Definition 6.1.1. The circulation.
We define the circulation around a closed loop \(C\) in the flow as
\begin{equation*}
\Gamma = \oint_C \mathbf{u} \cdot \de\mathbf{x}.
\end{equation*}
Section 6.1 Vorticity and circulation
Subsection 6.1.1 Vorticity
The fluid vorticity \(\bomega=\nabla\times\bu\) was introduced in Definition 3.5.2. Many people struggle to understand the concept of vorticity when they first encounter it, so you should not worry if it takes a while to sink in. The excellent movie `Vorticity, Part 1’ by Ascher Shapiro available at this website gives a clear and comprehensive introduction to the phenomenon. Also, [1] gives a simple illustration of the vorticity. To recap:
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Vorticity, \(\bomega\text{,}\) is a vector field and it is a function of space and time (like the fluid velocity).
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At each point in the fluid, \(\bomega\) points in the direction of the axis of the local fluid rotation.
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The magnitude of \(\bomega\) is twice the local angular velocity.
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In the special case of a two-dimensional flow (that is \(\bu(\bx,t)=(u(x,y,t),v(x,y,t),0)\)), the vorticity is unidirectional, pointing in the \(z\)-direction with magnitude\begin{equation*} \pd{v}{x}-\pd{u}{y}\text{.} \end{equation*}
\begin{equation*}
\DD{\bomega}{t} = (\bomega \cdot \nabla)\bu.
\end{equation*}
In two-dimensional irrotational flow, the right-hand side is zero, so the vorticity of each fluid particle is constant in time. This is a special case of the more general result known as the Helmholtz principle, which states that in an inviscid fluid, the vorticity of each fluid particle is conserved.
Subsection 6.1.2 The circulation in a flow
We introduce a definition that was briefly covered in the line vortex example Example 4.2.10.
Theorem 6.1.2. Kelvin’s circulation theorem.
Assume that a fluid is inviscid and incompressible and that a conservative body force (e.g. gravity) acts on it. If a loop \(C\) is fixed in the fluid (and thus moves around in the Lagrangian sense as the fluid moves) then the circulation around \(C\) is constant in time.
Proof of Kelvin’s circulation theorem.
A proof of this theorem is given in Exercise 6.5.1.
Remark 6.1.3. Importance of Kelvin’s circulation theorem.
This theorem explains why the circulation is an important physical concept. In particular, if a flow is initally irrotational, then the circulation around any loop is zero at \(t=0\text{.}\) The theorem then implies that the circulation around any loop is zero for all time. Therefore the flow at any time is irrotational.
Remark 6.1.4. Relationship between circulation and vorticity.
Using Stokes’ theorem
\begin{equation}
\Gamma=\oint_C{\bu}\cdot \de{\bx}=\int_S\left(\bnabla\times{\bu}\right)\cdot{\bn}\,{\de}S
=\int_S\bomega\cdot{\bn}\,{\de}S,\tag{6.1.1}
\end{equation}
where \(S\) is any surface spanning the loop \(C\text{.}\) Thus the circulation equals the flux of vorticity through the spanning surface.
Remark 6.1.5. Independence of spanning surface.
Note that, for a given closed curve \(C\text{,}\) in the argument given in Remark 6.1.4 we could choose any surface \(S\) spanning \(C\text{,}\) and thus the circulation equals the vorticity flux through any spanning surface.
Subsection 6.1.3 Vortex lines and surfaces
Definition 6.1.6. Vortex lines.
Vortex lines are curves that point in the same direction as the vorticity vector. They satisfy the differential equation
\begin{equation*}
\dd{\bx}{s}=\bomega(\bx(s),t),
\end{equation*}
where \(t\) is fixed. Just as the streamlines give a picture of the flow field, so the vortex lines give a picture of the vorticity field. Note that if the vorticity is zero at some point, the vortex lines are not defined there.
Remark 6.1.7. Analogy with streamlines.
The vortex lines are to the vorticity of the fluid as the streamlines are to its velocity.
Remark 6.1.8. Vortex lines in a two-dimensional flow.
In a two-dimensional flow, the vorticity is unidirectional, so the vortex lines are straight lines parallel to the vorticity vector. For example, for a flow in the \((x,y)\)-plane, the vorticity points in the \(z\)-direction, and the vortex lines are straight lines parallel to the \(z\)-axis.
Definition 6.1.9. Vortex surface.
A vortex surface is a surface such that \(\bomega\) is tangent to the surface at all points on the surface.
Remark 6.1.10. Notes on vortex surfaces.
A couple of important points about vortex surfaces:
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Vortex lines cannot pass through a vortex surface.
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In particular, for any given vortex line, it is possible to choose two different vortex surfaces such that the vortex line is the intersection of the two vortex surfaces. This will be important in the next section.
