Stokes' law

This article is about the expression for frictional force. For the acoustics theorem, see Stokes' law (sound attenuation).

Not to be confused with Stokes' theorem.

File:Stokes sphere.svg

Creeping flow past a sphere: streamlines, drag force F_d and force by gravity F_g.

In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers (e.g., very small particles) in a continuous viscous fluid. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the generally unsolvable Navier–Stokes equations:^[1]

\[F_d = 6 \pi\,\mu\,R\,v_s\,\]

where:

F_d is the frictional force acting on the interface between the fluid and the particle (in N),
μ is the dynamic viscosity (N s/m²),
R is the radius of the spherical object (in m), and
v_s is the particle's settling velocity (in m/s).

If the particles are falling in the viscous fluid by their own weight due to gravity, then a terminal velocity, also known as the settling velocity, is reached when this frictional force combined with the buoyant force exactly balance the gravitational force. The resulting settling velocity (or terminal velocity) is given by:^[2]

\[v_s = \frac{2}{9}\frac{\left(\rho_p - \rho_f\right)}{\mu} g\, R^2\]

where:

v_s is the particles' settling velocity (m/s) (vertically downwards if ρ_p > ρ_f, upwards if ρ_p < ρ_f ),
g is the gravitational acceleration (m/s²),
ρ_p is the mass density of the particles (kg/m³), and
ρ_f is the mass density of the fluid (kg/m³).

Stokes' law makes the following assumptions for the behavior of a particle in a fluid:

Laminar Flow
Spherical particles
Homogeneous (uniform in composition) material
Smooth surfaces
Particles do not interfere with each other

Note that for molecules Stokes' law is used to define their Stokes radius.

The CGS unit of kinematic viscosity was named "stokes" after his work.

Applications

Stokes's law is the basis of the falling-sphere viscometer, in which the fluid is stationary in a vertical glass tube. A sphere of known size and density is allowed to descend through the liquid. If correctly selected, it reaches terminal velocity, which can be measured by the time it takes to pass two marks on the tube. Electronic sensing can be used for opaque fluids. Knowing the terminal velocity, the size and density of the sphere, and the density of the liquid, Stokes' law can be used to calculate the viscosity of the fluid. A series of steel ball bearings of different diameters are normally used in the classic experiment to improve the accuracy of the calculation. The school experiment uses glycerine or golden syrup as the fluid, and the technique is used industrially to check the viscosity of fluids used in processes. Several school experiments often involve varying the temperature and/or concentration of the substances used in order to demonstrate the effects this has on the viscosity. Industrial methods include many different oils, and polymer liquids such as solutions.

The importance of Stokes' law is illustrated by the fact that it played a critical role in the research leading to at least 3 Nobel Prizes.^[3]

Stokes' law is important to understanding the swimming of microorganisms and sperm; also, the sedimentation, under the force of gravity, of small particles and organisms, in water.^[4]

In air, the same theory can be used to explain why small water droplets (or ice crystals) can remain suspended in air (as clouds) until they grow to a critical size and start falling as rain (or snow and hail). Similar use of the equation can be made in the settlement of fine particles in water or other fluids.

Stokes flow around a sphere

Steady Stokes flow

In Stokes flow, at very low Reynolds number, the convective acceleration terms in the Navier–Stokes equations are neglected. Then the flow equations become, for an incompressible steady flow:^[5]

\[ \begin{align} &\nabla p = \eta\, \nabla^2 \mathbf{u} = - \eta\, \nabla \times \mathbf{\boldsymbol{\omega}}, \\ &\nabla \cdot \mathbf{u} = 0, \end{align} \] where:

p is the fluid pressure (in Pa),
u is the flow velocity (in m/s), and
ω is the vorticity (in s^-1), defined as \(\boldsymbol{\omega}=\nabla\times\mathbf{u}.\)

By using some vector calculus identities, these equations can be shown to result in Laplace's equations for the pressure and each of the components of the vorticity vector:^[5]

\[\nabla^2 \boldsymbol{\omega}=0\] and \(\nabla^2 p = 0.\)

Additional forces like those by gravity and buoyancy have not been taken into account, but can easily be added since the above equations are linear, so linear superposition of solutions and associated forces can be applied.

Flow around a sphere

For the case of a sphere in a uniform far field flow, it is advantageous to use a cylindrical coordinate system ( r , φ , z ). The z–axis is through the centre of the sphere and aligned with the mean flow direction, while r is the radius as measured perpendicular to the z–axis. The origin is at the sphere centre. Because the flow is axisymmetric around the z–axis, it is independent of the azimuth φ.

