Water flows through a $$10$$ $$mm$$ diameter and $$250$$ $$m$$ long smooth pipe at an average velocity of $$0.1$$ $$m/s.$$ The density and the viscosity of water are $$997kg/{m^3}$$ and $$855 \times {10^{ - 6}}$$ $$N.s/{m^2},$$ respectively. Assuming fully-developed flow, the pressure drop (in $$Pa$$) in the pipe is ______________

Consider laminar flow of water over a flat plate of length $$1$$ $$m.$$ If the boundary layer thickness at a distance of $$0.25$$ $$m$$ from the leading edge of the plate is $$8$$ $$mm,$$ the boundary layer thickness (in $$mm$$), at a distance of $$0.75$$ $$m,$$ is __________________

A fluid of dynamic viscosity $$2 \times {10^{ - 5}}\,\,kg/m.s$$ and density $$1kg/{m^3}$$ flows with an average velocity of $$1$$ $$m/s$$ through a long duct of rectangular $$\left( {25\,\,mm \times \,\,15\,\,mm} \right)$$ cross-section. Assuming laminar flow, the pressure drop (in $$Pa$$) in the fully developed region per meter length of the duct is __________________.

The velocity profile in fully developed laminar flow in a pipe of diameter $$D$$ is given by $$u = {u_0}\left( {1 - 4{r^2}/{D^2}} \right),$$ where $$r$$ is the radial distance from the center. If the viscosity of the fluid is $$\mu ,$$ the pressure drop across a length $$L$$ of the pipe is