Stoke's Law: Formula, Derivation and Application

Stokes' law is an empirical law that describes the frictional force, also known as drag force or viscous drag, exerted on spherical objects in a viscous fluid with very small Reynolds numbers.

• Viscous drag is the drag force experienced by an object traveling through a fluid due to the viscosity of the fluid.
• When a body falls on a liquid, the viscous drag opposes the motion of the body.
• This viscous drag is directly proportional to the velocity of the body that falls on the liquid.
• Stoke’s law is a mathematical equation derived by George Gabriel Stokes in 1851.
• The Stokes law only applies to liquids flowing in a laminar flow.
• It does not apply to tumultuous/turbulent flow.

According to Stoke’s law, the viscous drag felt by an object falling into a liquid is given by

F = 6πμrv

Where

• F is the drag force or viscous drag.
• μ is the coefficient of viscosity of the liquid.
• r is the radius of the object (sphere).

Key Terms: Viscosity, Coefficient of viscosity, Stoke’s law, Drag force, Viscous drag, Terminal velocity, Density, Buoyant force

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Stoke’s Law

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Stoke's Law is a mathematical equation that explains the velocity of small spherical particles in a fluid.

• The law is developed by considering the forces acting on a specific particle as it falls into a liquid column under the influence of gravity.
• The force that retards a sphere traveling through a viscous fluid is proportional to the velocity and radius of the sphere, and viscosity of the fluid.

When a body falls through a liquid, various forces acting on it are

• Viscous drag (F) in the upward direction
• Upward thrust (T) or buoyant force in the upward direction
• Weight of the body (W) in the downward direction

When the upward viscous force and buoyant force together balance the weight of the body, then the net force acting on the body becomes zero and the body moves down with constant velocity due to the inertia of motion. This constant velocity is known as Terminal velocity

Stoke’s Law

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Stoke’s Law Formula

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According to Stoke’s law, the viscous drag acting on a spherical body falling in a liquid is given by

F = 6πμrv

Where

• F is the drag force or viscous drag.
• μ is the coefficient of viscosity of the liquid.
• r is the radius of the object (sphere).

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Stoke’s Law Derivation

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The viscous force (F) exerted on a sphere falling in a liquid is directly proportional to the following parameters

• The velocity of the object (v)
• The radius of the sphere (r)
• Coefficient of viscosity of the liquid (μ)

This can be represented as

F ∝ μ^a r^b v^c

The proportionality sign can be removed by adding a constant term k, therefore

F = k μ^a r^b v^c   ….(i)

The dimensional formula of each parameter in equation (i) is given by

• Dimensional formula of viscous force, [F] = [M L T^-2]
• k is a constant, hence dimensionless.
• Dimensional formula of coefficient of viscosity, [μ] = [M L^-1 T^-1]
• Dimensional formula of radius, [r] = [L]
• Dimensional formula of velocity, [v] = [L T^-1]

Equation the dimensions of equation (i), we get

[M L T^-2] = [M L^-1 T^-1]^a [L]^b [L T^-1]^c

Simplifying the above equation, we get

[M L T^-2] = [M]^a [L]^-a+b+c [T]^-a-c

Comparing the superscripts of mass, length, and time on both sides of the above equation, we get

• a = 1
• -a+b+c = 1
• -a-c = -2

On solving the above three equations, we get

• a = 1
• b = 1
• c = 1

Substituting these values in equation (i), we get

F = k μ¹ r¹ v¹

⇒ F = k μrv

Experimentally the value of k is obtained as 6π, therefore the equation becomes

F = 6πμrv

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Terminal Velocity Formula

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The maximum constant velocity acquired by a body while falling through a viscous medium is called its terminal velocity.

The formula of terminal velocity is given by

\(v_t=\frac{2}{9}\frac{r^2}{\mu}(\rho-\sigma)g\)

Where

• v_t is the terminal velocity
• r is the radius of the sphere
• μ is the coefficient of viscosity
• σ is the density of the liquid
• ρ is the density of the body
• g is the acceleration due to gravity

Terminal velocity is directly proportional to the square of radius, that’s why bigger raindrops fall with a larger velocity as compared to the smaller drop.

Terminal velocity is inversely proportional to the viscosity of the medium i.e. for two similar bodies terminal velocity is larger in the medium which has a low viscosity.

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Stoke's Law Applications

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The following are the applications of Stoke’s law

• To measure the viscosity of a fluid:  It can be used to measure the viscosity of a fluid by determining the terminal velocity of a sphere falling through it.
• Settling of sediment in freshwater: According to Stokes' law, particle sedimentation velocity is proportional to the density difference between the solid and liquid phases, inversely proportional to liquid viscosity, and proportional to the square of particle diameter.
• To separate drilling mud: Stokes' law is also used to separate drilling mud from desired drilling material, such as limestone, in settling pits.
• API separator: The API separator (oil-water separator) is a gravity separation device that uses Stokes' law to determine the rise velocity of oil droplets based on density, size, and water properties.

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Things to Remember

• Stokes' law describes the viscous force exerted on spherical objects in a viscous fluid.
• Viscous drag is the drag force experienced by an object passing through a fluid due to the viscosity of the fluid.
• George Gabriel Stokes developed Stoke's law in 1851.
• According to Stoke’s law, the viscous drag acting on a body is given by F = 6πμrv
• Terminal velocity is the maximum constant velocity acquired by a body while falling through a viscous medium.
• The formula of terminal velocity is given by \(v_t=\frac{2}{9}\frac{r^2}{\mu}(\rho-\sigma)g\)