Velocity gradient exists in a cross-sectional area due to the effect of fluid flow and the conservation of mass. When a fluid flows through a pipe or any conduit with varying cross-sectional area, the velocity of the fluid changes.
According to the principle of conservation of mass, the mass of fluid entering a section must be equal to the mass of fluid leaving that section. Since mass is conserved, the product of mass and velocity at any given point must remain constant along the flow.
Consider a fluid flowing through a pipe that has a decreasing cross-sectional area in the direction of the flow. As the cross-sectional area decreases, the same amount of fluid mass must pass through a smaller area, resulting in an increase in velocity to maintain mass conservation.
Due to the increase in velocity, a velocity gradient is established along the cross-sectional area. The fluid near the walls of the conduit experiences lower velocity, while the fluid near the center of the conduit experiences higher velocity. This velocity gradient exists to ensure that the mass flow rate remains constant throughout the conduit.
The velocity gradient is a consequence of the relationship between fluid flow, cross-sectional area, and the conservation of mass. It indicates how the velocity of the fluid changes as it passes through different regions of varying cross-sectional area.