Applications of Bernoulli's Equation

Bernoulli's Equation is derived for ideal fluid flow along a streamline. This equation is valid only if the conditions that were assumed during its derivation hold good while it is applied to a problem. Bernoulli's theorem states that the total energy of the fluid, under particular conditions as stated in the previous article, along a streamline remains constant. From this statement we can use this equation between two points on a streamline to find the unknown parameters.

Bernoulli's Equation between any two points on a streamline can be written as

z₁ + p₁/ρg + v₁²/2g = z₂ + p₂/ρg + v₂²/2g

To find Pressure

In certain problems in fluid flows we know the velocities at two points of the streamline and pressure at one point. The unknown is the pressure of the fluid at the other point. In such cases (if they satisfy the required condition for Bernoulli's Equation ) we can use Bernoulli's Equation to find the unknown pressure. One such example is

Flow through a Nozzle: It’s a converging nozzle. Flow enters the nozzle at low speed, accelerates and leaves the nozzle at atmospheric pressure. We have to find the pressure at inlet. We can simply apply Bernoulli's Equation between inlet and outlet points and calculate the unknown pressure assuming that the change in elevation in zero.

In this example there is no change in elevation. The converging nozzle causes fluid to accelerate. From the energy balance feature of the equation we can say the increase in velocity results in the drop in the pressure at the outlet of the nozzle.

To find Velocity

In problems where the pressure and elevation at two points and velocity at one point is known, and we have to find the unknown velocity, Bernoulli's Equation is applied to calculate the required velocity. One such example is

Flow through a Siphon: Siphon is used to drain a fluid from a reservoir at a higher level to a lower level. Here it is required to find the velocity with which the fluid leaves the siphon. We apply Bernoulli's Equation between the reservoir surface and the exit point of the siphon where the fluid leaves the tube. Pressure at both points is same (atmospheric), velocity at the reservoir is negligible because the reservoir is large. Velocity at the exit point can be calculated by using the values of elevation at the two points.

In this example we can say the decrease in elevation or the potential head manifests as the velocity of the fluid at the exit point of the siphon tube.

In some fluid flow patterns the conditions for applicability of Bernoulli's Equation are not satisfied in the static frame. But for the same case for some moving reference frame the required condition are satisfied. For such case we can apply Bernoulli's Equation from the moving inertial reference frame. A fitting example of application of Bernoulli's Equation in a moving reference frame is finding the pressure on the wings of an aircraft flying with certain velocity. In this case the equation is applied between some point on the wing and a point in free air.

These were few applications of Bernoulli's Equation. There are many more such applications which will be taken up from time to time in the coming articles.