Bernoulli’s Equation

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A stream tube is a volume within a flow field that is bounded by stream lines.  Fluid entering the stream tube at one end must exit the stream tube at the other end.  Flow cannot enter or exit the stream tube through the sides since the flow is tangent to the streamlines that define sides.

By Samuel.hapak (Own work) [Public domain], via Wikimedia Commons

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.

Learning Objectives:

The assumptions used in developing the Bernoulli equation.

Other Forms of Bernoulli’s Equation

Bernoulli’s equation can be written in several different forms that facilitate the solving of specific problems.

Total Pressure

Bernoulli’s equation gives rise to the concept of the total pressure.

Relation to Energy Equation

There are various ways at which to arrive at Bernoulli’s equation. You can start by applying the momentum equation along a streamline as was done previously. You can also arrive at Bernoulli’s equation from the Pressure-Energy Equation or Euler’s Equation which are discussed in the publication, “Fundamentals of Gas Dynamics”. You can also arrive at Bernoulli’s equation using the Conservation of Energy and Mass Equations.

Example Problem

Nozzles are important devices that are used in a wide variety of applications.