Editing Terminal velocity (section)

==Terminal velocity in the presence of buoyancy force==
{{see also|Sediment_transport#Settling_velocity}}

[[Image:Settling velocity quartz.png|thumb|Settling velocity W<sub>s</sub> of a sand grain (diameter d, density 2650&nbsp;kg/m<sup>3</sup>) in water at 20&nbsp;°C, computed with the formula of Soulsby (1997).]]

When the buoyancy effects are taken into account, an object falling through a fluid under its own weight can reach a terminal velocity (settling velocity) if the net force acting on the object becomes zero. When the terminal velocity is reached the weight of the object is exactly balanced by the upward [[buoyancy force]] and drag force. That is
{{NumBlk|:|<math> W = F_b + D </math>|{{EquationRef|1}}}}
where
*<math>W</math> is the weight of the object,
*<math>F_b</math> is the buoyancy force acting on the object, and
*<math>D</math> is the drag force acting on the object.

If the falling object is spherical in shape, the expression for the three forces are given below:
{{NumBlk|:|<math> W = \frac{\pi}{6} d^3 \rho_s g,</math>|{{EquationRef|2}}}}
{{NumBlk|:|<math>F_b = \frac{\pi}{6} d^3 \rho g,</math>|{{EquationRef|3}}}}
{{NumBlk|:|<math> D = C_d \frac{1}{2} \rho V^2 A,</math>|{{EquationRef|4}}}}
where
*<math>d</math> is the diameter of the spherical object,
*<math>g</math> is the gravitational acceleration,
*<math>\rho</math> is the density of the fluid,
*<math>\rho_s</math> is the density of the object,
*<math>A = \frac{1}{4} \pi d^2</math> is the projected area of the sphere,
*<math>C_d</math> is the drag coefficient, and
*<math>V</math> is the characteristic velocity (taken as terminal velocity, <math>V_t </math>).

Substitution of equations ({{EquationNote|2}}–{{EquationNote|4}}) in equation ({{EquationNote|1}}) and solving for terminal velocity, <math>V_t</math> to yield the following expression
{{NumBlk|:|<math> V_t = \sqrt{\frac{4 g d}{3 C_d} \left( \frac{\rho_s - \rho}{\rho} \right)}. </math>|{{EquationRef|5}}}}

In equation ({{EquationNote|1}}), it is assumed that the object is denser than the fluid. If not, the sign of the drag force should be made negative since the object will be moving upwards, against gravity. Examples are bubbles formed at the bottom of a champagne glass and helium balloons. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up.