1. 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)

2. 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).

3. 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

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

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

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

7. 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:

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

9. 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?

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

11. 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