MAS2803 FLUID DYNAMICS LECTURE 2 - Andrew...
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MAS2803 FLUID DYNAMICS LECTURE 2
DR ANDREW BAGGALEY
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• 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
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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?
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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
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2.4 DIMENSIONS & UNITS - CONVERTING BETWEEN UNITS
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• 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
≈
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• 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
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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)
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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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• 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
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• 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
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• 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?
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• Then, for some volume V with varying density the mass is,
2.6 DENSITY
M =
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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.