In fluid dynamics, Stokes's law gives the frictional force – also called drag force – exerted on spherical objects moving at very small Reynolds numbers in a viscous fluid. It was derived by George Gabriel Stokes in 1851 by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.
Statement of the law The force of viscosity on a small sphere moving through a viscous fluid is given by:
F → d = − 6 π μ R v → {\displaystyle {\vec {F}}_{\rm {d}}=-6\pi \mu R{\vec {v}}}
where (in SI units):
F → d {\displaystyle {\vec {F}}_{\rm {d}}} is the frictional force – known as Stokes's 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);
v → {\displaystyle {\vec {v}}} is the body velocity vector, not the flow velocity relative to the object (meters per second). Note the minus sign in the equation, the drag force points in the opposite direction to the relative velocity: drag opposes the motion. Stokes's 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's law is used to define their Stokes radius and diameter. 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's 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's law is illustrated by the fact that it played a critical role in the research leading to at least three Nobel Prizes. Stokes's 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 settling of fine particles in water or other fluids.
Terminal velocity of sphere falling in a fluid
At terminal (or settling) velocity, the excess force Fe due to the difference between the weight and buoyancy of the sphere (both caused by gravity) is given by:
F e = ( ρ p − ρ f ) g 4 3 π R 3 , {\displaystyle F_{e}=(\rho _{p}-\rho _{f})\,g\,{\frac {4}{3}}\pi \,R^{3},}
where (in SI units):
ρp is the mass density of the sphere [kg/m3] ρf is the mass density of the fluid [kg/m3] g is the gravitational acceleration [m/s2] 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's 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:
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