Question

define terminal velocity. show that the terminal velocity v of a sphere of radius r, density ρ falling through a viscous fluid of density ς and coefficient of viscosity η is given by v = 2 (ρ-ς) r2 g/ 9η

Answer 1

Answer:

Show that the terminal velocity v of a sphere of radius r, density ρ falling through a viscous fluid of density ς and coefficient of viscosity η is given by. v = 2 (ρ-ς) r2 g/ 9η

Explanation:

hope it helps

Answer 2

Terminal Velocity :

• It is a constant velocity with which an object passing through a fluid moves under the effect of various forces.

Now,

• Let the density of the object be $\rho$ and density of the fluid be $\sigma$ and the coefficient of viscosity be $\eta$.

At terminal velocity, the sphere is at translational equilibrium [the weight is balanced by the buoyant force and strokes force]

$W = F_{b} + F_{stroke}$

$\implies \dfrac{4}{3} \pi {r}^{3} \rho g= \dfrac{4}{3} \pi {r}^{3} \sigma g+ 6\pi \eta rv$

$\implies \dfrac{4}{3} \pi {r}^{3} (\rho - \sigma) g= 6\pi \eta rv$

$\implies v = \dfrac{2}{9 \eta} {r}^{2} (\rho - \sigma) g$

[Hence derived]