Before you read this post, I suggest you read post 17.5.
A fluid is anything that flows – like a liquid or a gas (post 16.37). In post 17.5, we saw that a fluid remains stationary when there is a surrounding pressure that acts equally in all directions. However, when the pressure is greater in one direction, the fluid flows.
When we look at a river, we see two types of flow depending on whether the water is moving fast or slow. If we drop a stick on a slow-moving river, we see it move in a straight line. This is because the water flows along lines called streamlines. A fast-flowing river is completely different: the water tumbles in lots of directions (see picture above) and we may see features like whirlpools. This irregular movement is called turbulence. The first type of flow is called streamline flow or laminar flow: the second is called turbulent flow. We can understand laminar flow by simple application of Newton’s laws of motion (post 16.2) but turbulent flow is chaotic and much more difficult to understand.
The British applied mathematician, Sir Horace Lamb (1849-1934), who was Beyer Professor of Mathematics at the University of Manchester and an expert on hydrodynamics (the study of fluid flow) once said, “When I die and go to heaven there are two matters on which I hope for enlightenment. One … is the turbulent motion of fluids.” More recently, the Nobel prize-winning physicist Richard Feynman described turbulence as, “the most important unsolved problem of classical physics”.
So, in this post, we will only think about laminar flow. We will also suppose that mechanical energy is conserved (post 16.21) so that no energy is dissipated, for example, as heat (posts 16.21 and 16.35). This assumption is similar to the idea of neglecting friction when thinking about the movement of solid objects (post 16.19); in fluids, the effect that resembles friction, by creating a force to oppose the applied force, is called viscosity and will be the subject of a future post. We will make one further assumption: that applied pressure does not compress the fluid, in other words, that it does not push its molecules closer together and so decrease its volume. This is a reasonable approximation for liquids (post 17.5) and for gases if they are flowing at much less than the speed of sound. (The relationship between pressure and the transmission of sound in a gas will be the subject of a future post.)
Let’s think about a pressure, p, acting perpendicular to a cross-section area A of fluid; then the force acting on this area is pA (post 17.5). As a result, the fluid flows with a speed, v. In a very short interval of time, δt, it moves a distance vδt, perpendicular to A. The idea is that δt is so short that any change in v is negligible; see post 17.4 for more details. So, the work done is the force multiplied by the distance moved which is pAvδt (post 16.20). But Avδt is the volume, V, of the flow created in time δt , so the work done by the pressure is pV = pm/ρ, where m is the mass of fluid and ρ is its density (post 16.44). Note that, if the fluid is incompressible, ρ remains constant (post 17.5).
The work done on the fluid will increase its kinetic energy (leading to flow) and can increase its potential energy (post 16.21). So let’s suppose that a pressure p1 creates a flow of speed v1 perpendicular to an area A1; some time later, the pressure in the fluid is p2, acting perpendicular to A2 and the fluid speed is v2. Using the definitions and ideas from the previous paragraph, the work done in moving the fluid during this time is
W = p1m1/ρ– p2m2/ρ.
This work leads to an increase in kinetic energy of
K = m2v22/2 – m1v12/2.
If the fluid rises from a vertical height h1 to h2, there will also be an increase in potential energy of
U = m2gh2 – m1gh1,
where g is the magnitude of the earth’s gravitational field (post 16.16).
Since the work done leads to the increase in energy of the fluid, we can write
p1m1/ρ – p2m2/ρ = m2v22/2 – m1v12/2 + m2gh2 – m1gh1,
which is the same as
p1m1/ρ + m1v12/2 + m1gh1 = p2m2/ρ + m2v22/2 + m1gh1.
This simply means that the quantity
pm/ρ + mv2/2 + mgh
is conserved during fluid flow. Dividing this quantity by m gives
p/ρ + v2/2 + gh.
The conservation of this quantity is usually written in the form
p/ρ + v2/2 + gh = C,
where C is a constant. The result is called Bernoulli’s equation after the Swiss mathematician Daniel Bernoulli (1700-1782) who first derived it.
In the derivation above, I have assumed that gravitational potential energy is the only form of potential energy. But we have seen that there are other forms of potential energy – like the energy stored in a stretched spring (post 16.49). Suppose we had a magnetic fluid; then some of the work done might be used in moving the fluid against an external magnetic field (see post 16.25). So sometimes Bernoulli’s equation is written more generally as
p/ρ + v2/2 + Ψ = C,
where Ψ represents the potential energy per unit mass of fluid.
It is important to realise that Bernoulli’s equation is simply a result of Newton’s laws of motion (post 16.2) applied to a fluid. The derivation involves the concepts of pressure (post 17.5), work (post 16.20) and energy (post 16.21) that are all defined from ideas derived from the concepts of mass, length and time (post 16.13) when they obey Newton’s laws.
Bernoulli’s equation is then simply a statement of the conservation of mechanical energy for fluids (post 16.21). We also expect that a flowing fluid will obey the conservation of mass (post 16.21). Returning to the flow of fluid across areas A1 and A2, the mass of fluid flowing across A1 is A1v1ρ.δt and the mass flowing cross A2 is A2v2ρ.δt. If mass is conserved
A1v1ρ.δt = A2v2ρ.δt;
Dividing both sides of this equation by ρ.δt gives the result
A1v1 = A2v2.
This result is called a continuity equation and together with Bernoulli’s equation can be used to solve many problems in fluid flow.
If energy is not conserved in the flow process, in other words if viscosity is not negligible, we need to replace Bernoulli’s equation with the Navier-Stokes equation which is much more complicated.
In the next post, we will see how Bernoulli’s equation can be used to explain how planes fly.