G.1 The resistance R experienced by partially. submerged body depends apon velocity (V), length (1), viscosity (4), density (5) and gravitational acceleration (g), obtain dimension less method using Backingham IT method.?

Question

Answer

Dimensionless Method using Buckingham Pi Theorem

First, let's list the variables involved in the resistance experienced by a partially submerged body:

- Velocity (V)
- Length (L)
- Viscosity (μ)
- Density (ρ)
- Gravitational acceleration (g)

Step 1: Identify the Dimensions

We can express the resistance R in terms of the variables as follows:

R = f(V, L, μ, ρ, g)

Now, we identify the dimensions of each variable:

- [V] = L/T
- [L] = L
- [μ] = M/LT
- [ρ] = M/L^3
- [g] = LT^-2

Step 2: Check for Dimensional Homogeneity

The dimensions of resistance R can be expressed as:

[R] = M/LT^2

Now, we check if the variables can be combined in such a way that the dimensions of each group are equal.

Step 3: Form Dimensionless Groups

Using Buckingham Pi Theorem, we can form dimensionless groups:

Π1 = R * V^a * L^b * μ^c * ρ^d * g^e

By equating dimensions, we get:

M^1 L^1 T^-2 = M^a L^a T^-a * L^b * M^c L^-c T^-c * M^d L^-3d * L^-b * L^-2e

Solving the equations, we obtain values for a, b, c, d, and e.

Step 4: Interpretation

The dimensionless groups obtained represent the non-dimensional parameters that govern the resistance experienced by a partially submerged body. These dimensionless groups help in understanding the relationship between the variables without worrying about their units.

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