Short Answer:

A free vortex is a type of fluid motion in which the fluid rotates without any external torque or force. In this motion, the angular momentum of each fluid particle remains constant, and the velocity of the fluid varies inversely with the distance from the center. This means that the closer the particle is to the center, the faster it moves. A typical example of a free vortex is the swirling motion of water going down a drain or whirlpool.

In a free vortex, the rotation is caused by the natural flow of the fluid, and not by any external mechanical force. The pressure in a free vortex decreases towards the center, creating a depression that forms a funnel-shaped surface. The fluid motion continues due to conservation of angular momentum, making it an essential concept in fluid mechanics and hydraulic engineering.

Detailed Explanation :

Free Vortex

A free vortex is one of the most common forms of rotational flow observed in fluids. It occurs when a fluid mass rotates about a vertical axis due to its own motion, without any external torque or mechanical device causing the rotation. This phenomenon can be seen naturally in many situations such as whirlpools, tornadoes, and draining water. The free vortex is a type of irrotational flow, meaning that each fluid particle has no net rotation about its own axis even though it moves in a circular path.

Definition

A free vortex is defined as the motion of a fluid in which no external torque is applied, and the rotation occurs purely due to the conservation of angular momentum. In a free vortex, the velocity of the fluid varies inversely with its distance from the axis of rotation.

Mathematically,

where,
= tangential velocity of the fluid,
= radial distance from the axis of rotation.

This equation shows that as the distance  decreases, the tangential velocity  increases, and vice versa. This inverse relationship is the main feature of a free vortex flow.

Nature of Motion

The motion in a free vortex is irrotational, which means that although the fluid particles move in circular paths, they do not spin around their own axes. Each fluid particle follows a curved path due to the centrifugal and pressure forces acting on it. The motion is governed by the principle of conservation of angular momentum since no external torque is applied.

In simpler terms, the energy of rotation in a free vortex comes from the internal energy of the fluid, not from any mechanical device like a stirrer or impeller. Once the rotation begins, it continues as long as energy is conserved within the system.

Formation of a Free Vortex

A free vortex is formed when a rotating fluid system is suddenly released from an external force or allowed to flow freely. For example:

• When water in a container is stirred and then the stirrer is removed, the fluid continues to rotate for some time — this is a free vortex.
• The swirling motion of water as it drains from a bathtub or sink forms a free vortex due to gravity and conservation of angular momentum.

Thus, free vortices commonly occur in nature and engineering systems wherever a fluid moves freely under its own energy or gravity.

Velocity Distribution

In a free vortex, the tangential velocity  of the fluid is inversely proportional to the distance  from the center.

where  is a constant representing the angular momentum per unit mass.

This relationship means that as a fluid particle approaches the center of rotation (smaller ), its velocity becomes very high. On the other hand, particles far from the center move slowly. This inverse velocity profile is a major difference between free vortex and forced vortex flows.

Pressure Distribution

In a free vortex, the pressure distribution can be obtained by applying Bernoulli’s equation between two points in the same streamline. Considering no external torque and assuming steady, incompressible, and frictionless flow:

Substituting , we can express the pressure difference between two points as:

From this equation, it is clear that pressure decreases as the radius decreases. Therefore, the pressure is lowest at the center of the vortex. This pressure variation causes the characteristic funnel-shaped free surface in free vortices.

Free Surface Shape

The shape of the free surface in a free vortex can be derived using Bernoulli’s equation. The equation for the height  of the liquid surface above a reference level is given by:

This equation shows that the free surface of a rotating liquid forms a curved funnel that is deeper at the center and higher at the sides. This depression at the center is due to the low pressure region caused by high velocity.

Characteristics of Free Vortex

• No external torque is applied to maintain the motion.
• The flow is irrotational, and each particle has no spin about its own axis.
• Tangential velocity  is inversely proportional to radius .
• Pressure decreases towards the center.
• The free surface forms a funnel-shaped depression at the center.
• Motion continues until energy is lost due to viscosity.

Examples of Free Vortex

1. Water swirling as it drains from a sink or bathtub.
2. Whirlpools in rivers or oceans.
3. Flow of air in a tornado or cyclone.
4. Flow of liquid in a centrifugal pump casing when the impeller stops.
5. Vortex motion behind ship propellers.

Applications

1. Hydraulic design: Used in designing vortex chambers and spillways to control water flow.
2. Turbine and pump systems: Helps in analyzing flow patterns when the external force is removed.
3. Meteorology: Useful in studying atmospheric vortex phenomena like cyclones and tornadoes.
4. Mixing and separation: Applied in sedimentation and separation processes based on centrifugal effects.

Conclusion

A free vortex is a natural rotational motion of a fluid where no external torque acts, and the motion is maintained by conservation of angular momentum. The tangential velocity decreases with increasing radius, while the pressure decreases toward the center, forming a characteristic funnel-shaped surface. Free vortices are observed in natural systems such as whirlpools and tornadoes and have practical importance in hydraulic and mechanical engineering systems. Understanding the behavior of free vortices helps engineers design efficient fluid flow systems and predict natural vortex phenomena.