6.a Assuming the drag force, F, exerted on a body is a function of the
following
fluid density r
fluid viscosity m
diameter d
velocity u
Show that the drag force can be expressed as
where f is some unknown function and Re is the Reynolds number.
(10 marks)
6.b The diameter of a ball is 0.1m and it is required to predict the force it will experience when exposed to a flow of water travelling at 5 m/s. A sphere of 1.0 m diameter is available and can be exposed to a flow of air. To obtain the dynamically similar conditions at what velocity should this flow of air opperate?
(8 marks)
6.c Under these conditions the 1m diameter sphere exeperiences a force of 14 200 N. What force would you expect on the 0.1m diameter ball when in water flow?
(7 Marks)
m water = 1.0 ´ 10-6 kg/ms
m air = 1.7 ´ 10-5 kg/ms
r water = 1000 kg/m3
r air = 12.5 kg/m3
6.a.
Assume the governing variables r , u, d
According to Buckingham's p theorem there are n-m groups
where
n = number of variables (5) and
m = number of dimensions (i.e. MLT, giving 3)
n-m = 5-3 = 2 groups
Dimensions of the variables are:
r = density (kg/m3) = ML-3
m = viscosity (kg/m/s) = ML-1 T-1
u = velocity (m/s) = ML-1
d = length (m) = L
F = newtons (kg m /s2) = MLT-2
For p 1
For p 2
Inverting p 1 gives 
Rearranging this gives 
6.b
For dynamic similarity the Reynolds numbers are equal for both the water and air situation
6.c.
With dynamic similarity 
So
Information provided by: http://www.efm.leeds.ac.uk