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# Mathematical Modelling

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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 &#961; (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)

https://brainmass.com/math/geometry-and-topology/solving-mathematical-modelling-problem-274098

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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 
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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