Using Buckingham's Pi Theorem, determine the functional relationship for the resistance of a supersonic plane.

The variables involved are:

• Resistance, $R$ (Force)
• Aircraft Length, $L$
• Velocity, $V$
• Air Viscosity, $\mu$ (Dynamic Viscosity)
• Air Density, $\rho$
• Bulk Modulus, $K$ (Pressure)

Their fundamental dimensions (in terms of Mass [M], Length [L], Time [T]) are:

• $R = [M L T^{-2}]$
• $L = [L]$
• $V = [L T^{-1}]$
• $\mu = [M L^{-1} T^{-1}]$
• $\rho = [M L^{-3}]$
• $K = [M L^{-1} T^{-2}]$

• Number of variables, $n = 6$
• Number of fundamental dimensions, $m = 3$ (M, L, T)

According to Buckingham's Pi Theorem, the number of dimensionless groups is $n - m = 6 - 3 = 3$.

We need to choose $m=3$ repeating variables that collectively contain all fundamental dimensions (M, L, T) and do not form a dimensionless group among themselves. A common and suitable choice for fluid dynamics problems is:

• Density, $\rho = [M L^{-3}]$
• Velocity, $V = [L T^{-1}]$
• Length, $L = [L]$

Each $\Pi$ group is formed by combining a non-repeating variable with the repeating variables raised to unknown powers ($a, b, c$):
$\Pi_i = (\text{non-repeating variable}) \times \rho^a V^b L^c$

$\Pi_1$ (using Resistance, $R$):

Substitute dimensions:

Equating powers for M, L, T:

• For M: $1 + a = 0 \implies a = -1$
• For T: $-2 - b = 0 \implies b = -2$
• For L: $1 - 3a + b + c = 0 \implies 1 - 3(-1) + (-2) + c = 0 \implies 1 + 3 - 2 + c = 0 \implies 2 + c = 0 \implies c = -2$ Therefore,

$\Pi_2$ (using Air Viscosity, $\mu$):

Substitute dimensions:

Equating powers for M, L, T:

• For M: $1 + a = 0 \implies a = -1$
• For T: $-1 - b = 0 \implies b = -1$
• For L: $-1 - 3a + b + c = 0 \implies -1 - 3(-1) + (-1) + c = 0 \implies -1 + 3 - 1 + c = 0 \implies 1 + c = 0 \implies c = -1$ Therefore,

(This is the inverse of the Reynolds number, $Re = \frac{\rho V L}{\mu}$)

$\Pi_3$ (using Bulk Modulus, $K$):

Substitute dimensions:

Equating powers for M, L, T:

• For M: $1 + a = 0 \implies a = -1$
• For T: $-2 - b = 0 \implies b = -2$
• For L: $-1 - 3a + b + c = 0 \implies -1 - 3(-1) + (-2) + c = 0 \implies -1 + 3 - 2 + c =