A liquid is flowing in a streamlined manner through a cylindrical pipe. Along a section containing the axis of the pipe, the flow profile will be

The correct answer is Option 4. In a streamlined flow through a cylindrical pipe, the velocity distribution of the liquid is governed by viscosity and the boundary conditions of the pipe walls.

According to the principle of viscous flow (Poiseuille's Law):

• No-Slip Condition: The layer of liquid in direct contact with the pipe walls has zero velocity due to friction and molecular adhesion.
• Maximum Velocity: The velocity increases as we move toward the center of the pipe, reaching its maximum at the central axis, where resistive frictional forces from the walls are minimal.
• Parabolic Profile: The mathematical relationship between the velocity and the radial distance from the axis is parabolic.

Option 4 correctly represents this parabolic curve, showing zero velocity at the edges and a smooth, curved peak at the center. Options 1, 2, and 3 are incorrect as they suggest uniform velocity (plug flow) or non-physical distributions that ignore the cumulative effect of internal fluid friction (viscosity).