Continuity Equation and Midterm Review

From fluid statics to fluid dynamics

Fluid statics – the study of stationary fluids (pressure of a static column of fluid, buoyancy)

Fluid dynamics – the study of fluids in motion (flow of water through a pipe system, airflow)

Let’s flow!

One crucial assumptionWater, oil, and almost all other liquids are incompressible.

No matter how much pressure is exerted on them, the total volume of a liquid will remain constant.

This is not true for gases (which can be compressed), but it is true for liquids for up to one part per billion.

What does this mean?

Water is incompressible

H2O

piston

F

No matter how hard you push on the piston, the volume of the water will not decrease!

(this is not exactly true of course, but it is true to a very close approximation)

Water and other liquids have a fixed volume, and this will factor in very strongly when analyzing the following scenario.

The same amount of water crosses this

plane per second…

This means that when water flows through a pipe, the same amount of water crosses any plane of the pipe in a given interval of time.

Flow must be steady and continuous

…as this one!

flow

flow

Flow ContinuityNo build-up or lack of water will exist when the

liquid reaches a steady, sustainable flow.

This leads us to a very interesting conclusion!

Volume that crosses plane A per unit time

Volume that crosses plane B per unit time

A B

A1 A2

The same concept applies to pipes that have a changing cross-sectional area!

flow

Volume that crosses plane A1 per unit time

Volume that crosses plane A2 per unit time

A1 A2

flow

In order to have a stable flow of liquid through the pipe, the same amount of volume per unit time must pass through

any cross-section of the pipe!

This is called flow continuity, and applies to all incompressible fluids (pretty much all liquids)

However, this does not say anything about the speed of the fluid through each section…

Whiteboard: Explain your childhood!

Why does this happen?

A1 A2

v1 v2

But, section 1 has a larger area, and can fit more water in it at any given moment!

How could it be possible for the same amount of water to pass through each section per unit time?

Answer: The water in the smaller section must have a greater speed!

ΔV2ΔV1

Δx1

Δx2

A1 A2

The same volume of water per unit time must pass

through both sections for continuous flow to occur.

ΔV1

Δx1

A1flow

Volume Rate of FlowWhen a fluid passes through a pipe, the amount of volume per unit time that passes any point on the pipe is called the volume rate of flow.

By using some simple geometry and kinematics, we can also express the volume rate of flow in terms of cross-sectional area and fluid speed.

Volume rate of flow

Flow ContinuityWhen an incompressible fluid passes through a pipe, the amount of volume per unit time that passes any point on the pipe must be the same.

Using the earlier expression for volume rate of flow, we can state this mathematically as the continuity equation.

A1 A2

v1 v2

Super Soaker Whiteboard

In the nozzle of a super soaker, water flows from a tube of radius 2R to a tube of radius R. If the speed of the water as it flows through the wider portion is v, what is the speed of the water as it shoots out the end?

Radius is not area!

If you divide the radius by two, you divide the area by four.

This will cause the smaller segment to have four times the fluid speed.

Area x Velocity = Constant

Area and velocity are inversely proportional