We'll examine an ideal fluid, making some simplifying assumptions. The flowing fluids we'll consider are:

1. Steady - no turbulence.
2. Incompressible - the density of the fluid does not change.
3. Non-viscous - no resistive force from objects or pipe walls.
4. Irrotational - the fluid won't make an object spin about its own axis.

When an incompressible fluid flows through a tube of varying cross-section, the rate at which mass flows past any point in the tube is constant. If this flow rate varied, fluid would build up at points where the flow rate is low.

The mass flow rate is the total mass flowing past a point in a given time interval, divided by that time interval.

At a point where the flow is in the x direction and the tube has a cross-sectional area A:

mass flow rate

=

Δm Δt

=

ρ ΔV Δt

=

ρA Δx Δt

=

ρ A v

The continuity equation reflects the idea that the mass flow rate is constant:

ρAv = constant         or         ρ₁A₁v₁ = ρ₂A₂v₂

In an incompressible fluid the density is constant, so the continuity equation is:

A₁v₁ = A₂v₂

Moral of the story: The fluid flows faster in narrow sections of the tube.

There are basically two ways to make fluid flow through a pipe.

1. Tilt the pipe and let gravity pull the fluid downhill.
2. Make the pressure higher at one end of the pipe than the other. A pressure difference is equivalent to a net force, accelerating the fluid toward lower-pressure regions.

Apply the master energy equation to a fluid flowing in a pipe.

U₁ + K₁ + W_nc = U₂ + K₂

This relates the energy at point 1 in the fluid to the energy at some other point 2. The potential energy we'll consider is gravitational. Any work done can be written as a force multiplied by a distance:

W = F_net Δx = ΔPAΔx

so:

mgy₁ + ½mv₁² + ΔPAΔx = mgy₂ + ½mv₂²

ΔP is the pressure difference between points 1 and 2:

ΔP = P₁ - P₂.

The energy expression becomes:

mgy₁ + ½mv₁² + P₁AΔx = mgy₂ + ½mv₂² + P₂AΔx

If we divide through by volume we get energy/volume, which is energy density. Mass over volume is mass density, and AΔx = volume, so the energy density relationship is:

ρgy₁ + ½ρv₁² + P₁ = ρgy₂ + ½ρv₂² +P₂

This is Bernoulli's equation. Combining this with the continuity equation allows us to relate pressures, speeds, and heights at any two points in a flowing fluid.