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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 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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    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 [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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