Velocity Profile in Capillary Flow

When a liquid flows through a narrow tube (capillary) under streamline motion, the flow is characterized as laminar. In such flow, the fluid moves in smooth layers, and the velocity is not constant across the tube's cross-section.

The velocity is highest at the center of the capillary and gradually decreases as it approaches the walls. This is due to viscous forces, which cause friction between fluid layers and between the fluid and the capillary wall. The "no-slip" condition dictates that the fluid velocity at the wall is zero.

Mathematical Description of Velocity Profile

The relationship between velocity ($v$) and the radial distance ($r$) from the center of the capillary for laminar flow is described by the Hagen-Poiseuille equation: $v(r) = \frac{\Delta P}{4 \eta L} (R^2 - r^2)$ Here, '$ R $' represents the radius of the capillary, '$ r $' is the distance from the center ($0 \le r \le R$), '$ \Delta P $' is the pressure difference driving the flow, '$ \eta $' is the fluid's dynamic viscosity, and '$ L $' is the capillary length.

The equation $v(r) \propto (R^2 - r^2)$ shows a quadratic dependence of velocity on the radius. Plotting this function results in a curve that is symmetrical and bows upwards, which is the shape of a parabola.

Conclusion

Therefore, the characteristic velocity profile for a liquid in streamline flow through a capillary is Parabolic.