Stokes' law

In

The force of viscosity on a small sphere moving through a viscous fluid is given by:[3]^[4]

${\displaystyle F_{\rm {d}}=6\pi \mu Rv}$

where (in

• F_d is the frictional force – known as Stokes' drag – acting on the interface between the fluid and the particle (newtons, kg m s⁻²);
• μ (some authors use the symbol η) is the
  dynamic viscosity (Pascal
  -seconds, kg m⁻¹ s⁻¹);
• R is the radius of the spherical object (meters);
• ${\displaystyle v}$ is the flow velocity relative to the object (meters per second).

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

• Laminar flow
• No inertial effects (zero Reynolds number)
• Spherical particles
• Homogeneous (uniform in composition) material
• Smooth surfaces
• Particles do not interfere with each other.

Depending on desired accuracy, the failure to meet these assumptions may or may not require the use of a more complicated model. To 10% error, for instance, velocities need be limited to those giving Re < 1.

For molecules Stokes' law is used to define their Stokes radius and diameter.

The

Applications

Stokes' law is the basis of the falling-sphere

The importance of Stokes' law is illustrated by the fact that it played a critical role in the research leading to at least three Nobel Prizes.[5]

Stokes' law is important for understanding the swimming of microorganisms and sperm; also, the sedimentation of small particles and organisms in water, under the force of gravity.^[5]

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).^[6] Similar use of the equation can be made in the settling of fine particles in water or other fluids.^{[citation needed]}

Terminal velocity of sphere falling in a fluid

At

• 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  ${\displaystyle {\boldsymbol {\omega }}=\nabla \times \mathbf {u} .}$

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:^[8]

${\displaystyle \nabla ^{2}{\boldsymbol {\omega }}=0}$   and   ${\displaystyle \nabla ^{2}p=0.}$

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

Transversal flow around a sphere

For the case of a sphere in a uniform

In this cylindrical coordinate system, the incompressible flow can be described with a Stokes stream function ψ, depending on r and z:^[9][10]

${\displaystyle u_{z}={\frac {1}{r}}{\frac {\partial \psi }{\partial r}},\qquad u_{r}=-{\frac {1}{r}}{\frac {\partial \psi }{\partial z}},}$

with u_r and u_z 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 2πψ and is constant.^[9]

For this case of an axisymmetric flow, the only non-zero component of the vorticity vector ω is the azimuthal φ–component ω_φ^[11][12]

${\displaystyle \omega _{\varphi }={\frac {\partial u_{r}}{\partial z}}-{\frac {\partial u_{z}}{\partial r}}=-{\frac {\partial }{\partial r}}\left({\frac {1}{r}}{\frac {\partial \psi }{\partial r}}\right)-{\frac {1}{r}}\,{\frac {\partial ^{2}\psi }{\partial z^{2}}}.}$

The Laplace operator, applied to the vorticity ω_φ, becomes in this cylindrical coordinate system with axisymmetry:^[12]

${\displaystyle \nabla ^{2}\omega _{\varphi }={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r\,{\frac {\partial \omega _{\varphi }}{\partial r}}\right)+{\frac {\partial ^{2}\omega _{\varphi }}{\partial z^{2}}}-{\frac {\omega _{\varphi }}{r^{2}}}=0.}$

From the previous two equations, and with the appropriate boundary conditions, for a far-field uniform-flow velocity u in the z–direction and a sphere of radius R, the solution is found to be^[13]

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

The solution of velocity in cylindrical coordinates and components follows as:

${\displaystyle {\begin{aligned}u_{r}(r,z)&={\frac {3R^{3}}{4}}\cdot {\frac {rzu}{{\sqrt {r^{2}+z^{2}}}^{5}}}-{\frac {3R}{4}}\cdot {\frac {rzu}{{\sqrt {r^{2}+z^{2}}}^{3}}}\\[4pt]u_{z}(r,z)&={\frac {R^{3}}{4}}\cdot \left({\frac {3uz^{2}}{{\sqrt {r^{2}+z^{2}}}^{5}}}-{\frac {u}{{\sqrt {r^{2}+z^{2}}}^{3}}}\right)+u-{\frac {3R}{4}}\cdot \left({\frac {u}{\sqrt {r^{2}+z^{2}}}}+{\frac {uz^{2}}{{\sqrt {r^{2}+z^{2}}}^{3}}}\right)\end{aligned}}}$

The solution of vorticity in cylindrical coordinates follows as:

${\displaystyle \omega _{\varphi }(r,z)=-{\frac {3Ru}{2}}\cdot {\frac {r}{{\sqrt {r^{2}+z^{2}}}^{3}}}}$

The solution of pressure in cylindrical coordinates follows as:

${\displaystyle p(r,z)=-{\frac {3\mu Ru}{2}}\cdot {\frac {z}{{\sqrt {r^{2}+z^{2}}}^{3}}}}$

The solution of pressure in

${\displaystyle p(r,\theta )=-{\frac {3\mu Ru}{2}}\cdot {\frac {\cos \theta }{r^{2}}}}$

The formula of pressure is also called dipole potential analogous to the concept in electrostatics.

