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.
The CGS unit of kinematic viscosity was named "stokes" after his work.
Applications
Stokes' 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 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
Creeping flow past a falling sphere in a fluid (e.g., a droplet of fog falling through the air): streamlines, drag force Fd and force by gravity Fg.
Requiring the force balance Fd = Fe and solving for the velocity v gives the terminal velocity vs. Note that since the excess force increases as R3 and Stokes' drag increases as R, the terminal velocity increases as R2 and thus varies greatly with particle size as shown below. If a particle only experiences its own weight while falling in a viscous fluid, then a terminal velocity is reached when the sum of the frictional and the buoyant forces on the particle due to the fluid exactly balances the gravitational force. This velocity v [m/s] is given by:[7]
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.
Transversal flow around a sphere
Streamlines of creeping flow past a sphere in a fluid. Isocontours of the ψ function (values in contour labels).
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 azimuthφ.
In this cylindrical coordinate system, the incompressible flow can be described with a Stokes stream functionψ, depending on r and z:[9][10]
with ur and uz 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 ωφ[11][12]
The Laplace operator, applied to the vorticity ωφ, becomes in this cylindrical coordinate system with axisymmetry:[12]
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]
Stokes-Flow around sphere with parameters of Far-Field velocity , radius of sphere , viscosity of water (T = 20°C) . Shown are the field-lines of velocity-field and the amplitudes of velocity, pressure and vorticity with pseudo-colors.
The solution of vorticity in cylindrical coordinates follows as:
The solution of pressure in cylindrical coordinates follows as:
Alternatively, in a more compact way, one can formulate the velocity field as follows:
,
where is the Hessian matrix differential operator and is a differential operator composed as the difference of the Laplacian and the Hessian. In this way it becomes explicitly clear, that the solution is composed from derivatives of a Coulomb-type potential () and a Biharmonic-type potential (). The differential operator applied to the vector norm generates the Stokeslet.
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 is identical to the Jacobian matrix. The matrix I represents the identity-matrix.
The force acting on the sphere is to calculate by surface-integral, where er represents the radial unit-vector of spherical-coordinates:
Rotational flow around a sphere
Stokes-Flow around sphere: , ,
Other types of Stokes flow
This section is about steady Stokes flow around a sphere. For the forces on a sphere in unsteady Stokes flow, see Basset–Boussinesq–Oseen equation.
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.
Lamb, H. (1994). Hydrodynamics (6th ed.). Cambridge University Press. ISBN978-0-521-45868-9. Originally published in 1879, the 6th extended edition appeared first in 1932.
^Robert Byron, Bird; Warren E., Stewart; Edwin N., Lightfoot (7 August 2001). Transport Phenomena (2 ed.). John Wiley & Sons, Inc. p. 61. ISBN0-471-41077-2.
^ abDusenbery, David (2009). Living at micro scale : the unexpected physics of being small. Cambridge, Mass: Harvard University Press. ISBN978-0-674-03116-6. OCLC225874255.
^Hadley, Peter. "Why don't clouds fall?". Institute of Solid State Physics, TU Graz. Archived from the original on 12 June 2017. Retrieved 30 May 2015.
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