Fluid Mechanics: Terminal Velocity with Viscous

When an object falls through a fluid, it experiences resistance due to the viscosity of the fluid. This resistance causes the object to reach a terminal velocity, which is the maximum velocity that the object can attain while falling through the fluid. The terminal velocity of an object in a viscous fluid can be calculated using the following equation:

v_t = (2mg)/(ρAC_d)^0.5

where:

• v_t is the terminal velocity of the object

• m is the mass of the object

• g is the acceleration due to gravity

• ρ is the density of the fluid

• A is the cross-sectional area of the object

• C_d is the drag coefficient of the object

The drag coefficient depends on the shape of the object and the Reynolds number of the flow. The Reynolds number is a dimensionless quantity that describes the ratio of inertial forces to viscous forces in the fluid. For laminar flow (low Reynolds number), the drag coefficient can be calculated using the following equation:

C_d = 24/Re

where Re is the Reynolds number.

For turbulent flow (high Reynolds number), the drag coefficient is more complex and depends on the shape of the object and the Reynolds number. In general, the drag coefficient increases with increasing Reynolds number.

In conclusion, the terminal velocity of an object falling through a viscous fluid can be calculated using the equation v_t = (2mg)/(ρAC_d)^0.5, where the drag coefficient depends on the shape of the object and the Reynolds number of the flow.