# Energy in fluid flow

#### Prerequisites

Fluids and their properties play a critical role in essentially all biological systems. Although we have analyzed some properties of fluid flow in terms of forces, we have seen in our treatment of solid systems that an energy perspective adds insight and provides powerful tools (such as a conservation law) for making sense of motion and change. The same is true for fluids.

But fluids are a level of complexity above solids. The state of an object that is approximated by the (simplified model of) a rigid solid* can be specified by 9 numbers: the 3D position of its center of mass and its 3D velocity — and by 3 angles (and their rates of change) — to say what orientation it has and how it's rotating. (If the object doesn't get reoriented we don't even have to pay attention to those angles at all, and we can get away with our "point mass" models). For a fluid you have to — in principle — specify the position and velocity of every bit of the fluid, an extremely difficult task.

For that reason, we tend to restrict our considerations to fluids that are nearly incompressible (such as water at typically observable pressures) and that are moving smoothly (without turbulence). This simplifies things dramatically and lets us build useful mathematical models. These models are very good as a starting point for considering the fluid flows that happen in biology.

In our considerations of energy in fluids, we'll start with the general work-energy theorem, the connection between forces (and Newton's laws) and energy. We'll then make a similar chain of models as we did in our consideration of the energy of motion of solid objects. There, we started with "point masses" and ignored friction. The result was the conservation of mechanical energy. Then we added friction and talked about the transfer of mechanical energy to thermal energy.

In the case of fluids, applying the work-energy theorem leads to three valuable results.

• Bernoulli's principle: We'll first apply our general purpose work-energy theorem by considering an incompressible fluid (whose density can be treated as a constant) and ignore resistive forces like viscosity or inertial drag. We'll then build up the analog of the conservation of mechanical energy for a moving fluid: an energy-conservation like equation in density form.
• Archimedes' Principle: One interesting result of that is that since we have included gravitational potential energy in Bernoulli's principle, we can apply it to a non-flowing fluid and pull out the equation for the increase of pressure in a fluid with depth that leads to Archimedes' Principle.
• The Hagen-Poiseuille equationWe'll then see how adding internal resistance (viscosity) as in our treatment of the HP equation results in the transfer of energy from the coherent motion of the fluid to the random (thermal) motion of its molecules.

As in the case of solid objects, going to energy adds a set of highly useful tools!

* Treating an object as a solid is an approximation since all solids can be deformed, at least a bit. For many situations the deformations of solid objects can be ignored. If you couldn't do this, the treatment of a solid body would have to track the motion of every bit of the object, making the description as complicated as for a fluid.

Joe Redish 7/21/17

Article 444