Bernoulli’s Equation is an important and widely used equation in fluid dynamics. It can be derived directly from Newton’s 2nd Law. Bernoulli’s equation is not based on a control volume, but is derived for a fluid particle that moves along a streamline. Although derived differently, Bernoulli’s equation does satisfy the three governing equations: continuity, energy, and momentum.
Consider a small particle of fluid that follows a streamline. A streamline is a line drawn through the flow field that is everywhere tangent to the velocity. The forces acting on the fluid particle are gravity and pressure. The viscous shear stresses are not included, so the fluid must be inviscid (negligibly small viscosity).
A local coordinate system is shown in the figure. The s-direction is tangent to the streamline. The n-direction is normal to the streamline, and the w-direction is perpendicular to and coming out of the page.
There is a pressure gradient, dP/ds, along the direction of the stream line. There is also a pressure gradient, dP/dn, normal to the streamline if the streamline is curved. The Bernoulli equation is only concerned with the pressure gradient in the s-direction. The pressure at each end of the fluid element can be written in terms of the pressure at the center of the fluid element and the s-direction pressure gradient. Note that the assumption used in the figure is that a positive pressure gradient exists in the positive s-direction.
Although the Bernoulli equation is not concerned with the pressure gradient normal to the streamline, it does not mean that this pressure gradient is unimportant. The lift on air foils, etc. is controlled by this gradient.
Newton’s Second Law states that the sum of the forces acting on a particle equals the particle’s mass times its acceleration. Since we are tracking a particular fluid particle, Newton’s second law can be applied directly. Mathematically, this is written as
Newton’s Second Law applied to the s-direction
The acceleration in the s-direction is the time rate of change of the velocity in the s-direction. The velocity also depends on where the particle is along the streamline. Hence, the velocity is a function of s and time, t. This allows the acceleration component in the s-direction to be written as
Limiting the derivation to steady-state conditions allows the s-direction acceleration to be written as
Applying Newton’s Second Law in the s-direction and simplifying yields the equation
This equation can be integrated with an indefinite integral if the density is constant (incompressible). The resulting equation is Bernoulli’s Equation
Bernoulli’s equation states that the sum of the three terms shown in the equation is the same at any point along the length of the streamline. Assumptions used during the derivation are: 1) steady-state, 2) inviscid, and 3) incompressible.
The assumptions used in developing the Bernoulli equation.