Tutorial 1* (Solution)

1

1. The variables controlling the motion of a floating vessel through water are the drag force

gcF, the speed v, the length L, the density ρ and the viscosity μ of the water and the

gravitational acceleration g. Derive an expression for gcF by Dimensional Analysis.

1. (solution): First we set up the dimensional matrix.

gcF v L ρ μ g

M 1 0 0 1 1 0

L 1 1 1 -3 -1 1

t -2 -1 0 0 -1 -2

There are six parameters and the rank is three, so from Buckingham's π theorem, we will have

three dimensionless groups. We will isolate gc F, μ and g. (The reason for making this selection

is because that μ is a very important parameter in Fluid Mechanics and we wish to specifically

study how this parameter is affected by other parameters. gc F is a force and Fluid Mechanics is

part of mechanics and thus worthwhile to be studied. g being a factor that will contribute its

effect when we are involved in 3D. If we isolate this term and later work only in 2D, then we

may be able to discard the dimensionless group which is formed by g.)

Tutorial 1* (Solution)

2

2. The capillary rise h of a liquid in a tube varies with tube diameter d, gravity g, fluid

density ρ, surface tension σ, and the contact angle θ.

a) Find a dimensionless statement of this relation.

b) If h = 3cm in a given experiment, what will h be in a similar case if diameter and

surface tension are half as much, density is twice as much, and the contact angle

is the same?

2. (Solution): First we set up the dimensional matrix.

h d g ρ σ θ

M 0 0 0 1 1 0

L 1 1 1 -3 0 0

t 0 0 -2 0 -2 0

There are six parameters and the rank is three, so from Buckingham's π theorem, we will have

three dimensionless groups. As h and d are having the same dimension, so we have to isolate one

of them. We will isolate h, σ and θ.

We have that is

If σp = 2 σm , dp = 2 dm , ρp = 0.5 ρm , θp = θm . Assuming both cases are conducted under same

gravitational field, then gp = gm,

Tutorial 1* (Solution)

3

So the parameters in the function have not been changed at all, so the dimensionless group π1

should remain the same.