Fluid dynamics

  

Upthrust and Viscosity

Density

·         Density, ρ is defined as the mass per unit volume. It is measured in
Kgm^-3.

Density: Mass per unit volume

Upthrust

·         A fluid will exert a force upward on a body if it is partly or wholly submerged within it. This is because the deeper into a fluid you go, the greater the weight of it and so the greater the pressure. This difference in pressure between the top and the bottom of the object produces an upward force on it. This is called Upthrust.

·         According to Archimedes' Principle, the upthrust on an object in a fluid is equal to the weight of the fluid displaced. So the volume of the object multiplied by the density of the fluid.

Upthrust = Weight of Fluid Displaced

Viscosity

·         In a fluid, each 'layer' experts a force of friction of each other 'layer'. This frictional force is also present when solid object moves through a liquid. This force is termed Viscous Drag. Viscous Drag is greater in Turbulent Flow than Laminar Flow.

·         The size of the Viscous Drag in a fluid depends on the (coefficient of)Viscosity of that fluid. Viscosity is given the letter η and is measured inKgm^-2s or Pa s. The greater the Viscosity, the greater the Viscous Drag.

·         In most liquids, Viscosity decreases as temperature increases, whereas in most gases, Viscosity increases as temperature increases. It is therefore important to always measure the temperature of a fluid when measuring Viscosity.

·         It is possible to calculate the drag force exerted on a spherical object in a fluid using Stoke's Law:

F = 6πηrv

·         Stoke's Law assumes Laminar Flow, and so low velocities.

·         In this equation, v represents Terminal Velocity. This means that the forces acting on the object are balanced. This means that is it possible to form an equation be equating Weight with Upthrust and Viscous Drag (or, in the case of Upward Motion, Upthrust with Weight and Viscous Drag). Laminar flow With solids, friction is the force that opposes motion between two surfaces pressed together. In fluid flow, viscosity is the force that opposes motion in and of a fluid. The easiest case to consider is laminar flow.

Laminar flow occurs when a fluid can be pictured as split into thin layers which slide smoothly over each other.
The thin layers (or laminas) are held back by viscous drag between the surfaces of the layers.

For example, if two flat solid plates are separated by a viscous fluid, an external force is needed to slide the top plate at constant speed over the fixed lower plate. Viscous drag acts not only between the fluid and the upper plate but also between adjacent laminas of fluid.

The velocity of the laminas differs by a small amount from the layers on each side as shown in the diagram. 
The velocity decreases uniformly from the upper plate speed to zero at the lower plate.

Laminar flow of a fluid in a tube

Laminar flow in a tube can be thought of as co-axial tubes sliding past each other. The velocity profile for the sliding tubes has the fluid moving slowly on the outside and quickly at the centre.

Even though the fluid elements travel in a straight line, the flow is rotational because a small paddle wheel placed in the tube (anywhere but the exact centre) will rotate. The rotation is due to the difference in velocities.

Newton's law of Viscosity

Shear stress is the tangential force (in the plane of the lamina) divided by the area across which the force acts (the area vector is at right angles to the plane of the lamina). Shear strain is the distance moved in the direction of the force divided by the perpendicular distance that separates the opposing forces producing the "twist".

For solids, shear stress divided by shear strain gives an elastic modulus.

For viscous liquids, since the strain is increasing all the time, shear stress divided by the rate of shear strain gives the viscosity coefficient.

In particular, for the simple two flat plates geometry

The force depends directly on the area of the plates and the velocity gradient between the plates. This is known as Newton's Law of Viscosity. Force/Area has the units of pressure, i.e. Pascal (Pa). Velocity/distance has the units of s^-1. Therefore, the viscosity coefficient has the unit of Pa.s. There is an older unit called the poise, 1 poise = 0.1 Pa.s.

Viscosity coefficients for some fluids

Fluid	Temperature	Coefficient
CO2	20°C	15 μPa.s
Air	20°C	18 μPa.s
Petrol	20°C	290 μPa.s
Water	90°C	320 μPa.s
Water	20°C	1 mPa.s
Blood	37°C	2 mPa.s
Motor Oil	20°C	0.03 Pa.s
Motor Oil	0°C	0.11 Pa.s
Glycerine	20°C	1.5 Pa.s
Typical Polymers*	Tg+20°C	10 GPa.s
	Tg+10°C	100 GPa.s
	Tg	100 TPa.s
Soda Lime Glass	800°C	1 MPa.s
	515°C	100 TPa.s

*Tg: See below. The glass transition temperature is defined for the situation where there is a smooth transition from liquid to solid. At the glass transition temperature the material is considered to become solid.

