Peter's Physics Pages

Physics for Civil Engineering

This is an introduction to Electricity, Strength of Materials and Waves.

Lecture 5 (Fluids: Flowing and Static)

In this lecture the following are introduced:
• Solids, Fluids, Plastics & Plasmas
• Two techniques of calculation with fluids
• Four concepts of fluid flow
• Streamlines
• Equation of continuity
• Bernoulli's equation
• Fluid statics

Solids, Fluids, Plastics & Plasmas

Generally,
A solid holds its shape.
A fluid flows to take the shape of its container.
A plastic can be moulded or shaped (clay).
A plasma is an ionised gas (eg flame, stars).

Two techniques of calculation with fluids

J.L.Lagrange (1736 - 1813)
You divide the fluid into small volume elements, called fluid particles.
You find the forces on each particle.
You solve for the position and velocity of all particles as functions of time (tedious!).

L. Euler (1707 - 1783)
You specify the density ρ and the velocity v(x,y,z,t) at each point in space at one instant of time.
The focus is on a point in space rather than a particle, and this is implied in what follows. (easier!)

Four Parameters for fluid flow

Steady flow: the pressure, density, flow velocity at every point are constant in time (but can be different constants at each point).
Turbulent flow: 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 so are assumed incompressible).

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

4. Rotational or irrotational.
Irrotational flow: the 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.
Rotational flow: the volume elements rotate about their centre of mass.

Streamlines

 In steady flow, since the velocity at a point doesn't change in time, the trajectory of every particle injected at that point follows the same path called a streamline. 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.

Equation of continuity

 For a tube with an incompressible fluid flowing through it, the volume is conserved. This means that the volume of fluid entering area A1 (shown as a slice of width x1) is the same as the volume of fluid exiting area A2 through the slice of width x2. The rate of flow is called the Volume Flux This leads to the Equation of Continuity, which says that the Volume Flux is constant

Example F1
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

Bernoulli's equation

 Assume a steady, incompressible, nonviscous, irrotational fluid flow in a pipe. The pipe can vary in diameter and height as shown in the diagram on the right. The work done by pressure difference is the product of pressure and volume. There is no energy loss in the fluid, so the net work from pressure equals the change in the total mechanical work. Total mechanical work is the sum of potential and kinetic energy. Dividing by the volume on both sides: Re-arranging the terms gives Bernoulli's equation.

In words, the sum of pressure, potential energy density and kinetic energy density is a constant for the flow.

Example F2
The overpressure in a horizontal fire hose with diameter 64 mm is 350 kPa and the speed of the water through 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 between the hose and the tip, to find the speed of water in the tip. The speed of the flow in the tip has increased by more than 6×, while the diameter has reduced to just under half of the hose size. Use Bernoulli's equation between the hose and the tip, to find the pressure in the tip. Overpressure is pressure above atmospheric (e.g. P1 - P0), and atmospheric pressure is effectively subtracted from both sides of the equation. The speed of the flow in the tip has increased by more than 6×, while the diameter has reduced to just under half of the hose size. Use Bernoulli's equation between the tip and normal atmospheric pressure outside to find the speed just outside the tip. The velocity has increased only slightly from 26.2 m.s-1 to 26.8 m.s-1. Use the equation of continuity between the tip and the outside air to find the diameter of flow just outside the tip. The diameter has decreased slightly, but then it will increase because of air resistance.

Example F3
An incompressible, 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 then separates into two narrower pipes of radius 50 mm. Find the change in pressure from the larger pipe to one of the narrower pipes.

 Use the equation of continuity between the wider pipe and the narrower pipes to find the flow speed in the narrower pipes. The speed has doubled in each of the two narrower pipes. Use Bernoulli's equation between the wider pipe and the narrower pipes to find the pressure difference between the wider and narrower pipes.

Fluid statics

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

For the ocean, this means that the pressure under water increases with depth.

Example F4
The ocean has water of density 1030 kg.m-3 and normal atmospheric pressure at the surface is 101.3 kPa. Find the depth below the surface where the water pressure is 2 atmospheres.

The human body can withstand a 1 atmosphere increase in pressure. What happens when you step out of a spaceship into a vacuum with no protective clothing?

Summarising:

Solids hold their shape; fluids take the shape of their container; plastics can be moulded; plasmas are ionised gases.
Lagrange's technique solves for the position and velocity of all fluid particles as functions of time.
Euler's technique specifies the density and velocity at each point in space.
Four parameters: Steady/Turbulent, Compressible/Incompressible, Viscous/Nonviscous, Rotational/Irrotational.
In steady flow the velocity at a point doesn't change in time, the trajectory of every particle follows the same path called a streamline.
Equation of continuity:

Bernoulli's equation:

A static fluid has zero velocity everywhere, so

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