Today we continue with Physics, and more specifically the branch of "Classical Mechanics", in order to get into Bernoulli’s Equation.
So, without further ado, let's dive straight into it!
Energy Conservation and Flow
In order to satisfy flow continuity the velocity of ideal fluid (uncompressed fluid with no viscosity) changes along the path it flows. A narrower path means higher velocity, which in turn yields an increase in kinetic energy. What is constant in any case is the flow rate.
So, where does that energy come from? Energy conservation comes into mind. The energy must come from work done by some force that acts upon the fluid. Its possible to express this using the work-energy principle:
Let's simplify this analysis even further by considering only friction-less laminar flow, with no energy loss due to dissipative forces.
Of course, no external force is applied upon the system, which means that the force must come from some portion of the fluid itself. The work is thus done by pressure from surrounding fluid. This pressure exerts force that does work and causes the fluid to speed up.
So, as the net work done by the pressure increases the fluid's kinetic energy, the pressure of the fluid drops. The pressure is thus different in different regions of the fluid, as otherwise the net force would be zero. As the fluid accelerates it always moves from a higher-pressure region to a lower-pressure region.
Lastly, we should also not forget that pressure in fluid is affected by the height or depth of it. The difference in height causes an additional pressure, which is completely unrelated to the flow speed. It's related to the conservative force of gravity.
All this leads to a very useful relationship between the pressure, the flow speed and the height in a fluid, which is known as Bernoulli's equation.
Bernoulli’s Equation
In order to end up with Bernoulli's equation, we will apply the work-energy theorem at some portion of a fluid pipe. Consider a fluid pipe with varying diameter and height as shown below.
During a period of time dt the same volume of fluid dV flows through any given section of the fluid pipe. So, the sections with cross-section A1 and cross-section A2 respectively are related as follows:
The total work done during that same period of time due to pressure and in turn the forces F1 and F2 can be expressed as:
The work done is also equal to the change in kinetic and potential energy. The change in kinetic energy can be expressed as follows:
The corresponding potential energy is:
Combining these equations, the total work is:
Rearranging the equation gives us Bernoulli's equation:
or
Static Fluids
For static fluids (v1 = v2 = 0) and a reference height of h2 = 0:
which yet again shows that pressure increases with depth.
Constant Depth
Another common situation is constant depth, h1 = h2. In that case only the pressure and velocity are now present. It's common to call this form of the equation Bernoulli's principle:
It has lots of applications that we will cover in the examples post.