Example F5
An airtrack supports a cart that rides on a thin cushion of air 1 mm thick and 0.04 m² in area. The viscosity of air is 18 x 10^-6 Pa.s. Find the force required to move the cart at a constant speed of 0.2 m.s^-1

Viscosity at the Atomic Level

At the atomic level, viscous movement occurs when tightly bonded molecular units flow past each other. For this to occur, there has to be weak bonding between the units. This means that viscosity is important in materials with secondary bonding between molecular units. Also, materials with a large number of defects will have similarly weakened bonding.

Materials with secondary bonding:

Material	Bonding
polar liquid (water)	Hydrogen
molten metals	Ion/Electron
polymers	Van der Waals

Viscosity and Temperature

The bonds between atoms in the solid state are due to a large negative potential energy and a small positive kinetic energy.

At absolute zero (0K, -273°C) there is no thermal motion, i.e. no kinetic energy, to assist bond breaking.
As the temperature increases, atoms are given more and more kinetic energy so the total energy of the bond becomes less negative. The bonds between atoms weaken with increasing temperature.

When the potential energy and kinetic become comparable a solid can change phase into a liquid.

When the kinetic energy dominates then there can be a change of phase into a gas.

The phase changes are usually abrupt, but in some situations there can be a gradual change and a smooth transition between solid and liquid.

Above 1600° SiO₂ is a liquid with its tetrahedral base units in random motion. If liquid SiO₂ cools quickly then the tetrahedral units do not have time to move to their lowest energy configuration and the tetrahedrons are linked in random orientations by secondary bonds, thus forming a glass. If liquid SiO₂ cools slowly then the tetrahedrons can jostle past each other into their lowest energy configurations and link with long range order to form a crystal.

The change to a crystal is abrupt with a sudden increase in the viscosity coefficient. The change to a glass is smooth with a gradual increase in viscosity. At some point there will be a large increase in viscosity and this is called the glass transition temperature (T_g in the above graph) and the material is then said to be a solid.

Polymers and Temperature

Polymers are long chains of atoms joined together with primary bonds but cross linked between chains with secondary bonds. Temperature has a stronger effect on the secondary bonds than the primary bonds.

At low temperatures, polymers like most materials, are brittle. There is an elastic region ending in brittle fracture. At normal temperatures polymers have a non-linear elastic region with a smaller modulus (slope) and a yield point followed by limited plasticity before fracture. At warm temperatures the modulus decreases further and there is a larger easy glide region due to secondary bonds breaking and reforming. The chains tend to align with the applied stress. At high temperatures there is viscous flow.

Heavily cross-linked polymers, like rubber, may have different properties such as non-linear elasticity and high yield points.

Viscosity and Time

Time is not usually a factor for solids, but provides opportunities for bond breaking when secondary bonds are important. If the atoms are moving apart at a particular instant of time then the probability of the bond between them breaking is enhanced.

Example F6
A glass slab at room temperature, with dimensions 140 mm × 60 mm × 20 mm, has a shear force of 8.4 kN across its largest faces. The viscosity of this glass at room temperature is 10⁺¹⁶ Pa.s and the average separation between molecules is 0.5 nm. Find the time for two neighboring molecules to slide past each other.

Answer F6

Note that this is independent of the atom spacing.

Drag Force and Terminal Velocity

An object moving through a viscous fluid has a resistivity drag force exerted on it by the fluid. This prevents the object's velocity from increasing without limit (e.g. cars, boats) as it eventually reaches the maximum applied force. It means there is a terminal speed which is the maximum speed for the given conditions.

In general, , where x starts at 1 for low speeds and increases to 2 ,4 etc for high speeds.

The drag force is affected by shape and speed, through the drag coefficient, b and, v. 

For a car at typical speeds drag force is given by:

• CD is the drag coefficient produced      by the shape of the car. • A is the "front-on" cross-sectional area. • ρ is the density of the fluid. • v is the speed of the car.

For most cars C_D lies between 0.2 and 0.5 (the Model T Ford was 0.7).

Drag coefficients for some passenger vehicles 

Vehicle (class)	CD	CD×A (m2)
VW Polo (class A)	0.37	0.636
Ford Escort (class B)	0.36	0.662
Opel Vectra (class C)	0.29	0.547
BMW 520i (class D)	0.31	0.649
Mercedes 300SE (class E)	0.36	0.785

Including drag, the resultant force on a car is given by:

The initial acceleration causes the speed to increase from zero, which in turn causes the acceleration to decrease. Equilibrium is achieved when the drag force equals the force the engine can provide, the acceleration goes to zero and a constant (terminal) speed is reached.

