52.8: Stokes’s Law
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Stokes's law gives the resistive force on a sphere moving through a viscous fluid. It was developed by the 19th century English physicist and mathematician George Stokes. Stokes's law states that the resistive force on the sphere is given by
FR=6πμrv,
where FR is the resistive force on the sphere, r is its radius, μ is the dynamic viscosity of the fluid, and v is the relative velocity between the fluid and the sphere. This is generally valid for low Reynolds numbers ( Re<1 ).
Notice that the Stokes's law force is of the form of a Model I resistive force described in Chapter 22 ( FR∝v ), with the resistance coefficient b=6πμr. By Eq. 22.1.21 the terminal velocity for Model I is v∞=mg/b; so for a sphere moving through a viscous fluid, we have by Stokes's law
v∞=mg6πμr.
What is the terminal velocity of a steel ball of diameter 1 cm falling through a jar of honey? Solution. Taking the density of steel as ρ=7.86 g/cm3, we find the mass of the steel ball as
m=ρV=ρ(43πr3)=4.115 g=4.115×10−3 kg.
Solution
From Table 52.6.1, the dynamic viscosity μ of honey is 5 Pa s ; the terminal velocity is then given by Eq. 52.8.2:
v∞=mg6πμr=(4.115×10−3 kg)(9.80 m/s2)6π(5 Pa s)(0.5×10−2 m)=8.56 cm/s.