# Fluid Dynamics and Its Biological and Medical Applications

## The Most General Applications of Bernoulli’s Equation


### Learning Objectives

By the end of this section, you will be able to:
1. Calculate using Torricelli’s theorem.
2. Calculate power in fluid flow.

### Torricelli’s Theorem

 shows water gushing from a large tube through a dam. What is its speed as it emerges? Interestingly, if resistance is negligible, the speed is just what it would be if the water fell a distance  from the surface of the reservoir; the water’s speed is independent of the size of the opening. Let us check this out. Bernoulli’s equation must be used since the depth is not constant. We consider water flowing from the surface (point 1) to the tube’s outlet (point 2). Bernoulli’s equation as stated in previously is
Both  and  equal atmospheric pressure ( is atmospheric pressure because it is the pressure at the top of the reservoir.  must be atmospheric pressure, since the emerging water is surrounded by the atmosphere and cannot have a pressure different from atmospheric pressure.) and subtract out of the equation, leaving
Solving this equation for , noting that the density 
 
   cancels (because the fluid is incompressible), yields
We let ; the equation then becomes
where  is the height dropped by the water. This is simply a kinematic equation for any object falling a distance  with negligible resistance. In fluids, this last equation is called Torricelli’s theorem. Note that the result is independent of the velocity’s direction, just as we found when applying conservation of energy to falling objects.
All preceding applications of Bernoulli’s equation involved simplifying conditions, such as constant height or constant pressure. The next example is a more general application of Bernoulli’s equation in which pressure, velocity, and height all change. (See .)

### Power in Fluid Flow

Power is the rate at which work is done or energy in any form is used or supplied. To see the relationship of power to fluid flow, consider Bernoulli’s equation:
All three terms have units of energy per unit volume, as discussed in the previous section. Now, considering units, if we multiply energy per unit volume by flow rate (volume per unit time), we get units of power. That is, . This means that if we multiply Bernoulli’s equation by flow rate , we get power. In equation form, this is
Each term has a clear physical meaning. For example,  is the power supplied to a fluid, perhaps by a pump, to give it its pressure . Similarly,  is the power supplied to a fluid to give it its kinetic energy. And  is the power going to gravitational potential energy.

### Test Prep for AP Courses


### Summary

1. Power in fluid flow is given by the equation  where the first term is power associated with pressure, the second is power associated with velocity, and the third is power associated with height.

### Conceptual Questions


### Problems & Exercises
