A spherical balloon is inflated by pumping gas in through a narrow nozzle at its centre

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Kinematics, conservation laws and ideal flow

Fluid Dynamics, Ch Kinematics in three dimensions and Cauchy's equation of motion

Problem

A spherical balloon is inflated by pumping gas in through a narrow nozzle at its centre, modelled as a point source, at a constant mass rate m˙=6.00 g s−1\dot m = 6.00\,\mathrm{g\,s^{-1}}. Its radius increases at a constant rate, R(t)=R0+UtR(t) = R_0 + Ut with R0=0.100 mR_0 = 0.100\,\mathrm{m} and U=0.0500 m s−1U = 0.0500\,\mathrm{m\,s^{-1}}. The membrane is impermeable, so no gas crosses it, and it moves with the local gas velocity at r=R(t)r=R(t). At t=0t=0 the gas inside has uniform density ρ0=1.20 kg m−3\rho_0 = 1.20\,\mathrm{kg\,m^{-3}}. Assume that at every later instant the density inside is still spatially uniform (pressure equalises much faster than the balloon expands), although it changes with time, and that the flow inside is spherically symmetric with radial velocity ur(r,t)u_r(r,t). ((a)) Using conservation of the total mass inside the balloon, find the density ρ\rho at t=1.00 st = 1.00\,\mathrm{s}. ((b)) Using the differential form of mass conservation with ρ\rho spatially uniform, show that ∇⋅u\nabla\cdot\mathbf{u} is spatially uniform inside the balloon, equal to some D(t)D(t), and evaluate DD at t=1.00 st=1.00\,\mathrm{s}. ((c)) Integrate ∇⋅u=D(t)\nabla\cdot\mathbf{u} = D(t) for the spherically symmetric field ur(r,t)u_r(r,t), using the condition that the mass flux through a small sphere about the centre tends to m˙\dot m as its radius shrinks to zero, and evaluate uru_r at r=0.0750 mr = 0.0750\,\mathrm{m}, t=1.00 st = 1.00\,\mathrm{s}.

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