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.
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.
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.
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.
with ρp and ρf the mass densities of the sphere and fluid, respectively, and g the gravitational acceleration. Requiring the force balance Fd = Fg 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:
(vertically downwards if ρp > ρf, upwards if ρp < ρf ), where:
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.
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.
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 2π ψ and is constant.
For this case of an axisymmetric flow, the only non-zero component of the vorticity vector ω is the azimuthal φ–component ωφ
The Laplace operator, applied to the vorticity ωφ, becomes in this cylindrical coordinate system with axisymmetry:
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
The formula of pressure is also called dipole-potential in analogues to electrostatics.
A more general formulation, with arbitrary far-field velocity-vector , in cartesian coordinates follows with:
Stokes-Flow around sphere with parameters of Far-Field velocity , radius of sphere , viscosity of water (T = 20°C) . Shown are the filed-lines of velocity-field and the amplitudes of velocity, pressure and vorticity with pseudo-colors.
The following formula describes the viscous-stress-tensor for the special case of stokes-flow. It's 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 represents the identity-matrix.
The force acting on the sphere is to calculate by surface-integral, where represents the radial unit-vector of spherical-coordinates: