Lecture 2

Properties of FluidsUnits and Dimensions

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What is a fluid?• Is formed by molecules...

– that move, as in any other type of matter, above 0 kelvin;– the difference between a fluid and a solid is that in the

fluid the molecules can change their relative positions allowing them to get the shape of the containers;

– fluids can be liquids or gases.

• In gases molecules have free relative movement• In liquids molecules form groups with relative free

movement (allowing them to get the shape of the container) which dimension depends on temperature (influencing their viscosity)

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Why is Fluid Mechanics distinct from Solid Mechanics?

• In a fluid each molecule (or group of molecules) have relative movement freedom and in solids not. The consequence is that tangential stress deforms the fluids. Or in other words, if there is tangential stress there is movement.

• Normal stress compress the fluid, that can remain at rest. Tangential shear moves the fluid in layers creating velocity gradients.

Shear is proportional to the rate of deformation

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Elementary Volume• Is large enough to include a large number of molecules and

small enough to have uniform properties.

• If the elementary volume is of the order of the size of a molecule, the density value is uncertain due to the number of molecules it contains. If it is too large the density can change from one region to another.

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Continuum Hypothesis

• The elementary volume is much larger than 10 nm• Necessary because we cannot assess the movement of

individual molecules (too many and the Heisenberg principle)

• But they move individually.... – The unknown molecule movement will be dealt as diffusion

in the equations.

• When do we have velocity in a fluid?– When there is net mass transport across a surface.

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What is the velocity?• Velocity is the flux of volume per unit of area. • The Velocity is defined at a point and thus is the flow per unit of

area, when the area tends to zero:

• A surface can have 3 orientations in a tridimensional space and thus velocity can have up to 3 components.

• The velocity component in one direction is the internal product of the velocity vector by the unitary vector along that direction. Using the surface normal n :

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Discharge / Advective FluxKnowing the 3 velocity components and knowing that the velocity is the discharge per unit of area when the area tends to zero ( the velocity is defined at a point), we can compute the discharge across an area integrating the velocity along the whole area:

Defining a specific property as its value per unit of volume (when the volume tends to zero)

And the flux of M across a surface is:

We can say that the flux of M across an elementary surface is:

If c is uniform along the surface then:

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Density Dimensions and Units

• Dimensions

• Units:

SI C.G.S Gravitational (MKpS)

kg m-3 g cm-3 UMM m-3

A dimension is the measure by which a physical variable is expressed quantitatively. A unit is a particular way of attaching a number to the quantitative dimension.

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Specific Weight, Dimensions and Units

• Dimensions

• Units

SI C.G.S Gravitational (MKpS)

N m-3 dine cm-3

(gcms-2)cm-3

(10-3kg*10-2ms-2)(10-2m)-3

dine cm-3

= 10 Nm-2= 1 kgm-2

kg m-3

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Other properties

• Temperature, is a measure of molecule kinetic energy

• Specific Gravity (dimensionless)

SI C.G.S Gravitational (MKpS)

k k k

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Viscosity

When a fluid is sheared, it begins to move at a strain rate inversely proportional to a property called its coefficient of viscosity μ

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Viscosity is responsible for the shear stress which inhibits the relative movement of adjacent elementary fluid volumes. Is null if there is no velocity gradient, so it destroys gradients, i.e. generates momentum diffusion.

Viscosity

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Viscosity Units

SI C.G.S Gravitational (MKpS)

N m-2s=kgm-1s-1

(Poiseuille)

dine cm-2s = gcm-

1s-1

(Poise=10 Poiseuille)

kg m-2s=UMMm-1s-1

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Kinematic Viscosity

SI C.G.S Gravitational (MKpS)

m2s-1 cm2s-1 m2s-1

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No Slip Condition

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Flow between two plates

Shear decreases as h increases! Consequences for aquaplaning?

Why Constant?

hV

dydu

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Viscosity variability

• Varies much with temperature and weekly with

pressure.

• In liquids:

– Decreases with temperature

• In gases:

– Increases with temperature

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Surface Tension

• Exists because intermolecular attraction forces between liquid molecules are higher than between the liquid molecules and air.

• Is responsible for drop formation and for pressure increase inside the drop.

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Spherical droplet

• Surface tension decreases with temperature. This is why solubility increases with temperature.

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Capilarity

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Summary

• We have seen the main fluid properties and the

respective dimensions and units.

• Viscosity is relevant when there is movement and

velocity gradients are present.

• In the next lecture we will see properties of the flow

field.

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Velocity, momentum, kinetic energy

Momentum is the result of the action of a Force applied to a volume for a certain time. A small force acting for long time can generate a large momentum.

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Momentum and kinetic energy

• Momentum is often called impulse. If F is constant in time:

12

2

1

2

1

2

1

vvMvMddtdtvd

MdtFv

v

t

t

t

t

• If the initial velocity was null and this quantity was computed per unit of volume one would get:

vvMdVol

dtFVol

Vt

t

0

2

1

11

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Other properties computed from velocity

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Dimensions consistency/homogeneity

• In one equation all terms must have the same

dimensions.

• In Fluid mechanics this norm is usually respected.

• In many empirical laws (often in natural sciences) it is

not, being a source of errors.

• Manning formula for friction is a tricky example of a

non-homogeneous formula.

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Summary

• Knowing the velocity at each instant and point one can

compute all other flow properties, including forces,

using the Newton Law.

• This is why in fluid mechanics a major goal is to

describe the flow pattern, i.e. the velocity distribution.