It is found experimentally that the terminal settling velocity u0 of a spherical particle in a fluid is a function of the following quantities: particle diameter, d; buoyant weight of particle (weight of particle _ weight of displaced fluid), W; fluid density, ρ, and fluid viscosity, μ. Obtain

Question

It is found experimentally that the terminal settling velocity [latex]u_0[/latex] of a spherical particle in a fluid is a function of the following quantities: particle diameter, d; buoyant weight of particle (weight of particle _ weight of displaced fluid), W; fluid density, ρ, and fluid viscosity, μ.
Obtain a relationship for [latex]u_0[/latex] using dimensional analysis.
Stokes established, from theoretical considerations, that for small particles which settle at very low velocities, the settling velocity is independent of the density of the fluid except in so far as this affects the buoyancy. Show that the settling velocity must then be inversely proportional to the viscosity of the fluid.

Accepted Answer

If: [latex]u_0=k d^a W^b \rho^c \mu^d[/latex], then working in dimensions of M, L and T:

[latex]( L / T )=k\left( L ^a\left( M L / T ^2\right)^b\left( M / L ^3\right)^c( M / L T )^d\right)[/latex]

Equating dimensions:

M : 0 = b + c + d
L : 1 = a + b - 3c - d
T : 1 = 2b - d

Solving in terms of b:

[latex]a=-1, c=(b-1)[/latex], and [latex]d=(1-2 b)[/latex]

∴ [latex]u_0=k(1 / d)\left(W^b\right)\left(\rho^b / \rho\right)\left(\mu / \mu^{2 b}\right)[/latex] where k is a constant,

or: [latex]u_0=k(\mu / d \rho)\left(W \rho / \mu^2\right)^b[/latex]

Rearranging:

[latex]\left(d u_0 \rho / \mu\right)=k\left(W \rho / \mu^2\right)^b[/latex]

where [latex]\left(W \rho / \mu^2\right)[/latex] is a function of a form of the Reynolds number.
For [latex]u_0[/latex] to be independent of ρ, b must equal unity and [latex]\underline{\underline{u_0=k W / d \mu}}[/latex]

Thus, for constant diameter and hence buoyant weight, the settling velocity is inversely proportional to the fluid viscosity.

Question 1.P.5: It is found experimentally that the terminal settling veloci...

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