Answer:
To obtain the expression for the velocity of the stream-line flow, we can use dimensional analysis. Let's consider the following variables:
Velocity (v)
Radius (r)
Density (ρ)
Coefficient of viscosity (η)
The dimensions of these variables are:
[v] = L/T
[r] = L
[ρ] = M/L^3
[η] = M/LT
Using the Buckingham Pi theorem, we can form the following dimensionless groups:
π1 = v/(r^a ρ^b η^c)
where a, b, and c are unknown exponents that we need to determine. To do this, we can equate the dimensions of both sides of the equation:
L/T = L^1-a M^b L^-3b T^-c
Equating the dimensions of length, mass, and time, we get:
1 - a = 0 -> a = 1
b = 0
-c = 1 -> c = -1
Therefore, the dimensionless group becomes:
π1 = v/(r ρ η^-1)
We can rewrite this as:
v = K r ρ η^-1
where K is a constant that we need to determine. To find K, we can use the given equation:
K = 2ρ/4η = 1/2η
Substituting this value of K in the expression for v, we get:
v = (r/2) (η/ρ)
Therefore, the expression for the velocity of the stream-line flow is:
v = (r/2) (η/ρ)
where v is the velocity, r is the radius of the tube, η is the coefficient of viscosity of the fluid, and ρ is the density of the fluid.