Question

If the net rotation of fluid particles about their mass centre remains zero, then such type of flow is known as

• A. Rotational flow
• B. Irrotational flow
• C. Laminar flow
• D. Turbulent flow

Hint:

To test mathematically for irrotational flow, use the curl of the velocity vector (Vorticity, \(\vec{\zeta}\)): \[ \vec{\zeta} = \nabla \times \vec{V} = 2\vec{\omega} \] For irrotational flow, \(\nabla \times \vec{V} = 0\). This condition allows for the definition of a scalar velocity potential function (\(\phi\)), where \(\vec{V} = -\nabla\phi\).

Answer 1

The Correct Option is B

Concept: Kinematics of fluid flow classifies fluid fields based on the angular motion of the fluid elements as they move along their streamlines. Consider a small fluid element moving through a flow field. As it travels from one position to another, it can experience translation, deformation (both linear and shear), and rotation. The rotation of a fluid element about a given axis is defined as the average angular velocity of two mutually perpendicular linear differential segments meeting at that point. Analytically, the angular velocity vector \(\vec{\omega}\) in a three-dimensional Cartesian flow field with velocity components \((u, v, w)\) is given by: \[ \vec{\omega} = \omega_x \hat{i} + \omega_y \hat{j} + \omega_z \hat{k} \] Where the component rotations about the \(x\), \(y\), and \(z\) axes are: \[ \omega_x = \frac{1}{2}\left(\frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}\right), \quad \omega_y = \frac{1}{2}\left(\frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}\right), \quad \omega_z = \frac{1}{2}\left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\right) \]

Step 1: Evaluating the condition where net rotation is zero.
The problem states that the net rotation of the fluid particles about their center of mass is zero during motion. Mathematically, this condition requires all components of the angular velocity vector to vanish simultaneously: \[ \vec{\omega} = 0 \quad \Rightarrow \quad \omega_x = 0, \; \omega_y = 0, \; \omega_z = 0 \]

Step 2: Defining Irrotational Flow.
When a fluid element moves along a path in such a way that its net orientation does not rotate about its mass center (i.e., its axes remain parallel to their original directions, despite any shear deformation), the flow is classified as

irrotational flow.

Step 3: Comparing with other options.

• Rotational flow: Fluid elements rotate about their mass centers (\(\vec{\omega} \neq 0\)), typically due to shear stresses caused by viscosity near solid boundaries.

• Laminar flow: Fluid particles move in smooth, parallel layers or laminas, without macroscopic mixing.

• Turbulent flow: Fluid particles exhibit chaotic, random, and fluctuating three-dimensional motion.
Since the zero-rotation condition refers uniquely to irrotational flow fields, Option (2) is the correct choice.