MAS2803 FLUID DYNAMICS LECTURE 2

DR ANDREW BAGGALEY

• A physical dimension is a fundamental physical variable, for example, length and time. We quote the value of a physical dimensions in units, for example, metres and seconds.

• We should quote units where appropriate (usually SI units), otherwise the information is incomplete.

• From the basic dimensions/units of mass, length and time we can construct all of the other dimensions/units used in this module, for example, speed, acceleration and density.

Example:

Speed is length divided by time, so it has dimensions of

and

length

time

[speed ] =[length]

[time]=m

s

2.4 DIMENSIONS & UNITS - DIMENSIONAL ANALYSIS

• “Dimensional analysis" allows us to check physical equations.

• The equation should have the same dimensions/units on both sides.

4 oranges = 1 pig?

2.4 DIMENSIONS & UNITS - DIMENSIONAL ANALYSIS

• Example: An object’s position is given by x=at where a is it’s acceleration and t is time. Is this dimensionally correct?

x=at

• LHS: dimensions of length L

• RHS: dimensions of (L/t2)×t=L/t ⇒ The equation is dimensionally incorrect

(a dimensionally correct equation would be x=vt) • A useful sanity check if we are developing a new

model

2.4 DIMENSIONS & UNITS - CONVERTING BETWEEN UNITS

• In different situations we might want to (and we will in this course) use different units. • For cells and bacteria we might want to work in terms of nanometers

(1nm = 10-9 m) • For galaxies we might work in terms of light years (1 light year 9

trillion km) • If we move to America we might want to work with pints and feet,

not litres and metres. • Converting between units can be done straightforwardly and

algebraically

2.4 DIMENSIONS & UNITS - CONVERTING BETWEEN UNITS

≈

• Example: Let’s specify an area of 3m2 in nm2:

• We multiplied the original unit (m) by the fraction desired unit/desired unit (nm/nm).

• We manipulated and introduced the numerical conversion factor (1m/1nm=109).

• This approach can be extended as required to convert between any units, even utterly ridiculous ones….

2.4 DIMENSIONS & UNITS - CONVERTING BETWEEN UNITS

2.5 COORDINATES - CARTESIAN COORDINATES

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• Position vector:

• Units vectors:

• Line element vector:

• Surface element for a surface of constant z:

dS = dx dy • Volume element:

dV = dx dy dz

!x = (x, y , z)

x , y , z

d!" = dx x + dy y + dz z= (dx, dy , dz)

2.5 COORDINATES - CYLINDRICAL POLAR COORDINATES

• Position vector:

• Units vectors:

• Cartesian-polar relations:

x = r cosθ, y = r sinθ

which lead on to the useful relations:

x2+y2=r2, tanθ = y/x

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!x = (r, ", z)

r , !, z

• Line element vector:

• Surface element for surface of constant z:

dS = r dr dθ • Surface element of constant r:

dS = r dθ dz • Volume element

dV = r dr dθ dz

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d!" = dr r + rd# # + dz z= (dr, rd#, dz)

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2.5 COORDINATES - CYLINDRICAL POLAR COORDINATES

• Density is a measure of how much mass is in a given volume.

• Note that these are average densities; the density of substances can vary in space and time (particularly gases, which are easy to compress)

2.6 DENSITY

Density of some common substances

• Density is a measure of how much mass is in a given volume.

• Note that these are average densities; the density of substances can vary in space and time (particularly gases, which are easy to compress)

2.6 DENSITY

Density of some common substances

Oil floats on water, does alcohol?

• Then, for some volume V with varying density the mass is,

2.6 DENSITY

M =

!!!

V!("x) dV

! = !("x)

The mass dM of an infinitesimal volume dV is dM = ρ dV, where ρ is the density.

• If ρ is constant then we simply have that M = ρV.