26. Stokes' Theorem

Let \(S\) be a nice surface in \(\mathbb{R}^3\) with a nice properly oriented boundary, \(\partial S\), and let \(\vec{F}\) be a nice vector field on \(S\). Then \[ \iint_S \vec{\nabla}\times\vec{F}\cdot d\vec{S} =\oint_{\partial S} \vec{F}\cdot d\vec{s} \] Each piece of the boundary of the surface must be traversed counterclockwise as seen from the tip of the normal vector to the surface.

d. Applications

5. Circulation and Vorticity of a Fluid

Circulation

In fluid dynamics, the circulation of a fluid around a closed curve \(C\) is the line integral of the fluid velocity field, \(\vec{V}\), around the curve. It is a scalar quantity that can be thought of as the net amount of fluid flowing around the closed loop. It is defined mathematically as \[ \text{Circulation}=\oint_C \vec{V}\cdot d\vec{s} \] where \(d\vec{s}=\vec{v}\,dt\) is the vector differential of arc length along the curve \(C\).

Be careful not to get confused between \(\vec{V}\) which is the velocity of the fluid and \(\vec{v}\) which is the velocity or tangent vector of the parametric curve \(\vec{r}(t)\).

You always need to specify the orientation of the curve, i.e. the direction it is traversed. If the circulation comes out positive, then the fluid is flowing in the direction of the orientation of the curve; if negative, then the fluid is flowing opposite to the orientation of the curve. If you reverse the orientation of the curve, the sign of the circulation will change.

The air inside an oven is flowing with velocity \(\vec{V}=\langle-2y,xz,yz\rangle\). Find the circulation of the air around the circle \(x^2+y^2=4\) at \(z=2\) traversed counterclockwise as seen from the positive \(z\)-axis.

The first step is to parameterize the circle. \[ \vec{r}(\theta) =\langle 2\cos\theta,2\sin\theta,2\rangle \qquad \text{for} \quad 0 \lt \theta \lt 2\pi \] This is counterclockwise; so the orientation is correct. From this, we find the tangent vector (or velocity) \(\vec{v}\): \[ \vec{v}=\langle-2\sin\theta,2\cos\theta,0\rangle \] Next we evaluate the fluid velocity on the curve and its dot product with the tangent vector of the curve: \[ \left.\vec{V}\right|_{\vec{r}(\theta)}=\langle-2y,xz,yz\rangle =\langle-4\sin\theta,4\cos\theta,4\sin\theta\rangle \] \[\begin{aligned} \vec{V}\cdot\vec{v} &=\langle-4\sin\theta,4\cos\theta,4\sin\theta\rangle \cdot\langle-2\sin\theta,2\cos\theta,0\rangle \\ &=8\sin^2\theta+8\cos^2\theta+0=8 \end{aligned}\] Thus the circulation is: \[ \text{Circulation}=\oint_C \vec{V}\cdot d\vec{s} =\int_0^{2\pi} \vec{V}\cdot\vec{v}\,d\theta =\int_0^{2\pi} 8\,d\theta=16\pi \]

Vorticity of a Fluid

The vorticity of a fluid is the vector field that describes the local rotational motion of a fluid at each point. In other words, vorticity describes the spinning of the fluid near the point. Mathematically it is defined as the curl of the fluid velocity vector: \[ \vec{W}=\vec{\nabla}\times\vec{V} \] where \(\vec{W}\) is the vorticity and \(\vec{V}\) is the fluid velocity vector. The direction of the vorticity is the axis of the rotation and the magnitude of the vorticity gives the rate of rotation about that axis.

Vorticity is a useful concept that helps us understand and model various phenomena such as tornados, smoke rings, blowing out candles, the lift on helicopters and airplanes, and even vortex rings that are formed by blood flow in the human heart.

The air inside an oven is flowing with velocity \(\vec{V} =\langle-2y,xz,yz\rangle\). Find the vorticity of the gas.

The vorticity is \[\begin{aligned} \vec{W}&=\vec{\nabla}\times\vec{V}= \begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ \partial_x & \partial_y & \partial_z \\ -2y & xz & yz \end{vmatrix} \\ &=\hat{\imath}(z-x)-\hat{\jmath}(0)+\hat{k}(z+2) =\langle z-x,0,z+2\rangle \end{aligned}\]

Relation by Stokes' Theorem

Stokes' Theorem can be used to relate the circulation to the vorticity of a fluid. If the closed path \(C\) is the boundary curve of a surface \(S\) then the circulation of the fluid around \(C\) is also the flux of the vorticity through the surface \(S\). To see this, we apply Stokes' Theorem to the formula for the circulation to change the line integral into a surface integral and then use the definition of vorticity: \[ \text{Circulation}=\oint_C \vec{V}\cdot d\vec{s} =\oint_{\partial S} \vec{V}\cdot d\vec{s} =\iint_S \vec{\nabla}\times\vec{V}\cdot d\vec{S} =\iint_S \vec{W}\cdot d\vec{S} \] Thus, the circulation around a closed curve, \(C\), is the flux of vorticity through any the surface, \(S\), whose boundary is \(C\). Consequently, the vorticity is interpreted as the circulation per unit area.

