Bernoulli’s Equation

Bernoulli's Equation

The Bernoulli equation states that,




? points 1 and 2 lie on a streamline,

? the fluid has constant density,

? the flow is steady, and

? there is no friction.

Although these restrictions sound severe, the Bernoulli equation is very useful, partly because it is very simple to use and partly because it can give great insight into the balance between pressure, velocity and elevation.

Pressure/velocity variation

Consider the steady, flow of a constant density fluid in a converging duct, without losses due to friction (figure 1). The flow therefore satisfies all the restrictions governing the use of Bernoulli's equation. Upstream and downstream of the contraction we make the one-dimensional assumption that the velocity is constant over the inlet and outlet areas and parallel.

CV_force_1   Figure 1. One-dimensional duct showing control volume.

When streamlines are parallel the pressure is constant across them, except for hydrostatic head differences (if the pressure was higher in the middle of the duct, for example, we would expect the streamlines to diverge, and vice versa). If we ignore gravity, then the pressures over the inlet and outlet areas are constant. Along a streamline on the centerline, the Bernoulli equation and the one-dimensional continuity equation give, respectively,


These two observations provide an intuitive guide for analyzing fluid flows, even when the flow is not one-dimensional. For example, when fluid passes over a solid body, the streamlines get closer together, the flow velocity increases, and the pressure decreases. Airfoils are designed so that the flow over the top surface is faster than over the bottom surface, and therefore the average pressure over the top surface is less than the average pressure over the bottom surface, and a resultant force due to this pressure difference is produced. This is the source of lift on an airfoil. Lift is defined as the force acting on an airfoil due to its motion, in a direction normal to the direction of motion. Likewise, drag on an airfoil is defined as the force acting on an airfoil due to its motion, along the direction of motion.

Example 1.

Suppose a ball is spinning clockwise as it travels through the air from left to right The forces acting on the spinning ball would be the same if it was placed in a stream of air moving from right to left, as shown in figure 2.


Magnus   Figure 2. Spinning ball in an airflow.

A thin layer of air (a boundary layer) is forced to spin with the ball because of viscous friction. At A the motion due to spin is opposite to that of the air stream, and therefore near A there is a region of low velocity where the pressure is close to atmospheric. At B, the direction of motion of the boundary layer is the same as that of the external air stream, and since the velocities add, the pressure in this region is below atmospheric. The ball experiences a force acting from A to B, causing its path to curve. If the spin was counterclockwise, the path would have the opposite curvature. The appearance of a side force on a spinning sphere or cylinder is called the Magnus effect, and it well known to all participants in ball sports, especially baseball, cricket and tennis players.

Example 2.

From Bernoulli's principle, the pressure on the upper surface where the flow is moving faster is lower than the pressure on the lower surface. The pressure difference thus creates a net aerodynamic force, pointing upward and downstream to the flow direction. The component of the force normal to the free stream is considered to be lift





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