# Deriving Stoke's Law for Settling Velocity

### Key Takeaways

If particles or components are present in a fluid, they accelerate until the friction equals the net force of gravity.

The magnitude of settling velocity gives an idea of the relative velocity of the particle and the fluid.

According to Stoke’s law for settling velocity, when the density of the particle is greater than the fluid density, it falls.

*The movement of sediments or particles in a fluid is associated with its settling time*

Components or particles present in fluids are sometimes separated; some examples of this include oil and gas separation and the separation of sediments from a liquid. The movement of sediments or particles in a fluid is associated with settling time. The settling time is of great importance in the process of separation. The settling time of particles determines how much time is required for them to rise or fall a given distance. Based on the settling velocity of the particles, the method of separation changes.

The settling velocities are governed by four laws based on the particle diameter and the type of system. Stoke’s law for settling velocity is associated with smaller particle diameters.

## Separating Components in a Fluid

Separating components present in a fluid can be either complex or simple, depending on the color, shape, size, etc. Physical differences in the components are mostly utilized for manual separations. The significance of physical as well as chemical differences is considered for equipment-based separation.

## Settling Velocity

Settling velocity is of great significance when it comes to the separation of components present in a fluid. If particles or components are present in a fluid, they accelerate until the friction equals the net force of gravity. Whenever the drag force or the friction force acting on the moving particle in the fluid equals the particle's weight, the particle starts falling at a constant rate. The particle ceases accelerating and moves with constant velocity. This constant velocity at which the particle moves in a fluid is called settling velocity or terminal fall velocity. At settling velocity, the acceleration of the particle is zero.

## Size of Particles and Settling Velocity

The size of the particles influence the settling velocity. When the particles have smaller dimensions, they reach their settling velocity faster. As the size increases, the time taken to reach settling velocity increases. Settling velocity is a measure to determine how much time is taken by the particle to either rise or fall a given distance.

The magnitude of settling velocity gives an idea of the relative velocity of the particle and the fluid. The size of the particle suspended in the fluid directly influences the settling velocity. For example, consider suspended particles less than 2 microns. The settling velocity may be reached faster and the value is extremely low. The low settling velocity makes 2-micron particles permanent suspensions.

From the discussion so far, it can be concluded that the settling velocity is dependent on the particle size. The settling velocity can be determined using different laws, and determining which law should be applied depends on the particle diameter.

The laws governing settling velocity calculation in the order of decreasing particle size are:

**Newton’s Law****Intermediate Law****Stoke’s Law****Stoke’s Cunningham Law**

Let’s explore Stoke’s law for settling velocity.

## Stoke’s Law

Stoke’s law is derived based on the forces acting on a component or particle in the fluid as it moves through the fluid under gravitational force. Stoke’s law expresses the drag force or the frictional force that prevents the fall of the particle (mostly spherical) through a fluid.

### Stoke’s Law for Settling Velocity

To quantify settling velocity, the fluid’s drag force is balanced with the weight of the particle suspended in it. The precise determination of the settling velocity requires knowledge about the fluid’s drag. Stoke’s law describes the fluid drag and gives the mathematical expression to calculate drag. According to Stoke’s law, the fluid drag force acting on a spherical particle of radius r can be given as:

is the viscosity of the fluid and v is the free stream velocity.

The suspended particles in the fluid accelerate until the net forces equal zero. The forces acting on a particle moving through a viscous fluid are drag force, gravitational force, and buoyant force.

is the density of the particle and g is the acceleration due to gravity.

σ is the density of the fluid.

The equilibrium condition is reached when:

Substituting equations 1, 2, and 3 into equation 4 and rearranging, we get the settling velocity:

Equation 5 describes Stoke’s law for settling velocity. When the density of the particle is greater than the fluid density, it falls; otherwise, it rises. The settling velocity derived from Stoke’s law is directly proportional to the square of the radius of the particle and inversely proportional to the viscosity of the fluid.

Stoke’s law for settling velocity is essential to use in applications such as separating sediments from fresh water, oil and gas separation, and measuring the viscosity of fluids. However, Stoke’s law for settling velocity is limited in the presence of turbulence. To determine the settling velocity under the influence of turbulence, you should rely on other equations and methods.

Subscribe to our newsletter for the latest CFD updates or browse Cadence’s suite of CFD software, including Fidelity and Fidelity Pointwise, to learn more about how Cadence has the solution for you.