Curl Operator/Examples/Motion of Fluid

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