For a fully developed laminar flow of water (dynamic viscosity $$0.001$$ $$Pa$$-s) through a pipe of radius $$5$$ $$cm,$$ the axial pressure gradient is $$-10$$ $$Pa/m$$. The magnitude of axial velocity (in $$m/s$$) at a radial location of $$0.2$$ $$cm$$ is ____________

The head loss for a laminar incompressible flow through a horizontal circular pipe is $${h_1}$$. Pipe length and fluid remaining the same, if the average flow velocity doubles and the pipe diameter reduces to half its previous value, the head loss is $${h_2}.$$ The ratio $${h_2}/{h_1}$$ is

Within a boundary layer for a steady incompressible flow, the Bernoulli equation

If a foam insulation is added to a $$4cm$$ outer diameter pipe as shown as figure, the critical radius of insulation (in $$cm$$) is ________.