The Bernoulli Equation

Key Concept: Bernoulli’s equation relates pressure, speed, and elevation of a fluid particle along

a streamline. Fluid particles travel along their streamlines since by definition the velocity of each

particle is tangent to its streamline. i.e. P + ½ ρ V² + ρ g z = Constant along streamline

In a Nutshell: The Bernoulli equation holds along each streamline. It governs the accelerated

motion of the fluid particle under the assumptions that viscosity is negligible, the density is

constant, and the flow is steady (independent of time).

Bernoulli Equation:

The strategy in developing this equation is to take the Lagrangian viewpoint and determine the

accelerated motion of a general fluid particle using Newton’s 2^nd , F = m a Integration of the

this differential equation of fluid particle motion yields Bernoulli’s equation.

The figure below shows a free body diagram of a typical fluid element using intrinsic coordinates,

s and n where s denotes the coordinate along the streamline and n the coordinate normal to the

streamline. Forces on the fluid particle come from pressure acting on the fluid particle and on

gravity giving rise to the elemental weight, dW, of the fluid particle.

Now the acceleration of the fluid particle using intrinsic coordinates is Note: v = v(s)

a = dv/dt = a_s e_s + a_n e_n = dv/dt e_s + v² /R e_n

a_s = dv/dt = ∂v/∂s ds/dt (using the chain rule) and ds/dt = v

So the components of acceleration of the fluid particle are a_s = v ∂v/∂s and a_n = v² /R The element of mass, dm = ρ dsdbdn. The equations describing the motion of the fluid particle are

Σ F_s = dm a_s and Σ F_n = dm a_n .

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