Bernoulli's Equation for Flow Normal to a Streamline

Pressure is higher on the inner side of a curve because outward centrifugal forces push fluid away from the center of curvature. To balance these forces, pressure must increase toward the inner side. This pressure gradient acts perpendicular to the streamline, ensuring the fluid follows the curved path without diverging outward.

Pressure differences depend on both fluid velocity and the radius of curvature. Sharp curves with small radii create pronounced pressure variations, while nearly straight streamlines produce minimal pressure differences. High-velocity flows through sharp bends, such as vortices or pipe bends, experience the most significant pressure variations across streamlines.

Bernoulli's equation normal to the streamline describes how pressure varies perpendicular to flow direction in curved paths. Assuming steady, inviscid, incompressible flow, this equation relates pressure differences to centrifugal acceleration. It expresses the balance between pressure gradients and the outward forces acting on fluid elements in curved streamlines.

In curved dams and reservoirs, water flows along curved surfaces where pressure differences across streamlines must be accurately predicted. Designers must account for pressure variations to ensure structures withstand forces exerted by water, especially during high-velocity flows. Incorrect pressure calculations may lead to structural failure or instability.

Wind flows around civil structures like buildings and bridges create curved streamlines that generate pressure differences. These pressure variations can cause instability or damage during high winds. Understanding how curved streamlines produce pressure differences helps engineers design structures that remain stable under varying wind conditions.

Pipe systems with bends experience significant pressure variations due to curved streamlines. Accurate pressure predictions prevent excessive forces that could cause pipe failure. Engineers must account for these pressure differences when designing curved channels and hydraulic structures to ensure structural integrity under operational flow conditions.

These assumptions simplify the derivation of Bernoulli's equation normal to streamlines by eliminating time-dependent effects, viscous losses, and density variations. Steady flow means conditions don't change over time; inviscid flow ignores friction; incompressible flow assumes constant density. Together, they allow engineers to focus on pressure-centrifugal force balance in curved flows.