# Mathematical Modelling

Suppose that the lift force F (M L T-2) on a missile depends on characteristic length scales D (L) and r (L) of the missile.

Additionally, F may depend on the air density ρ (M L-3),

the viscosity µ (M L-1 T-1) and missile velocity v (L T-1) .

a) Develop a model for the lift force F.

b) Find two other possibilities to represent the standardized lift force F in terms of non-dimensional products. (Buckingham's Theorem)

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Solution:

The lift force F on a missile, depends on

1. D [L]

2. r [L]

3. Air Density ρ [ML-3]

4. Viscocity μ [ML-1T-1]

5. Velocity v [LT-1]

Let,

(i) F (alpha) D^a

(ii) F (alpha) r^b

(iii) F (alpha) ro^c

(iv) F (alpha) u^d

(v) F (alpha) v^e

Combining all these factors, we get F = k D^a*r^b*ro^c*u^d*v^e [1]

where k is a dimensionless constant of proportionality

Writing down the dimensions on either side, we get

[M^1L^1T^-2] = [L]a [L]b [M1 L-3T0]c [M1 L-1T-1]d [M0 L1 T-1]e

[M^1L^1T^-2] = [M^(c+d) L^(a+b-3c-d+e) T^(-d-e)]

Equating powers of M, L and ...

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