4. The Differential Equations for a Sphere Moving through a Fluid
We will now develop the differential equations for a sphere moving vertically through a fluid under the influence of gravity and a drag force. Let y[t] and v[t] be the height and velocity, respectively, of the sphere at time t. Recall that the direction of the drag force is always opposite the direction of motion. Therefore, an upwardly moving sphere will have a downward drag force (see Figure 2a) and a downwardly moving sphere will have an upward drag force (see Figure 2b). Gravitational force is always downward; hence, it is negative.
Figure 2:
Up to The Drag Force on a Sphere
4.1 The differential equation for Re < 0.5