## Dimensions and Homogeneity

Consistency in equations means the units in formulae have to be the same on both sides. For example:

distance=speed*time

m=m/s*s=m so the units on both sides are the same

The equation distance=speed/time is not possible because the units on both sides are not the same.

distance=speed/time

m<>m/s/s=m/s ^{2 }

Some equations can be written down with this in mind.

For instance the Reynolds number, important in studies of fluid motion, could be written in many different ways, but only equations of the form R=U*x*ρ/μ are homogeneous.

U=speed in m/s or m s ^{-1 }

x=distance in m

ρ=density in Kg m ^{-3 }

μ=viscosity in N s m ^{-2 }=Kg m s ^{-2 }s m ^{-2 }=Kg m ^{-1 }s ^{-1 }

Sometimes it is necessary to change units to the standard units of mass (kg), length (m) and time (seconds) and temperature (K) as I did above to write N as Kg m s ^{-2 } to make the equations homogeneous but this is also consistent.

Scientists might not know the equation for the Reynolds number, but they can do experiments to find the factors it depends on. So suppose they do these experiments and they find it depends on the speed U in m/s, the distance x in m, the density μ in Kg/m ^{3 } and the viscosity μ in Kg m ^{-1 } s ^{1 }.

They might assume an equation of the forum R=U ^{a }*x ^{b }*ρ ^{c }*μ ^{d }. Then a, b, c and d are found using dimensional analysis. We replace each quantity with their units,

U becomes m/s

x becomes m

ρ becomes Kg m ^{-3 }

μ becomes Kg m ^{-1 }s ^{-1 }

Since R has no units 0= (m s ^{-1 }) ^{a }*m ^{b }*(Kg m ^{-3 }) ^{c }(Kg m ^{-1 }s ^{-1 }) ^{d }. We can write down the equations, by equating powers of units,

For m, 0=a+b-3c-d

For s, 0=-a-d

For Kg, 0=c+d

Solving these equations gives us the powers.