The Bernoulli Equation • It is an approximate relation between pressure, velocity and elevation • It is valid in regions of steady, incompressible flow where net frictional forces are negligible • Viscous effects are negligible compared to inertial, gravitational and pressure effects. • Applicable to inviscid regions of flow (flow regions outside of boundary layers) • Steady flow (no change with time at a specified location)

Steady flow • The value of a quantity may change from one location to another. In the case of a garden hose nozzle, the velocity of water remains constant at a specified location but it changes from the inlet to the exit (water accelerates along the nozzle).

Acceleration of a Fluid Particle • Motion of a particle in terms of distance “s” along a streamline • Velocity of the particle, V = ds/dt, which may vary along the streamline • In 2-D flow, the acceleration is decomposed into two components, streamwise acceleration as, and normal acceleration, an. • For particles that move along a straight path, an =0

Fluid Particle Acceleration • Velocity of a particle, V (s, t) = function of s, t • Total differential • In steady flow, • Acceleration,

Derivation of the Bernoulli Equation (1) • Applying Newton’s second law of conservation of linear momentum relation in the flow field

Derivation of the Bernoulli Equation (2) Integrating For steady flow For steady incompressible flow,

Bernoulli Equation • Bernoulli Equation states that the sum of kinetic, potential and flow (pressure) energies of a fluid particle is constant along a streamline during steady flow. • Between two points:

Example 1 Figure E3.4 (p. 105) Flow of water from a syringe

Example 2 • Water is flowing from a hose attached to a water main at 400 kPa (g). If the hose is held upward, what is the maximum height that the jet could achieve?

Example 3 • Water discharge from a large tank. Determine the water velocity at the outlet.

Limitations on the use of Bernoulli Equation change in flow conditions • Frictional effects can not be neglected in long and narrow flow passage, diverging flow sections, flow separations • No shaft work