Stokes' Law and Terminal Speeds

Stokes calculated the drag force on a sphere at low speeds as , where η is the viscosity, r is the radius, and v, is the speed.

Terminal speed for a sphere falling under gravity in a medium such as air or water.

As indicated before, when the acceleration goes to zero, the speed goes to a constant (terminal) value.

Terminal Speeds in Air

Object	Terminal Speed [m.s-1]	95% distance [m]
7kg shot put	145	2500
skydiver	60	430
baseball	42	210
tennis ball	31	115
basketball	20	47
ping-pong ball	9	10
1.5mm rain drop	7	6
parachutist	5	3

The "95% distance" is the distance to achieve 95% of the terminal speed.

Example F7
An oil drop has a density of 930 kg.m^-3. The terminal velocity of a spherical drop of this oil falling in air at 20°C is 0.18 m.s^-1. At 20°C, air density is 1.2 kg.m^-3 and its viscosity is 18 μPa.s. Find the radius of the droplet.

Answer F7

Poiseuille's Law and Laminar flow in a tube

The French prounciation of this "law" is something like "Pwasweeyer".

Consider a solid cylinder of viscous fluid, (viscosity η), flowing inside a hollow cylindrical pipe of length, L, and internal radius, R, as shown below. The flow is driven by a pressure difference, ΔP, and can be modelled as a number of thin co-axial cylinders flowing past each other. There will be a stationary thin cylinder at the outer edge and the maximum speed cylinder will be at the centre. The velocity profile will be parabolic.

The volume flux (flow rate) is given by:

In this course you will not be asked to derive this formula. .

Note that this depends directly on the pressure difference and inversely on the length. It also depends on the 4th power of the radius, so that will be a dominant factor.
• The larger the pressure difference, the greater the flux.
• The larger the cross-sectional area, the greater the flux.
• The shorter the length, the greater the flux.

Note: blood is a viscous fluid but it does not follow Poiseulle's equation because it has platelets in its plasma.

Example F8
A small pipe has an inner radius of 4 mm. A fluid with a viscosity of 4 x 10^-3 Pa.s flows through it at a rate of 10^-6 m³.s^-1. Find the pressure difference across a 2 m length of the pipe.

Reynold's number and Turbulent flow

In turbulent flow the fluid paths change abruptly and unpredictably with time.

This picture on the right shows water flows from a tap that are laminar on the left and turbulent on the right.
This picture on the right shows cigarette smoke rising. It starts as laminar flow and then changes to turbulent flow.

Poiseuille's law only holds for laminar flow. For turbulent flow you need to find experimentally what rules apply to the specific situation.

Reynolds’s number

Reynolds’s number represents the ratio of driving force to viscous force.

From Newton's law of motion, the definition of density and the equation of continuity, the driving force on the fluid is given by:

From Newton's law of viscosity, the viscous drag force is given by:

The ratio of these gives Reynold's number:

In turbulent flow, the driving force will dominate and in laminar flow the viscous force will dominate.

Thus Reynold's number  as a ratio of these, gives an indication as to whether a fluid flow is laminar or turbulent.

A "rule of thumb" for laminar vs turbulent flow.

NR< 2000	Laminar flow
2000 <NR< 3000	Unstable, may flip between laminar and turbulent
NR > 3000	Turbulent flow

Example F9

A pipe with diameter 300mm has water (density 1000 kg.m^-3, viscosity 1 mPa.s), flowing through it. Find Reynolds’s number for the flow when there is a:
(a) flow speed of 3 mm.s^-1 
(b) flow speed of 30 mm.s^-1 

Answer F9

A ten-fold increase in speed changes the flow from laminar to turbulent.

Summarizing:

Laminar flow occurs when a fluid can be pictured as split into thin layers which slide smoothly over each other.
Laminar flow can be in planes or cylinders
Newton's law of Viscosity:
Viscosity is important in materials with secondary bonding between tightly bound molecular units.
The bonds between atoms weaken with increasing temperature.
Time provides opportunities for bond breaking.
Viscous Drag force for a car at typical speeds
Stokes' Law:
Terminal speed for a sphere falling under gravity:
Poiseuille's Law:
Reynold's number: