This section will introduce perhaps the most fundamental principle in our studies of fluid mechanics, Bernoulli’s principle. It is a direct application of conservation of mechanical energy density and proves useful for both fluid dynamics and fluid statics. Bernoulli’s equation and the continuity equation are usually the two most useful tools when approaching a fluid mechanics problem.
This OpenStax physics trailer provides real life examples of Bernoulli's equation in action.
Watch the pre-lecture videos and read through the OpenStax text before doing the pre-lecture homework or attending class.
How often are weather reports accurate? Sometimes it feels like they can never get the forecast right. Why is this? Well, the dynamics of our atmosphere fall under the realm of fluid dynamics, which means predicting the future (predicting the weather) requires solving fluid dynamic equations. Just like our early studies of kinematics, fluid dynamic equations require initial conditions, however, unlike kinematics, a very small change in the initial conditions can lead to a very different final answer in fluid dynamics. Basically, the study of fluid dynamics is very complicated.
Luckily we are able to make a few approximations which will allow us to quantify some aspects of fluids in motion. Our first approximation is restricting ourselves to only study fluids that move in a certain way. Consider two locations A and B within a tube of moving fluid as shown in the left image of the figure below. Our model of the moving fluid will assume that any particle that passes through location A will follow a unique path to get to location B. That unique path is represented by the dashed line between location A and B.
This is a representation of different kinds of flow. There are two tubes that has some smaller area in the front part of the tube that expands into a greater area and both are identical. The first tube is shows Laminar flow where the flow of the fluid is direct from one part of the tube to the other without any interference. The second tube shows turbulent flow where there is no direct flow and there is a lot of interference of the motion of the particle.
Similarly, if we look at other locations such as C and D, a particle passing through each will also follow unique paths and no paths cross each other. This type of fluid flow is called laminar flow (or smooth flow). For comparison, the right image of the figure above shows turbulent flow where particles in the fluid that pass through each point follow chaotic paths that change in time, this makes the future position of each particle extremely hard to predict.
Furthermore, if we make another assumption and ignore viscous forces (internal friction between "layers" of fluid and external friction between the fluid and the walls of the tube) then the velocity profile (the speed at every location along the cross section of a cylindrical of the tube) is constant. If this were not the case, then kinetic energy of the fluid near the surfaces of the tube would be lost to thermal energy because of the viscous forces (this would mean larger velocities in the center of the tube and velocities approaching zero at the edges). Below is a simplified velocity profile comparison between inviscid flow (without viscous forces) and viscous flow (with viscous forces).
This is a representation of different kinds of flow. There are two tubes that has some smaller area in the front part of the tube that expands into a greater area and both are identical. The first tube shows velocity profile without viscous forces where the vector arrows are all equal in direction and magnitude where the only difference is the length of the arrows are shorter at a greater cross-sectional area. The second tube shows velocity profile with viscous forces where the vector arrows are greater at the center of the tube and smaller near the edges of the tube and the length of the arrows are collectively shorter at a greater cross-sectional area.
Our third assumption is that the fluid is incompressible, or in other words, that the fluid's density is a constant.
When combined, all three of these approximations allow us to make the following statement, "the mass of fluid that enters a certain region in a given amount of time is equal to the mass of fluid that leaves another region in the same amount of time." This is known as the continuity of mass flow rate. Mathematically we write this as...
$\sum{\dot{m}}_{in}$ $=$ $\sum{\dot{m}}_{out}$ where $\dot{m} = \rho \, v \, A$ and $v$ is the speed, $A$ is cross sectional area
What the the continuity equation above says is, "what goes in at some rate, must come out at the same rate".
If there we restrict our studies to only one type of fluid in a tube at a time, then density is a constant and we use volume flow rate, $Q$...
$Q = \frac{\dot{m}}{\rho} = v \, A$
Then our continuity equation becomes...
$\sum{Q}_{in} = \sum{Q}_{out}$
Bernoulli’s equation relates the speed, relative height, and pressure at any location within a moving or stationary fluid. Bernoulli’s equation and the continuity equation are usually the two most useful tools when approaching a fluid mechanics problem. Before defining Bernoulli’s equation, a simple derivation will help demystify its appearance.
Consider the figure below which shows a moving fluid inside a tube that changes height and radius. We will consider an ideal fluid, (i.e. laminar flow, no viscous forces, and incompressible). The pressure is measured at the center of each cross sectional area as indicated by the red $P_1$ and $P_2$. The dashed blue line indicates the stream line that passes through the two locations we will consider.
This is an image of a pipe with where the first part of the tube has a smaller cross sectional area than the second part and the second part of the tube is elevated higher. There is a dashed line of the trajectory of the particle that travels through the center of the tube.
Let’s apply conservation of mechanical energy to this system.
$\sum{E}_1 + W_{nc} = \sum{E}_2$
Here we are ignoring any thermal energy since changes in temperature are typically very small in the scenarios we will study.
Some algebraic manipulations and we get our conservation of mechanical energy into the form below..
$\Delta KE + \Delta U^g = W_{nc}$
Each of the above terms have the following form…
$\Delta KE = \frac{1}{2} m \, \Delta v^2 = \frac{1}{2} \rho \, V \, \Delta v^2$ where $v$ is velocity and $V$ is volume.
$\Delta U^g = m \, g \, \Delta y = \rho \, V \, g \, \Delta y$ where $V$ is volume.
$W_{nc} = | \vec{F} | \, \Delta x = \Delta P \, A, \Delta x = \Delta P \, V$ where $A$ is cross sectional area and $P$ is pressure
In the expression for the non-conservative work, force is the force due to the horizontal pressure gradient within the fluid.
Note that all the expressions contain the volume $V$, so let’s substitute each expression back into the conservation of mechanical energy and divide by volume.…
$\frac{1}{2} \rho \, \Delta v^2 + \rho \, g \, \Delta y = \Delta P$
Note that each term has units of energy per volume, thus this is a now a statement of conservation of mechanical energy density. One more algebraic manipulation and we get the familiar form which we call Bernouill’s principle…
$P_{1} + \frac{1}{2} \rho \, v^{2}_{1} + \rho \, g \, y_{1} = P_{2} + \frac{1}{2} \rho \, v^{2}_{2} + \rho \, g \, y_{2} = constant$
Now, take a look at the pre-lecture reading and videos below.
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