قانون ستوكس

ينص قانون ستوكس Stokes' law علي أن " قوة مقاومة المائع لكرة تسقط سقوطاً حراً فيه تتناسب طردياً مع معامل لزوجة هذا المائع, وقطر الكرة وسرعتها الحدية"

$F_{d}=6\pi \,\mu \,R\,v\,$

where F_d is the frictional force – known as Stokes' drag – acting on the interface between the fluid and the particle (in N),μ is the dynamic viscosity (kg /m*s),R is the radius of the spherical object (in m), and v is the particle's velocity (in m/s).

وهذا القانون ينطبق فقط علي مدي معين للسرعة ، أماإذا زادت عن حد معين فإن قوة مقاومة المائع تصبح متناسبة تناسباً طردياً مع مربع السرعة.

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 السرعة النهائية لكرة تسقط في مائع

Creeping flow past a falling sphere( mostly in terminal velocity water droplet is taken ie fog in air etc) in a fluid: streamlines, drag force F_d and force by gravity F_g.

At terminal (or settling) velocity, the excess force F_g due to the difference of the weight of the sphere and the buoyancy on the sphere, (both caused by gravity:)

$F_{g}=\left(\rho _{p}-\rho _{f}\right)\,g\,{\frac {4}{3}}\pi \,R^{3},$

with ρ_p and ρ_f the mass density of the sphere and the fluid, respectively, and g the gravitational acceleration. Demanding force balance: F_d = F_g and solving for the velocity V gives the terminal velocity V_s. Note that since buoyant 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 the particle is falling in the viscous fluid under its own weight due to gravity, then a terminal velocity, or settling velocity, is reached when this frictional force combined with the buoyant force exactly balances the gravitational force. The resulting terminal velocity (or settling velocity) is given by:

$v_{s}={\frac {2}{9}}{\frac {\left(\rho _{p}-\rho _{f}\right)}{\mu }}g\,R^{2}$

where v_s is the particle's settling velocity (m/s) (vertically downwards if ρ_p > ρ_f, upwards if ρ_p < ρ_f ), g is the gravitational acceleration (m/s²), ρ_p is the mass density of the particles (kg/m³), ρ_f is the mass density of the fluid (kg/m³) and μ is the dynamic viscosity (kg /m*s).

 سريان ستوكس المنتظم

${\begin{aligned}&\nabla p=\eta \,\nabla ^{2}\mathbf {u} =-\eta \,\nabla \times \mathbf {\boldsymbol {\omega }} ,\\&\nabla \cdot \mathbf {u} =0,\end{aligned}}$

حيث:

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

 انظر أيضاً

• Stokes flow
• Einstein relation (kinetic theory)
• Scientific laws named after people
• Drag equation
• Viscometry
• Equivalent spherical diameter
• Deposition (geology)

 المراجع

• Batchelor, G.K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press. ISBN 0-521-66396-2.
• Lamb, H. (1994). Hydrodynamics (6th edition ed.). Cambridge University Press. ISBN 978-0-521-45868-9. {{cite book}}: |edition= has extra text (help) Originally published in 1879, the 6th extended edition appeared first in 1932.