Short Answer:

A forced vortex is a type of fluid motion in which the fluid is made to rotate by applying an external torque or force. In this motion, all fluid particles move in circular paths around a common axis with the same angular velocity, just like a solid body. The velocity of the fluid increases linearly with the distance from the center, and pressure rises outward due to the centrifugal force created by rotation.

In simple terms, a forced vortex occurs when an external agency, such as a rotating container or impeller, forces the fluid to rotate. Examples include the motion of water in a rotating cylindrical tank, liquid in a washing machine drum, or a stirred cup of coffee. When the external torque is removed, the forced vortex motion gradually stops due to fluid resistance.

Detailed Explanation :

Forced Vortex

A forced vortex is an important concept in fluid mechanics where a rotating motion is created in a fluid by the application of external torque. The motion continues only as long as the external force or energy is supplied. This type of motion is commonly seen in engineering devices like turbines, centrifuges, and mixing tanks. The forced vortex helps in understanding how rotational energy affects pressure and velocity in fluids.

Definition

A forced vortex is defined as the motion of a fluid in which the particles rotate about a fixed axis with a uniform angular velocity under the influence of an external torque. In such motion, each particle of the fluid behaves as if it is a part of a rigid body, rotating with constant angular velocity. The tangential velocity of any particle in a forced vortex is directly proportional to its distance from the center of rotation.

Mathematically,

where:
= tangential velocity,
= angular velocity,
= radial distance from the axis.

This equation shows that the velocity increases linearly with the radius — the farther a particle is from the center, the faster it moves.

Nature of Motion

In a forced vortex, the motion is rotational, and every fluid particle has the same angular velocity. The flow behaves like a rigid body where there is no relative motion between the layers of fluid. Because of this, no shear stresses exist within the fluid (as long as it rotates uniformly). The entire mass of the fluid rotates as a solid body.

If the external torque or force is suddenly removed, the fluid gradually slows down and the forced vortex motion ceases due to viscous friction.

Formation of Forced Vortex

A forced vortex is formed when an external torque is applied to the fluid through mechanical means. This can be achieved by:

• Rotating the container that holds the fluid.
• Stirring the fluid using a paddle or an impeller.
• Spinning the fluid using centrifugal motion.

As the container or stirrer rotates, the fluid near the boundary starts rotating first. Due to viscosity, this motion is gradually transferred to the inner fluid layers, resulting in the entire fluid body rotating with a constant angular velocity.

Velocity Distribution

For a forced vortex, the tangential velocity  at a distance  from the axis is given by

This means that the velocity increases directly with radius. Hence, at the center (r = 0), the velocity is zero, and at the boundary (r = R), the velocity is maximum. This linear velocity distribution is one of the key features of a forced vortex.

Pressure Distribution

In a forced vortex, the pressure increases from the center towards the outer edge due to centrifugal force. Consider a small fluid element of unit mass rotating in a circular path. The radial equilibrium condition can be written as:

Substituting , we get

Integrating both sides from the center (r = 0, p = p₀) to any point (r, p),

Thus, the pressure at any point in the vortex is given by:

This equation shows that the pressure increases quadratically with the radius. The outer surface of the rotating fluid becomes curved, forming a paraboloid shape due to the pressure variation.

Free Surface Shape

The free surface of a liquid in a forced vortex forms a paraboloid of revolution. This occurs because the centrifugal force causes the liquid to rise along the sides of the container while the center becomes depressed. The shape of the free surface can be expressed as:

where:
= height of liquid surface above the center,
= acceleration due to gravity.

This parabolic shape is often seen when you stir a liquid — the center dips down and the sides rise.

Characteristics of Forced Vortex Motion

• The motion is rotational and maintained by an external torque.
• Each fluid particle has the same angular velocity.
• The tangential velocity varies directly with the radius.
• The pressure increases outward from the center.
• The free surface forms a paraboloid shape.
• The motion stops when external torque is removed.

Examples of Forced Vortex

• Liquid in a rotating cylindrical container.
• Fluid inside a centrifuge used for separating components.
• Water rotating inside a washing machine drum.
• Coffee swirling when stirred with a spoon.
• Oil or gas rotating in a turbine or compressor.

Applications

1. Centrifugal separation – used for separating particles of different densities.
2. Mixing and stirring operations – in chemical and mechanical industries.
3. Hydraulic turbines and pumps – to impart angular momentum to fluids.
4. Aerodynamic and meteorological studies – to understand rotational flows.

Conclusion

A forced vortex is a motion of fluid in which every particle rotates with a uniform angular velocity due to an external torque. The flow is rotational, and the tangential velocity increases linearly with radius. The pressure also increases from the center toward the outer edge, forming a parabolic free surface. Forced vortex motion is widely used in engineering systems like turbines, centrifuges, and mixers to produce controlled rotational flow. Understanding this motion helps engineers analyze and design systems that depend on rotational energy transfer.