154.Fluid Dynamics

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Number 154

Fluid Dynamics Explanation

We can split our look at fluid dynamics into two principal sections: 1. the flow of a fluid (gas or liquid) through a passageway e.g. a pipe 2. the flow of fluid past an object e.g. an aeroplane wing

We are getting worryingly close to chemistry here, but there are attractive forces between molecules. These forces are much greater for longer, more complex, molecules. So the molecules of the ethyl alcohol tend to “cling” to each other more strongly than the molecules of the simpler methyl alcohol, leading to greater viscosity.

It is worth noting that we are talking about relative motion. There will be more mention of this later. In both parts of this discussion, the effects observed depend very largely on the viscosity of the fluid. We shall look at this first.

And the molecules in a gas, being distant from each other most of the time, are far less affected by intermolecular attraction. The viscosity of a gas is very small.

Viscosity

A further point not illustrated in this table is that the viscosities of liquids tend to decrease markedly as the liquids become warmer.

Viscosity is simply defined as a fluid’s resistance to flow. Water flows easily, engine oil is “thick” and flows much more slowly. The oil is more viscous than the water.

Example 1: Why should the viscosity of a warm liquid be less than that of a cold liquid?

Do not confuse viscosity and density. Oil floats on water – the water has a greater density. Oil is “thicker” than water – the oil has a larger viscosity.

Answer: The molecules in a warm liquid have greater thermal energy (random kinetic energy). They are more easily able to overcome attractive intermolecular forces. As an example, water is about three times more viscous at 20ºC than at 90ºC.

oil water

Fluid flow through a pipe There is attraction between the pipe wall and the molecules of the liquid touching the wall. This leads to slower fluid flow near the walls.

water more dense water slope

For liquids of low viscosity flowing slowly through pipes, the flow achieved is streamlined or laminar.

oil v

slope

v

oil more viscous

walls of pipe

Laminar flow

If we compare some viscosities: Gases

Viscosity

Liquids

Viscosity

(20oC)

(× ×10-3 kg m-1 s-1) (0oC)

methyl alcohol

0.6

hydrogen 8

ethyl alcohol

1.1

oxygen

Laminar means “in layers”. If these layers can flow easily past each other (low viscosity and low speed), there is little energy loss.

(× ×10-6 kg m-1 s-1)

However, if the viscosity is greater, the intermolecular attraction between the layers in the fluid is significant, and turbulent flow takes place.

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isopropyl alcohol 2.4 water

1.0

olive oil

84

Turbulent flow

heavy machine oil 233

1. Gases have much less resistance to flow (viscosity) than liquids. Note the difference in the power of the units. 2. Liquids with complex molecules have a much larger viscosity than liquids with simple molecular structure.

Example 2: Suggest two results of turbulent flow. Answer: Slower flow rate, and transfer of kinetic energy to thermal energy.

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Physics Factsheet

154. Fluid Dynamics Bernoulli effect in a pipe

The aerofoil may be tilted slightly to increase the upward force as it moves forward into the air. However if this angle of attack is made too large, laminar flow above the wing changes into turbulent flow.

When an incompressible fluid flows into a narrower section of pipe, it must accelerate (to maintain uniform flow, in kgs-1, through the pipe – i.e. to get the same mass of fluid through a smaller space every second). A

turbulent air

B

pC

pA

Lift is lost, and the aircraft stalls. vA vC For acceleration to take place at B, the forward pressure at A must be greater than the opposing pressure at C (remember that pressure acts in all directions).

A hydrofoil produces lift through water, but the angle of attack must be very small. Otherwise the viscosity of the water would very soon cause laminar flow to become turbulent.

So we can conclude that higher velocity means lower pressure: pA > pC , vA < vC

Many of the results for fluid flow are only valid when the flow is laminar.

The Bernoulli equation can be derived: pA + ½ρvA2 = pC + ½ρvC2

Example 4: Why are houses in very windy areas built with steep roofs?

This equation makes use of conservation of mechanical energy, and only works for horizontal flow.

Answer: To cause the airflow over the house to become turbulent when it goes over the peak of the roof. Laminar flow would lead to the Bernoulli effect lifting the roof off the house.

Example 3: Explain why this equation only works for (a) streamlined flow, and (b) horizontal flow.

The Magnus effect

Answer: (a) Turbulent flow causes heating. Mechanical energy is not conserved. (b) Vertical changes would introduce pressure changes due to depth ( ρgh).

A ball with a rough surface thrown through the air will curve, if it is spinning. This is an illustration of the Magnus effect (a special case of the Bernoulli effect). In addition to the air passing both sides of the ball (v1), there will be a boundary layer of air spinning with the ball (v2).