The air inside an oven is flowing with velocity \(\vec{V}=\langle-2y,xz,yz\rangle\). Recompute the circulation of the air counterclockwise around the circle \(x^2+y^2=4\)at \(z=2\) but this time using its vorticity.

We parameterize the disk as \[ \vec{R}(r,\theta)=(r\cos\theta,r\sin\theta,2) \text{ with }0 \le r \le 2\text{ and }0 \lt \theta \lt 2\pi \] The normal is \[\begin{aligned} \vec{N}&=\vec{e}_r\times\vec{e}_\theta= \begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ \cos\theta & \sin\theta & 0 \\ -r\sin\theta & r\cos\theta & 0 \end{vmatrix} \\ &=\hat{\imath}(0)-\hat{\jmath}(0)+\hat{k}(r\cos^2\theta+r\sin^2\theta) =\langle 0,0,r\rangle \end{aligned}\] The vorticity (previously found in Example 2) restricted to the surface is: \[ \vec{W}=\vec{\nabla}\times\vec{V} =\langle z-x,0,z+2\rangle =\langle 2-r\cos\theta,0,4\rangle \] So finally, the circulation is: \[\begin{aligned} \text{Circulation} &=\iint \vec{W}\cdot d\vec{S} =\int_0^{2\pi}\int_0^2 \langle2-r\cos\theta,0,4\rangle \cdot\langle0,0,r\rangle\,dr\,d\theta \\ &=2\pi\int_0^2 4r\,dr =2\pi\left[\rule{0pt}{10pt}2r^2\right]_0^2=16\pi \end{aligned}\]

In a hot tub, jets are pumping water out so that it is flowing around the circular tub with velocity vector \(\vec{V}=\langle-y,x,0\rangle\). Find the circulation of the water along the horizontal circle of radius \(5\) at height \(z=1\) in two different ways.

Calculate the circulation as the line integral of the velocity vector.

\(\displaystyle \text{Circulation}=\oint_C \vec{V}\cdot d\vec{s}=50\pi \)

The circle is parameterized as \(\vec{r}(\theta) =(5\cos\theta,5\sin\theta,1)\) with bounds of \(0 \lt \theta \lt 2\pi\). So the differential of arc length is \[ d\vec{s}=\langle-5\sin\theta,5\cos\theta,0\rangle\,d\theta \] Next we evaluate the water's velocity on the circle as \[ \left.\vec{V}\right|_{\vec{r}(\theta)}=\langle-y,x,0\rangle =\langle-5\sin\theta,5\cos\theta,0\rangle \] Then we compute the line integral: \[\begin{aligned} &\text{Circulation} =\oint_C \vec{V}\cdot d\vec{s} \\ &\quad=\int \langle-5\sin\theta,5\cos\theta,0\rangle \cdot\langle-5\sin\theta,5\cos\theta,0\rangle\,d\theta \\ &\quad=\int_0^{2\pi} (25\sin^2\theta+25\cos^2\theta)\,d\theta =25\int_0^{2\pi} d\theta=50\pi \end{aligned}\]
Calculate the circulation as the flux of the vorticity.

\(\vec{W}=\vec{\nabla}\times\vec{V}=\langle 0,0,2\rangle\)
\(\displaystyle \text{Circulation}=\iint_S \vec{W}\cdot d\vec{S} =50\pi \)

We parametrize the disk as \(\vec{R}(r,\theta)=(r\cos\theta,r\sin\theta,1)\) where \(0 \lt r \lt 5\) and \(0 \lt \theta \lt 2\pi\). Then the normal is \[\begin{aligned} \vec{N}&=\vec{e}_r\times\vec{e}_\theta= \begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ \cos\theta & \sin\theta & 0 \\ -r\sin\theta & r\cos\theta & 0 \end{vmatrix} \\ &=\langle 0,0,r\rangle \end{aligned}\] Next, the vorticity is the curl of the velocity field: \[\begin{aligned} \vec{W}&=\vec{\nabla}\times\vec{V}= \begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ \partial_x & \partial_y & \partial_z \\ -y & x & 0 \end{vmatrix} \\ &=\langle 0,0,2\rangle \end{aligned}\] Finally, we compute its flux to find the circulation: \[\begin{aligned} \text{Circulation} &=\iint_S \vec{W}\cdot d\vec{S} =\iint \vec{W}\cdot\vec{N}\,dr\,d\theta \\ &=\int_0^{2\pi}\int_0^5 \langle0,0,2\rangle\cdot\langle 0,0,r\rangle\,dr\,d\theta \\ &=\int_0^{2\pi}\int_0^5 2r\,dr\,d\theta =2\pi\left[\rule{0pt}{10pt}r^2\right]_0^5=50\pi \end{aligned}\]