In this cylindrical coordinate system, the incompressible flow can be described with a Stokes stream function ψ, depending on r and z:^[6]^[7]

\[ v = -\frac{1}{r}\frac{\partial\psi}{\partial z}, \qquad w = \frac{1}{r}\frac{\partial\psi}{\partial r}, \]

with v and w the flow velocity components in the r and z direction, respectively. The azimuthal velocity component in the φ–direction is equal to zero, in this axisymmetric case. The volume flux, through a tube bounded by a surface of some constant value ψ, is equal to 2π ψ and is constant.^[6]

For this case of an axisymmetric flow, the only non-zero of the vorticity vector ω is the azimuthal φ–component ω_φ^[8]^[9]

\[ \omega_\varphi = \frac{\partial v}{\partial z} - \frac{\partial w}{\partial r} = - \frac{\partial}{\partial r} \left( \frac{1}{r}\frac{\partial\psi}{\partial r} \right) - \frac{1}{r}\, \frac{\partial^2\psi}{\partial z^2}. \]

The Laplace operator, applied to the vorticity ω_φ, becomes in this cylindrical coordinate system with axisymmetry:^[9]

\[\nabla^2 \omega_\varphi = \frac{1}{r}\frac{\partial}{\partial r}\left( r\, \frac{\partial\omega_\varphi}{\partial r} \right) + \frac{\partial^2 \omega_\varphi}{\partial z^2} - \frac{\omega_\varphi}{r^{2}} = 0.\]

From the previous two equations, and with the appropriate boundary conditions, for a far-field uniform-flow velocity V in the z–direction and a sphere of radius R, the solution is found to be^[10]

\[ \psi = - \frac{1}{2}\, V\, r^2\, \left[ 1 - \frac{3}{2} \frac{R}{\sqrt{r^2+z^2}} + \frac{1}{2} \left( \frac{R}{\sqrt{r^2+z^2}} \right)^3\; \right]. \]

The viscous force per unit area σ, exerted by the flow on the surface on the sphere, is in the z–direction everywhere. More strikingly, it has also the same value everywhere on the sphere:^[1]

\[\boldsymbol{\sigma} = \frac{3\, \eta\, V}{2\, R}\, \mathbf{e}_z\]

with e_z the unit vector in the z–direction. For other shapes than spherical, σ is not constant along the body surface. Integration of the viscous force per unit area σ over the sphere surface gives the frictional force F_d according to Stokes' law.

Terminal velocity and settling time

At terminal velocity – or settling velocity – the frictional force F_d on the sphere is balanced by the excess force F_g due to the difference of the weight of the sphere and its buoyancy, both caused by gravity:^[2]

\[F_g = \left( \rho_p - \rho_f \right)\, g\, \frac{4}{3}\pi\, R^3,\]

with ρ_p and ρ_f the mass density of the sphere and the fluid, respectively, and g the gravitational acceleration. Demanding force balance: F_d = F_g and solving for the velocity V gives the terminal velocity V_s. If terminal velocity is reached relatively quickly, an average settling time can be calculated by dividing the height the particle will fall by its terminal velocity.

Notes

↑ ^1.0 ^1.1 Batchelor (1967), p. 233.
↑ ^2.0 ^2.1 Lamb (1994), §337, p. 599.
↑ Dusenbery, David B. (2009). Living at Micro Scale, p.49. Harvard University Press, Cambridge, Mass. ISBN 978-0-674-03116-6.
↑ Dusenbery, David B. (2009). Living at Micro Scale. Harvard University Press, Cambridge, Mass. ISBN 978-0-674-03116-6.
↑ ^5.0 ^5.1 Batchelor (1967), section 4.9, p. 229.
↑ ^6.0 ^6.1 Batchelor (1967), section 2.2, p. 78.
↑ Lamb (1994), §94, p. 126.
↑ Batchelor (1967), section 4.9, p. 230
↑ ^9.0 ^9.1 Batchelor (1967), appendix 2, p. 602.
↑ Lamb (1994), §337, p. 598.

References

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[Batch233-1] 1.0 ^1.1 Batchelor (1967), p. 233.

[Lamb599-2] 2.0 ^2.1 Lamb (1994), §337, p. 599.

[3] Dusenbery, David B. (2009). Living at Micro Scale, p.49. Harvard University Press, Cambridge, Mass. ISBN 978-0-674-03116-6.

[4] Dusenbery, David B. (2009). Living at Micro Scale. Harvard University Press, Cambridge, Mass. ISBN 978-0-674-03116-6.

[Batch229-5] 5.0 ^5.1 Batchelor (1967), section 4.9, p. 229.

[Batch78-6] 6.0 ^6.1 Batchelor (1967), section 2.2, p. 78.

[7] Lamb (1994), §94, p. 126.

[Batch230-8] Batchelor (1967), section 4.9, p. 230

[Batch602-9] 9.0 ^9.1 Batchelor (1967), appendix 2, p. 602.

[10] Lamb (1994), §337, p. 598.

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