Peter's Physics Pages

A Semester of First Year Physics with Peter Eyland

Lecture 2 (Fluid Dynamics)

In this lecture the following are introduced:
• Four concepts of fluid flow
• Equation of continuity
• Bernoulli's equation

Four concepts of fluid flow

Steady: pressure, density, flow velocity at every point are constant in time
(but can be different constants at each point).
Turbulent: velocities vary erratically in space and time.

2. Compressible or incompressible.

Compressible: the density changes with the applied pressure.
Incompressible: the density is constant in space and time.
(N.B. highly compressible gases can have small density variations and they can be assumed incompressible)

3. Viscous or nonviscous.
Nonviscous: The fluid flows with no energy loss.
Viscous: Energy dissipates into the fluid

4. Rotational or irrotational.
Irrotational: volume elements do not rotate about their centre of mass, but they can move in a circular path without rotating (like a Ferris wheel).
A vortex formed at a bathtub drain is irrotational.

This course deals with steady, incompressible, nonviscous, irrotational fluids. First, consider steady flow in a fluid.

Streamlines and tubes of flow
>
 In steady flow, since the velocity at a point doesn't change in time, the trajectory of every particle follows the same path. The direction of the velocity is a tangent to the streamline, but its size can change with position. Streamlines can't cross because the flow can't go in two directions at the same time.

Secondly, for a tube with an incompressible fluid flowing through it, (as shown below) any particular volume cannot change. (The volume is said to be "conserved").

Equation of continuity
 Since the volume of fluid entering area A1 (shown with a slice of width x1) is the same as the volume of fluid exiting area A2 through the slice of width x2, we have:. Equation of continuity:

Example:

Water flows at 2m.s-1 into a pipe where it has an area of 0.1m2. Find the flow speed where the pipe has a radius of 0.2m.

From the equation of continuity

Thirdly, for a nonviscous and irrotational fluid, there is no energy loss due to internal friction.

Bernoulli's equation
 Assuming no energy loss in a fluid, the net work from a pressure difference will equal the change in the total mechanical work, (the sum of potential and kinetic energy).

For a pipe of variable diameter and height.

Since energy is conserved:

dividing by the volume:

re-arranging the terms:

This is known as Bernoulli's equation. In words, the sum of pressure, potential energy density and kinetic energy density is a constant for the flow. Even though air is compressible, Bernoulli's equation works well in air.

Example:

The overpressure in a horizontal fire hose with diameter 64 mm is 350 kPa and the speed of the water through the length of the hose is 4.0 m.s-1. The fire hose ends with a metal tip with internal diameter 25 mm.
Find
(a) the speed, and pressure of water flowing in the tip, and
(b) the speed, and diameter of water flowing just outside the tip.

Use the equation of continuity to find the speed of water in the tip:

The speed of the flow has increased by more than 6x, while the diameter has reduced to just under 1/2.

Now use Bernoulli's equation to find the pressure within the region of the tip:

Also use Bernoulli's equation to find the speed just outside the tip where there is normal atmospheric pressure.

The velocity has increased slightly.

Now use the equation of continuity to find the diameter of flow just outside the tip.

The diameter first decreases slightly, but then it will increase because of interaction with the air outside.

Example:

A non-viscous fluid of density 1060 kg.m-3 flows at 0.2 m.s-1 along a pipe with radius 100 mm. The pipe ten separates into two narrower pipes of radius 50 mm.
Find the change in pressure from the larger pipe to one of the narrower pipes.

First find the flow speed in the narrow pipes.

Now use Bernoulli's equation

Fluid statics

Starting from Bernoulli's equation, if the fluid is static, the velocity is zero everywhere.

This is the usual static fluid equation.

Summarising:

Fluid flow can be:
• compressible or incompressible,
• viscous or nonviscous,
• rotational or irrotational.
The equation of continuity says that the volume flow rate is contant:

Bernoulli's equation conserves energy:

If the fluid is static, the velocity is zero everywhere, and

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