Question

The Stokes-Oseen formula for drag force F on a sphere of diameter D in a fluid stream of low velocity V, density p and viscosity μ is
F=3πμDV+9π/16∗pV2d2
Is this formula dimensionally homogenous?

Answer 1

Answer:

$\frac{ML}{T^2}=\frac{ML}{T^2}$

Hence it is proved that Stokes-Oseen formula is dimensionally homogenous.

Explanation:

For equation to be dimensionally homogeneous both side of the equation must have same dimensions.

For given Equation:

F= Force, μ= viscosity, D = Diameter, V = velocity, ρ= Density

Dimensions:

$F=\frac{ML}{T^2}$

$\mu=\frac{M}{LT}$

$D=L\\\\V=\frac{L}{T}\\ \\\rho=\frac{M}{L^3}$

Constants= 1

Now According to equation:

$\frac{ML}{T^2}=[\frac{M}{LT}][L] [\frac{L}{T}] + [\frac{M}{L^3}][\frac{L^2}{T^2}][L^2]$

Simplifying above equation, we will get:

$\frac{ML}{T^2}=2*\frac{ML}{T^2}$

Ignore "2" as it is constant with no dimensions. Now:

$\frac{ML}{T^2}=\frac{ML}{T^2}$

Hence it is proved that Stokes-Oseen formula  is  dimensionally homogenous.