Short Answer:

Vortex motion is a type of fluid motion in which the fluid particles move in circular paths around a fixed or moving axis. In this motion, every particle has some angular velocity, and the flow shows a rotational nature. Vortex motion commonly occurs in whirlpools, tornadoes, and water flowing down a drain. It is characterized by the spinning of the fluid about an axis due to the presence of circulation and vorticity.

In simple terms, vortex motion is the swirling motion of a fluid where velocity increases near the center and decreases toward the outer edge. This motion may be free or forced depending on how it is generated. Free vortex motion happens naturally due to pressure differences, while forced vortex motion occurs when an external force, such as a stirrer or rotating cylinder, is applied.

Detailed Explanation :

Vortex Motion

Vortex motion is a common and important concept in fluid mechanics that describes the circular motion of fluid particles around an imaginary or real axis. When a fluid rotates about an axis in such a way that every particle has some angular velocity, the motion is said to be vortex motion. This kind of motion can be observed in nature, such as in whirlpools, cyclones, tornadoes, and even in engineering devices like turbines and pumps.

Definition

A vortex is defined as the region in a fluid where the flow revolves around an axis line, which may be straight or curved. The motion in which the fluid elements move in concentric circular paths about this axis is called vortex motion. In this motion, the fluid possesses both tangential and sometimes radial velocity components.

Types of Vortex Motion

There are mainly two types of vortex motion, depending on how the motion is produced and maintained:

1. Free Vortex Motion
2. Forced Vortex Motion

1. Free Vortex Motion

In free vortex motion, no external torque or force is applied to the fluid after it starts moving. The motion is maintained purely by the conservation of angular momentum. In this case, the tangential velocity of the fluid varies inversely with the radial distance from the center, i.e.,

or

This means that the product of tangential velocity and radius remains constant throughout the motion. As a result, the velocity increases near the center and decreases away from it. An example of free vortex motion is the flow of water in a bathtub drain or a whirlpool in a river.

Characteristics of Free Vortex Motion:

• No external torque is applied.
• Flow is irrotational outside the vortex core.
• Pressure decreases towards the center.
• Angular momentum is conserved.

Examples:

• Whirlpools in rivers.
• Water draining through a hole.
• Atmospheric cyclones.

2. Forced Vortex Motion

In forced vortex motion, the fluid is made to rotate by the application of an external torque or force. Here, every particle of the fluid rotates with the same angular velocity as a solid body, and the tangential velocity varies directly with the radius, i.e.,

or

where  is the angular velocity of the fluid.

This motion is similar to the motion of a solid disk rotating about its axis. The motion stops when the external torque is removed. Examples include a rotating cylinder of liquid or a stirred cup of coffee.

Characteristics of Forced Vortex Motion:

• External torque is applied to maintain motion.
• Flow is rotational and behaves like a solid body.
• Pressure increases from the center outward.
• No slip occurs between adjacent fluid layers.

Examples:

• Rotating liquid in a centrifuge.
• Fluid in a rotating cup or vessel.
• Liquid in a washing machine drum.

Mathematical Representation

For a vortex motion, consider a fluid particle moving along a circular path of radius  with tangential velocity . The centrifugal force per unit mass acting on the particle is given by

In free vortex motion, substituting ,

This shows that the centrifugal force increases sharply near the center, leading to a decrease in pressure there.

In forced vortex motion, where ,

This force increases with radius, causing higher pressure at the outer boundary.

Comparison Between Free and Forced Vortex Motion

Property	Free Vortex	Forced Vortex
External torque	Not required	Required
Nature of motion	Irrotational (except at core)	Rotational
Velocity relation
Pressure variation	Decreases toward center	Increases toward center
Example	Water whirlpool	Rotating tank of fluid

(Note: Description here is textual for understanding, not tabular format in structure.)

Applications of Vortex Motion

• Centrifugal separators use forced vortex motion to separate solid particles from liquids.
• Vortex flow meters use vortex shedding to measure flow rates.
• Turbines and pumps involve vortex motion for energy conversion.
• Natural systems such as cyclones and tornadoes are examples of large-scale vortex motion.

Conclusion

Vortex motion is a fundamental phenomenon in fluid mechanics representing the circular movement of fluids around an axis. It is classified mainly into free and forced vortex motion based on whether external energy is applied. Free vortex occurs naturally due to angular momentum conservation, while forced vortex needs continuous external torque. Understanding vortex motion helps in analyzing natural flows like whirlpools and engineered systems like turbines and centrifuges. It plays a vital role in both theoretical and practical aspects of fluid dynamics.