A more general formulation, with arbitrary far-field velocity-vector ${\displaystyle \mathbf {u} _{\infty }}$, in cartesian coordinates ${\displaystyle \mathbf {x} =(x,y,z)^{T}}$ follows with:

$${\displaystyle {\begin{aligned}\mathbf {u} (\mathbf {x} )&=\underbrace {\underbrace {{\frac {R^{3}}{4}}\cdot \left({\frac {3\left(\mathbf {u} _{\infty }\cdot \mathbf {x} \right)\cdot \mathbf {x} }{\|\mathbf {x} \|^{5}}}-{\frac {\mathbf {u} _{\infty }}{\|\mathbf {x} \|^{3}}}\right)} _{\text{conservative: curl=0, div=0}}+\underbrace {\mathbf {u} _{\infty }} _{\text{far-field}}} _{\text{Terms of Boundary-Condition}}\;\underbrace {-{\frac {3R}{4}}\cdot \left({\frac {\mathbf {u} _{\infty }}{\|\mathbf {x} \|}}+{\frac {\left(\mathbf {u} _{\infty }\cdot \mathbf {x} \right)\cdot \mathbf {x} }{\|\mathbf {x} \|^{3}}}\right)} _{{\text{non-conservative: curl}}={\boldsymbol {\omega }}(\mathbf {x} ),{\text{ div=0}}}\\[8pt]&=\left[{\frac {3R^{3}}{4}}{\frac {\mathbf {x\otimes \mathbf {x} } }{\|\mathbf {x} \|^{5}}}-{\frac {R^{3}}{4}}{\frac {\mathbb {I} }{\|\mathbf {x} \|^{3}}}-{\frac {3R}{4}}{\frac {\mathbf {x} \otimes \mathbf {x} }{\|\mathbf {x} \|^{3}}}-{\frac {3R}{4}}{\frac {\mathbb {I} }{\|\mathbf {x} \|}}+\mathbb {I} \right]\cdot \mathbf {u} _{\infty }\end{aligned}}}$$

${\displaystyle {\boldsymbol {\omega }}(\mathbf {x} )=-{\frac {3R}{2}}\cdot {\frac {\mathbf {u} _{\infty }\times \mathbf {x} }{\|\mathbf {x} \|^{3}}}}$

${\displaystyle p\left(\mathbf {x} \right)=-{\frac {3\mu R}{2}}\cdot {\frac {\mathbf {u} _{\infty }\cdot \mathbf {x} }{\|\mathbf {x} \|^{3}}}}$

In this formulation the non-conservative term represents a kind of so-called Stokeslet. The Stokeslet is the Green's function of the Stokes-Flow-Equations. The conservative term is equal to the dipole gradient field. The formula of vorticity is analogous to the Biot–Savart law in electromagnetism.

The following formula describes the viscous stress tensor for the special case of Stokes flow. It is needed in the calculation of the force acting on the particle. In Cartesian coordinates the vector-gradient ${\displaystyle \nabla \mathbf {u} }$ is identical to the

The force acting on the sphere is to calculate by surface-integral, where e_r represents the radial unit-vector of spherical-coordinates:

${\displaystyle {\begin{aligned}\mathbf {F} &=\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset \;{\boldsymbol {\sigma }}\cdot {\text{d}}\mathbf {S} \\[4pt]&=\int _{0}^{\pi }\int _{0}^{2\pi }{\boldsymbol {\sigma }}\cdot \mathbf {e_{r}} \cdot R^{2}\sin \theta {\text{d}}\varphi {\text{d}}\theta \\[4pt]&=\int _{0}^{\pi }\int _{0}^{2\pi }{\frac {3\mu \cdot \mathbf {u} _{\infty }}{2R}}\cdot R^{2}\sin \theta {\text{d}}\varphi {\text{d}}\theta \\[4pt]&=6\pi \mu R\cdot \mathbf {u} _{\infty }\end{aligned}}}$

Rotational flow around a sphere

${\displaystyle {\begin{aligned}\mathbf {u} (\mathbf {x} )&=-\;R^{3}\cdot {\frac {{\boldsymbol {\omega }}_{R}\times \mathbf {x} }{\|\mathbf {x} \|^{3}}}\\[8pt]{\boldsymbol {\omega }}(\mathbf {x} )&={\frac {R^{3}\cdot {\boldsymbol {\omega }}_{R}}{\|\mathbf {x} \|^{3}}}-{\frac {3R^{3}\cdot ({\boldsymbol {\omega }}_{R}\cdot \mathbf {x} )\cdot \mathbf {x} }{\|\mathbf {x} \|^{5}}}\\[8pt]p(\mathbf {x} )&=0\\[8pt]{\boldsymbol {\sigma }}&=-p\cdot \mathbf {I} +\mu \cdot \left((\nabla \mathbf {u} )+(\nabla \mathbf {u} )^{T}\right)\\[8pt]\mathbf {T} &=\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset \mathbf {x} \times \left({\boldsymbol {\sigma }}\cdot {\text{d}}{\boldsymbol {S}}\right)\\&=\int _{0}^{\pi }\int _{0}^{2\pi }(R\cdot \mathbf {e_{r}} )\times \left({\boldsymbol {\sigma }}\cdot \mathbf {e_{r}} \cdot R^{2}\sin \theta {\text{d}}\varphi {\text{d}}\theta \right)\\&=8\pi \mu R^{3}\cdot {\boldsymbol {\omega }}_{R}\end{aligned}}}$

Other types of Stokes flow

Although the liquid is static and the sphere is moving with a certain velocity, with respect to the frame of sphere, the sphere is at rest and liquid is flowing in the opposite direction to the motion of the sphere.

See also

• Einstein relation (kinetic theory)
• Scientific laws named after people
• Drag equation
• Viscometry
• Equivalent spherical diameter
• Deposition (geology)
• Stokes number – A determinant of the additional effect of turbulence on terminal fall velocity for particles in fluids^[14]

Sources

• ISBN 0-521-66396-2
  .
• ISBN 978-0-521-45868-9
  . Originally published in 1879, the 6th extended edition appeared first in 1932.