Fluid flow past an object

V1

Relative motion

V2

We introduced this term earlier. It implies that it doesn’t matter if an object is moving through a stationary fluid, or the fluid is flowing past a stationary object.

V1+V2

L

F -V2

This is generally the case, and designers make use of this in wind tunnels. These can accurately predict the airflow past an object. The flow pattern is the same whether the air is flowing past the object, or the object is propelled through the air.

V1-V2

V2 airflows separated

H

combined airflows

The diagram shows how the resultant air speed above the ball will be greater than that below the ball, leading to lower pressure above the ball, and a resultant force. Object through stationary air

air blown past stationary object

The ball will curve.

Bernoulli effect with an aerofoil The aerofoil is shaped so that the air must travel a further distance over the top of the aerofoil, before joining up with the airflow below the aerofoil. This means it will travel faster above the aerofoil. L Lift H

Bernoulli’s ideas (explained earlier) predict that higher speed will mean lower pressure. The greater pressure below the wing produces an upward force (lift).

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Physics Factsheet

154. Fluid Dynamics Terminal speed

The resistive force depends on the shape and size of the object, its speed, and the viscosity,η, of the medium.

Objects fall at terminal speed through a medium when the gravitational force accelerating them downwards is balanced by the resistive force of the fluid. (This ignores upthrust on the object due to buoyancy – in general this is very small.)

For a sphere falling at low speed (to ensure laminar flow), Stokes Law gives the total resistive force: F = 6πηrv

F

At terminal velocity for the falling sphere, we have: mg = 6πηrvT This seems very precise,but remember we are ignoring buoyancy, limiting ourselves to a spherical shape, and assuming laminar flow. Many of the expressions we commonly use are only approximations to real-life situations.

mg TT

And now, some problems to attempt. The answers (but not necessarily the solutions) are provided below.

Practice Questions 1. It is noticed that turbulent flow is occurring when a fluid flows through a pipe. What could be done to reduce the turbulence and increase the flow rate? 2. The flow rate, in kgs-1, of a fluid through a pipe is given as: Flow rate = Aρv where A is the cross-sectional area of the pipe, ρ is the density of the fluid, and v is the average speed of the fluid. (a) A liquid has a density of 920kgm-3 and a maximum average speed without turbulence of 0.75ms-1 through a pipe of radius 1.4m. Find the flow rate of the liquid in kgs-1. (b) If the liquid is then warmed from 15oC to 35oC, its density decreases to 890kgm-3. Its viscosity also decreases, leading to a maximum average speed of 1.10ms-1. Find the new flow rate. (c) The system is rebuilt to operate at 35oC as before, but with a pipe of radius 2.0m. Nothing else changes. Find the new flow rate. (d) For the goodness of your soul, try to go from the result in (a) to that in (c) in one step, using ratios. 3. An incompressible liquid displays laminar flow through a horizontal tube decreasing in cross-section. If the density of the liquid is 1.20×103 kgm-3, and its speed increases from 1.3ms-1 to 1.6ms-1, find the change in pressure. 4. An oil drop of density 820kgm-3 and radius 0.012 mm falls through air (viscosity 1.8×10-5 kgm-1s-1). Assuming it has reached terminal velocity, how far would it fall in 5s?

mg 4. vT = 6πηr mg = volume × density × g = 7.24×10-12 × 820 × 9.81 =5.82×10-11 N. vr = 1.4 × 10-2 ms-1 = 14 mms-1 In 5s, it would fall 70mm 3. By considering the Bernoulli equation: pB – pA = 0.5 × ρ × (vA2 – vB2) = -520 Nm-2 (d) 4.2×103 ×

890 1.10 2.02 × × = 1.2×104 920 0.75 1.42

(c) Flow rate = 1.2×104 kgs-1 (b) Flow rate = 6.0×103 kgs-1 Acknowledgements: This Physics Factsheet was researched and written by Paul Freeman The Curriculum Press,Bank House, 105 King Street,Wellington, Shropshire, TF1 1NU Physics Factsheets may be copied free of charge by teaching staff or students, provided that their school is a registered subscriber. No part of these Factsheets may be reproduced, stored in a retrieval system, or transmitted, in any other form or by any other means, without the prior permission of the publisher. ISSN 1351-5136

2. (a) A = πr2 = 6.16m2 Flow rate = 4.2×103 kgs-1 1. Warm the fluid, decreasing the viscosity and increasing the probability of laminar flow.

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Answers