An air bubble of negligible mass rises steadily through a column of liquid of density 900 ~kg/m^3 . The coefficient of viscosity of the liquid is 16 Poise . If the steady rate of rise of the bubble is 0.5 ~cm/s , the diameter of the air bubble is ______ mm . (Take g = 10 ~m/s^2 )

Step-by-step solution

Since the air bubble rises steadily, its terminal velocity is constant and the net force acting on it is zero. The mass of the bubble is negligible, so its weight can be ignored. The upward buoyant force B is balanced by the downward viscous drag force F_v . Equating the two forces: B = F_v 4 3 r^3 g = 6 r v Rearranging for r^2 : r^2 = 9 v 2 g Convert the given values into SI units: Coefficient of viscosity, = 16 Poise = 1.6 ~Pa s Terminal velocity, v = 0.5 ~cm/s = 0.005 ~m/s Density, = 900 ~kg/m^3 Substitute these