When a sphere falls freely through a homogeneous fluid, it reaches a terminal velocity at which the weight of the sphere is balanced by the buoyant force and the frictional resistance of the fluid. Make a dimensional analysis of this problem and indicate how experimental data for this problem could be correlated. Neglect compressibility effects and the influence of surface roughness.

Question

GIVEN

A sphere falling freely through a homogeneous fluid

Terminal velocity occurs when weight is balanced by buoyant force and friction resistance of the fluid

FIND

Make a dimensional analysis and indicate how data may be correlated

ASSUMPTIONS

Compressibility effects are negligible

Influence of surface roughness is negligible

SKETCH

Answer 1

The variables which must be correlated and their dimensions are shown below



The density difference was chosen for variable 2 because we anticipate that this difference, rather than

the sphere density, will be an important parameter. Clearly, if ps = pf, then U = 0. The Buckingham ?

Theoremcan be used to correlate the variables. There are 6 variables and 3

primary dimensions. Therefore, 3 dimensionless groups will be found.



Substituting the primary dimensions into the equation



Equating the sum of the exponents of each primary dimension to zero:

For M: b + c + f = 0 [1]

For L: a – 3b – 3c + d + e – f = 0 [2]

For t: 2a + d + f = 0 [3]

There are 6 unknowns and only 3 equations, therefore, the value of the 3 exponents can be chosen for

each ?

For ?1, Let a = 0, b = 0 and c = 1

From equation [1] f = – 1

From equation [3] d = 1

From equation [2] e = 1



For ?2, Let a = 1, b = 1 and f = 0

From equation [1] c = – 1

From equation [3] d = – 2

From equation [2] e =



For ?3, Let a = 0, b = 1, and f = 0



But this dimensionless group already appears in ?2. (This redundancy could have been avoided had we

chosen the weight of the sphere in the liquid in place of the two variables (?s – ?f) and g.) Therefore,

the experimental data for this problem could be correlated by