the cylinder is not zero. At the same time, the line integral of $\FLPv$ around any closed path which does not include the cylinder is zero. We saw the same thing when we found the magnetic field around a wire. The curl of $\FLPB$ was zero outside of the wire, although a line integral of $\FLPB$ around a path which encloses the wire did not vanish. The velocity field in an irrotational circulation around a cylinder is precisely the same as the magnetic field around a wire. For a circular path with its center at the center of the cylinder, the line integral of the velocity is \begin{equation*} \oint\FLPv\cdot d\FLPs=2\pi rv. \end{equation*} For irrotational flow the integral must be independent of $r$. Let’s call the constant value $C$, then we have that \begin{equation} \label{Eq:II:40:20} v=\frac{C}{2\pi r}, \end{equation} where $v$ is the tangential velocity, and $r$ is the distance from the axis.  There is a nice demonstration of a fluid circulating around a hole. You take a transparent cylindrical tank with a drain hole in the center of the bottom. You fill it with water, stir up some circulation with a stick, and pull the drain plug. You get the pretty effect shown in Fig. 40-12. (You’ve seen a similar thing many times in the bathtub!) Although you put in some $\omega$ at beginning, it soon dies down because of viscosity and the flow becomes irrotational—although still with some circulation around the hole.  From the theory, we can calculate the shape of the inner surface of the water. As a particle of the water moves inward it picks up speed. From Eq. (40.20) the tangential velocity goes as $1/r$—it’s just from the conservation of angular momentum, like the skater pulling in her arms. Also the radial velocity goes as $1/r$. Ignoring the tangential motion, we have water going radially inward toward a hole; from $\FLPdiv{\FLPv}=0$, it follows that the radial velocity is proportional to $1/r$. So the total velocity also increases as $1/r$, and the water goes in along equiangular (or "logarithmic") spirals. The air-water surface is all at atmospheric pressure, so it must have—from Eq. (40.14)—the property that \begin{equation*} gz+\tfrac{1}{2}v^2=\text{const}. \end{equation*} But $v$ is proportional to $1/r$, so the shape of the surface is \begin{equation*} (z-z_0)=\frac{k}{r^2}. \end{equation*}  An interesting point—which is not true in general but is true for incompressible, irrotational flow—is that if we have one solution and a second solution, then the sum is also a solution. This is true because the equations in (40.19) are linear. The complete equations of hydrodynamics, Eqs. (40.9), (40.10), and (40.11), are not linear, which makes a vast difference. For the irrotational flow about the cylinder, however, we can superpose the flow of Fig. 40-11(a) on the flow of Fig. 40-11(b) and get the new flow pattern shown in Fig. 40-11(c). This flow is of special interest. The flow velocity is higher on the upper side of the cylinder than on the lower side. The pressures are therefore lower on the upper side than on the lower side. So when we have a combination of a circulation around a cylinder and a net horizontal flow, there is a net vertical force on the cylinder—it is called a lift force. Of course, if there is no circulation, there is no net force on any body according to our theory of “dry” water. 