Solve the velocity field of a rigid sphere falling under gravity within a larger sphere

Question

A rigid sphere of radius R₁ is falling at a velocity Ui under the action of gravity gi within a larger stationary hollow sphere of radius R_o. The inner sphere is at an instant in time centered on the outer sphere as shown in the attached Figure. The space between the spheres is filled with a fluid of viscosity μ and density ρ. Assuming creeping flow conditions:

a)Solve for the velocity field in the fluid, the dunamic pressure and the fall velocity U_i for a given fluid density ρ < ρ_solid . Define Δρ = ρ_solid - ρ .

b)Plot absolute value |U_i| / |U_i^stokes| vs R_o/R₁ where U_i^stokes is the Stokes sedimentation velocity for a sphere sedimenting in an unbound fluid viz.

U_i^stokes= (2 R₁² Δρg_i )/ 9μ . Complete the plot for R_o/R₁ ≥ 1.1. Briefly describe on how the confinement effects the sedimentation speed.

Answer 1

The given problem deals with the motion of a rigid sphere falling within a larger stationary hollow sphere under the influence of gravity. We are interested in determining the velocity field, dynamic pressure, and the fall velocity of the sphere within a fluid medium, considering creeping flow conditions.

According to Stokes' law, in the case of creeping flow, the drag force acting on a sphere is proportional to the velocity of the sphere, the viscosity of the fluid, and the size of the sphere. The equation that relates these variables is:

Fdrag = 6πμR1V,

where Fdrag is the drag force, μ is the fluid viscosity, R1 is the radius of the inner sphere, and V is the velocity of the sphere.

In the vertical direction, the gravitational force acting on the sphere is given by:

Fgravity = (4/3)πR1^3Δρg,

where Δρ = ρsolid - ρ is the density difference between the solid sphere and the fluid, g is the acceleration due to gravity, and ρsolid is the density of the solid sphere.

Under steady-state conditions, the drag force and gravitational force are balanced, resulting in the following equation:

6πμR1V = (4/3)πR1^3Δρg.

Simplifying the equation, we can solve for the velocity V:

V = (2R1^2Δρg) / (9μ).

This equation provides the fall velocity of the sphere within the fluid as a function of the density difference, gravitational acceleration, radius of the sphere, and fluid viscosity.

To analyze the effect of confinement, we consider the ratio Ro/R1, which represents the relative sizes of the outer and inner spheres. For Ro/R1 ≥ 1.1, the confinement effect becomes significant. As the confinement increases, the fall velocity of the sphere decreases. This is because the presence of the outer sphere restricts the motion of the inner sphere, resulting in a reduced velocity.