A bubble of radius R in water of density ρ is expanding uniformly at speed v . Given that water is incompressible, the kinetic energy of water being pushed is

Step-by-step solution

Let bubble of radius x expands and pushes water of density ρ by a distance d x . Then, kinetic energy of water of strip is d x is d K = 1 2 d m · v x 2 ⇒ d K = 1 2 4 π x 2 · d x · ρ · v x 2 or d K = 2 π R 2 · ρ · v x 2 · d x . . . ( i ) Considering laminar flow, for section 1 and 2 shown in figure, by equation of continuity A 1 v 1 = A 2 v 2 ⇒ 4 π x 2 v x = 4 π R 2 v ⇒ v x = R 2 x 2 . v . . . ( iii ) From Eqs. ( i ) and ( ii ) , we have d K = 2 π x 2 ρ R 2 x 2 · v · 2 d x ⇒ d K = 2 π R 4 ρ v 2 . d x x 2 So, kinetic