As depicted in Fig. P1.22, a spherical particle settling through a quiescent fluid is subj...

As depicted in Fig. P1.22, a spherical particle settling through a quiescent fluid is subject to three forces: the downward force of gravity (F_G), and the upward forces of buoyancy (F_B) and drag (F_D). Both the gravity and buoyancy forces can be computed with Newton’s second law with the latter equal to the weight of the displaced fluid. For laminar flow, the drag force can be computed with Stoke’s law,

where μ = the dynamic viscosity of the fluid (N s/m²), d = the particle diameter (m), and υ = the particle’s settling velocity (m/s). The mass of the particle can be expressed as the product of the particle’s volume and density, ρ_s (kg/m³), and the mass of the displaced fluid can be computed as the product of the particle’s volume and the fluid’s density, ρ (kg/m³). The volume of a sphere is πd³/6. In addition, laminar flow corresponds to the case where the dimensionless Reynolds number, Re, is less than 1, where Re = ρdυ/μ.

(a) Use a force balance for the particle to develop the differential equation for dυ/dt as a function of d, ρ, ρ_s, and μ.

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(b) At steady-state, use this equation to solve for the particle’s terminal velocity.

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(c) Employ the result of (b) to compute the particle’s terminal velocity in m/s for a spherical silt particle settling in water: d = 10 μm, ρ = 1 g/cm³, ρ_s = 2.65 g/cm³, and μ = 0.014 g/(cm·s).

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(d) Check whether flow is laminar.

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(e) Use Euler’s method to compute the velocity from t = 0 to 2⁻¹⁵ s with Δt = 2⁻¹⁸ s given the initial condition: υ(0) = 0.

Figure. P1.22