Curl Operator/Examples/Motion of Fluid
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Example of Curl Operator
Consider an infinitesimal volume of fluid $F$.
It may have $3$ kinds of motion:
- $(1): \quad$ Moving with a linear velocity as a whole
- $(2): \quad$ Changing its shape
- $(3): \quad$ In rotary motion.
At any instant, $F$ may be regarded as a rigid body.
Hence from Curl of Rotation of Rigid Body, the curl of the velocity of $F$ is twice its angular velocity where its axis of rotation at that instant is the same as that of the curl.
Consider the diagram above.
On the left, the element $E_1$ has itself rotated in moving to ${E_1}'$.
If every element of the body of fluid has rotated the same amount, $\curl \mathbf V$ would be twice the angular velocity about $O$.
On the right, on the other hand, the element $E_2$ has not actually rotated in moving to ${E_2}'$.
Hence there is no $\curl \mathbf V$ and its angular velocity is zero.
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {IV}$: The Operator $\nabla$ and its Uses: $5$. Simple Examples of Curl